$$ (\mathbf{J}_{v_{\varepsilon}, n_{\delta}, v_{\varepsilon}}^{x,t} [\mathcal{L}_{\varepsilon}(\dot{\omega}_{n_{\delta}, \varepsilon}(\tau), \omega(\tau)), \psi_{0}(\omega(t))])_{\varepsilon \in (0,1], \delta \in (0,1]} = \\ \left( \left[ \frac{v_{\varepsilon}^{n_{\delta}}}{((i - (v_{\varepsilon}^2))^{n_{\delta}})} \right]^{-d/2} \right)_{\varepsilon \in (0,1], \delta \in (0,1]} \times \\ \left( \int_{\Omega_t} \exp \left[ i \frac{t}{n_{\delta}} \sum_{j=1}^{n_{\delta}} \left[ \frac{m}{2} \dot{\omega}_{n_{\delta}, \varepsilon}^2 \left(j \frac{t}{n_{\delta}}\right) - V_{\varepsilon} \left(\omega\left(j \frac{t}{n_{\delta}}\right)\right) \right] \right] \times \\ \psi_{\varepsilon}(\omega(t)) \tilde{D}_{x}^{\mathbb{R}}[\omega(\tau), v_{\varepsilon}] \right)_{\varepsilon \in (0,1], \delta \in (0,1]} \quad (0.5.32) $$

for all $x \in \mathbb{R}^d \setminus \cup_{\varepsilon \in (0,1]} F_{v_{\varepsilon}}$, where $F_{v_{\varepsilon}}$ is a closed set in $\mathbb{R}^d$ which depend only on $V_{\varepsilon}$. By definition of the Colombeau-Wiener integral (0.3.25), this means that for all $x \notin \cup_{\varepsilon \in (0,1]} F_{v_{\varepsilon}}$

$$ \left( \left( U_{v_{\varepsilon}, n_{\delta}, v_{\varepsilon}}^{t} \psi_{\varepsilon}(x) \right)_{\varepsilon \in (0,1], \delta \in (0,1]} \right) = \left( \left( \Xi_{v_{\varepsilon}, n_{\delta}, v_{\varepsilon}}^{\frac{t}{n_{\delta}}, \frac{t}{n_{\delta}}} M_{v_{\varepsilon}}^{n_{\delta}} \psi_{\varepsilon}(x) \right)_{\varepsilon \in (0,1], \delta \in (0,1]} \right). \quad (0.5.33) $$

Here

$$ \left( U_{v_{\varepsilon}, n_{\delta}, v_{\varepsilon}}^{t} \right)_{\varepsilon \in (0,1], \delta \in (0,1]} = \left( \exp\{t[(i + v_{\varepsilon})\Delta[\phi_{n_{\delta}, \delta}] - iV_{\varepsilon}]\} \right)_{\varepsilon \in (0,1], \delta \in (0,1]}, \quad (0.5.34) $$ $$ \Xi_{v_{\varepsilon}, n_{\delta}}^{t} = \exp(t(i + v_{\varepsilon})\Delta[\phi_{n_{\delta}, \delta}]), $$

where we denote by $\Delta[\phi_{n_{\delta}, \delta}]$ smoothed Laplace operator, which defined by obvious way.

### I.0.6. Colombeau-Feynman Path Integral based on complex Colombeau-Wiener measure $\tilde{D}^{\mathbb{C}}[x(\tau), v_{\varepsilon}]$.
Let $B(\Omega)$ denote the class of all Borel sets of a topological space $\Omega$, i.e. $B(\Omega)$ is the $\sigma$-algebra of sets generated by all open sets. If $\Omega$ is locally compact we denote by $C_0(\Omega)$ the space of all continuous complex-valued functions on $\Omega$-vanishing at infinity.

Equipped with the uniform norm $||f|| = \sup_{x \in \Omega} |f(x)|$ this space is a Banach space.

**Theorem 0.6.1. (Riesz-Markov theorem)** Let $\Omega$ be a locally compact space, then the set $M(\Omega)$ of all finite complex regular Borel measures on $\Omega$ equipped with the norm

$$ \|\mu\| = \sup_{f \in C_0(\Omega), \|f\| \le 1} \left| \int_{\Omega} f(x) d\mu \right| \qquad (0.6.1) $$

is a Banach space, which coincides with the set of all continuous linear functionals on $C_0(\Omega)$.

Any complex $\sigma$-additive measure $\mu$ on $\mathbb{R}^d$ has a representation of the form

$$ \mu(dy) = f(y)M(dy) \qquad (0.6.2) $$

with a positive measure $M(dy)$ and a bounded complex-valued function $f(y)$. Moreover, the measure $M(dy)$ in (0.6.2) is uniquely defined under additional assumption that $|f(y)| = 1$ for all $y$. If this condition holds, the positive measure $M(dy)$ is called the total variation measure of the complex measure $\mu(dy)$ and is denoted by $|\mu|(dy)$. In general, if (0.6.2) holds, then

$$ \|\mu\| = \int |f(y)| M(dy) \qquad (0.6.3) $$

**Definition 0.6.1.** We say that a map $\mu^{\mathbb{C}} : \mathbb{R}^d \times B(\mathbb{R}^d) \to \mathbb{C}$ is a complex transition kernel, if for every $x$, the map $A \mapsto \mu^{\mathbb{C}}(x, A)$ is a finite complex measure on $\mathbb{R}^d$, and for every $A \in B(\mathbb{R}^d)$ the map $x \mapsto \mu^{\mathbb{C}}(x, A)$ is B-measurable.

**Definition 0.6.2.** A time homogeneous complex transition function (abbreviated CTF) on $\mathbb{R}^d$ is a family $\mu_t^{\mathbb{C}}, t \ge 0$ of complex transition kernels such that $\mu_0^{\mathbb{C}}(x, dy) = \delta(x - y)$ for all $x$, where $\delta(x - y)$ is the Dirac measure in $x$, and such that for every non-negative $s, t$, the Chapman-Kolmogorov equation
Let $B(\Omega)$ denote the class of all Borel sets of a topological space $\Omega$, i.e. $B(\Omega)$ is the $\sigma$-algebra of sets generated by all open sets. If $\Omega$ is locally compact we denote by $C_0(\Omega)$ the space of all continuous complex-valued functions on $\Omega$-vanishing at infinity.

Equipped with the uniform norm $\|f\| = \sup_{x \in \Omega} |f(x)|$ this space is a Banach space.

**Theorem 0.6.1. (Riesz-Markov theorem)** Let $\Omega$ be a locally compact space, then the set $M(\Omega)$ of all finite complex regular Borel measures on $\Omega$ equipped with the norm

$$ \|\mu\| = \sup_{f \in C_0(\Omega), \|f\| \le 1} \left| \int_{\Omega} f(x) d\mu \right| \qquad (0.6.1) $$

is a Banach space, which coincides with the set of all continuous linear functionals on $C_0(\Omega)$.

Any complex $\sigma$-additive measure $\mu$ on $\mathbb{R}^d$ has a representation of the form

$$ \mu(dy) = f(y)M(dy) \qquad (0.6.2) $$

with a positive measure $M(dy)$ and a bounded complex-valued function $f(y)$. Moreover, the measure $M(dy)$ in (0.6.2) is uniquely defined under additional assumption that $|f(y)| = 1$ for all $y$. If this condition holds, the positive measure $M(dy)$ is called the total variation measure of the complex measure $\mu(dy)$ and is denoted by $|\mu|(dy)$. In general, if (0.6.2) holds, then

$$ \|\mu\| = \int |f(y)| M(dy) \qquad (0.6.3) $$

**Definition 0.6.1.** We say that a map $\mu^{\mathbb{C}} : \mathbb{R}^d \times B(\mathbb{R}^d) \to \mathbb{C}$ is a complex transition kernel, if for every $x$, the map $A \mapsto \mu^{\mathbb{C}}(x, A)$ is a finite complex measure on $\mathbb{R}^d$, and for every $A \in B(\mathbb{R}^d)$ the map $x \mapsto \mu^{\mathbb{C}}(x, A)$ is B-measurable.

**Definition 0.6.2.** A time homogeneous complex transition function (abbreviated CTF) on $\mathbb{R}^d$ is a family $\mu_t^{\mathbb{C}}, t \ge 0$ of complex transition kernels such that $\mu_0^{\mathbb{C}}(x, dy) = \delta(x - y)$ for all $x$, where $\delta(x - y)$ is the Dirac measure in $x$, and such that for every non-negative $s, t$, the Chapman-Kolmogorov equation
$$ \int \mu_s^C(x, dy) \mu_t^C(y, A) = \mu_{s+t}^C(x, A) \qquad (0.6.4) $$

is satisfied.

**Definition 0.6.3.** A CTF will be called regular, if there exists a positive constant $K$ such that for all $x$ and $t > 0$, the norm $||\mu_t^C(x, \cdot)||$ of the measure $A \mapsto \mu_t^C(x, A)$ does not exceed $\exp(Kt)$.

**Remark 0.6.1.** We remain that if $T_t$ is a strongly continuous semigroup of bounded linear operators in $C_0(\mathbb{R}^d)$, then there exists a timehomogeneous CTF $\mu_t^C(x, dy)$ such that

$$ T_t f(x) = \int \mu_t^C(x, dy) f(y). \qquad (0.6.5) $$

**Remark 0.6.2.** In fact, the existence of a measure $\mu_t^C(x, \cdot)$ such that (2.2) is satisfied follows from the Riesz-Markov theorem, and the semigroup identity $T_s T_t = T_{s+t}$ is equivalent to the Chapman-Kolmogorov equation. Since $\int \mu_t^C(x, dy) f(y)$ is continuous for all $f \in C_0(\mathbb{R}^d)$, it follows by the monotone convergence theorem and the fact that each complex measure is a linear combination of four positive measures that $\mu_t^C(x, A)$ is a Borel function of $x$.

**Definition 0.6.4.** We say that the semigroup $T_t$ is regular, if the corresponding CTF is regular.

**Remark 0.6.3.** Clearly, this Definition 0.6.4 is equivalent to the assumption that $||T_t|| \le \exp(Kt)$ for all $t > 0$ and some constant $K$.

**Definition 0.6.5.** Let $\Omega$ be a locally compact space. Set $E_0(\Omega) = (C_0(\Omega))^I, I = (0,1]$

$$ \begin{aligned} E_M^0(\Omega) &= \left\{ (f_\epsilon)_\epsilon \in E_0(\Omega) \middle| \left( \forall \alpha \in \mathbb{N}_0^n \right) \left( \exists p \in \mathbb{N} \right) \sup_{x \in \Omega} |f(x)| = O(\epsilon^{-p}) \right\}, \\ N_0(\Omega) &= \left\{ (f_\epsilon)_\epsilon \in E_0(\Omega) \middle| \left( \forall \alpha \in \mathbb{N}_0^n \right) \left( \forall q \in \mathbb{N} \right) \sup_{x \in \Omega} |f(x)| = O(\epsilon^q) \right\}, \end{aligned} \qquad (0.6.6) $$

$$ G_0(\Omega) = E_M^0(\Omega)/N_0(\Omega). $$

Elements of $E_M^0(\Omega)$ and $N_0(\Omega)$ are called moderate, resp. negligible functions. Note that $E_M^0(\Omega)$ is an algebra with pointwise operations. It is the largest subalgebra of $E_0(\Omega)$ in which $N_0(\Omega)$ is ideal. Thus, $G_0(\Omega)$ is an associative, commutative algebra.
$$ \int \mu_s^C(x, dy) \mu_t^C(y, A) = \mu_{s+t}^C(x, A) \qquad (0.6.4) $$

is satisfied.

**Definition 0.6.3.** A CTF will be called regular, if there exists a positive constant $K$ such that for all $x$ and $t > 0$, the norm $||\mu_t^C(x, \cdot)||$ of the measure $A \mapsto \mu_t^C(x, A)$ does not exceed $\exp(Kt)$.

**Remark 0.6.1.** We remain that if $T_t$ is a strongly continuous semigroup of bounded linear operators in $C_0(\mathbb{R}^d)$, then there exists a timehomogeneous CTF $\mu_t^C(x, dy)$ such that

$$ T_t f(x) = \int \mu_t^C(x, dy) f(y). \qquad (0.6.5) $$

**Remark 0.6.2.** In fact, the existence of a measure $\mu_t^C(x, \cdot)$ such that (2.2) is satisfied follows from the Riesz-Markov theorem, and the semigroup identity $T_s T_t = T_{s+t}$ is equivalent to the Chapman-Kolmogorov equation. Since $\int \mu_t^C(x, dy) f(y)$ is continuous for all $f \in C_0(\mathbb{R}^d)$, it follows by the monotone convergence theorem and the fact that each complex measure is a linear combination of four positive measures that $\mu_t^C(x, A)$ is a Borel function of $x$.

**Definition 0.6.4.** We say that the semigroup $T_t$ is regular, if the corresponding CTF is regular.

**Remark 0.6.3.** Clearly, this Definition 0.6.4 is equivalent to the assumption that $||T_t|| \le \exp(Kt)$ for all $t > 0$ and some constant $K$.

**Definition 0.6.5.** Let $\Omega$ be a locally compact space. Set $E_0(\Omega) = (C_0(\Omega))^I, I = (0,1]$

$$ \begin{aligned} E_M^0(\Omega) &= \left\{ (f_\epsilon)_\epsilon \in E_0(\Omega) \middle| \left( \forall \alpha \in \mathbb{N}_0^n \right) \left( \exists p \in \mathbb{N} \right) \sup_{x \in \Omega} |f(x)| = O(\epsilon^{-p}) \right\}, \\ N_0(\Omega) &= \left\{ (f_\epsilon)_\epsilon \in E_0(\Omega) \middle| \left( \forall \alpha \in \mathbb{N}_0^n \right) \left( \forall q \in \mathbb{N} \right) \sup_{x \in \Omega} |f(x)| = O(\epsilon^q) \right\}, \end{aligned} \qquad (0.6.6) $$

$$ G_0(\Omega) = E_M^0(\Omega)/N_0(\Omega). $$

Elements of $E_M^0(\Omega)$ and $N_0(\Omega)$ are called moderate, resp. negligible functions. Note that $E_M^0(\Omega)$ is an algebra with pointwise operations. It is the largest subalgebra of $E_0(\Omega)$ in which $N_0(\Omega)$ is ideal. Thus, $G_0(\Omega)$ is an associative, commutative algebra.
**Definition 0.6.6.** Let $\Omega$ be a locally compact space. Set

$$E(\mathbf{M}(\Omega)) = (\mathbf{M}(\Omega))^I, I = (0,1]$$

$$ 
\begin{aligned}
E_M(\mathbf{M}(\Omega)) &= \left\{ (\mu_\epsilon)_\epsilon \in E(\mathbf{M}(\Omega)) \middle| (\forall \alpha \in \mathbb{N}_0^n)(\exists p \in \mathbb{N}) \sup_{B \subseteq \Omega} |\mu_\epsilon(B)| = O(\epsilon^{-p}) \right\}, \\
N_0(\mathbf{M}(\Omega)) &= \left\{ (\mu_\epsilon)_\epsilon \in E(\mathbf{M}(\Omega)) \middle| (\forall \alpha \in \mathbb{N}_0^n)(\forall q \in \mathbb{N}) \sup_{B \subseteq \Omega} |\mu_\epsilon(B)| = O(\epsilon^q) \right\},
\end{aligned}
\quad (0.6.7) $$

$$G(E_M(\mathbf{M}(\Omega))) = E_M(\mathbf{M}(\Omega))/N_0(\mathbf{M}(\Omega)).$$

Elements of $E_M^0(\Omega)$ and $N_0(\Omega)$ are called moderate, resp. negligible. Colombeau-Borel measures on $\Omega$.

**Theorem 0.6.2. (Generalized Riesz-Markov theorem)** Let $\Omega$ be a locally compact space, then the set $E_M(\mathbf{M}(\Omega))$ of all finite complex regular Colombeau-Borel measures on $\Omega$, which coincides with the set of all continuous linear Colombeau generalized functionals on $E_M^0(\Omega)$.

Any complex $\sigma$-additive Colombeau measure $(\mu_\epsilon)_\epsilon$ on $\mathbb{R}^d$ has a representation of the form

$$(\mu_\epsilon(dy))_\epsilon = (f_\epsilon(y)M_\epsilon(dy))_\epsilon \quad (0.6.8)$$

with a positive Colombeau measure $(M_\epsilon(dy))_\epsilon$ and a bounded for every $\epsilon \in (0,1]$ complex-valued generalized function $(f_\epsilon(y))_\epsilon$. Moreover, the Colombeau measure $(M_\epsilon(dy))_\epsilon$ in (0.6.8) is uniquely defined under additional assumption that $|f_\epsilon(y)|_\epsilon = 1$ for all $y$. If this condition holds, the positive measure $(M_\epsilon(dy))_\epsilon$ is called the total variation measure of the complex measure $(\mu_\epsilon^C(dy))_\epsilon$ and is denoted by $(|\mu_\epsilon^C(dy))_\epsilon$. In general, if (0.6.8) holds, then

$$(\|\mu_\epsilon^C\|)_\epsilon = \left(\int |f_\epsilon(y)| M_\epsilon(dy)\right)_\epsilon \quad (0.6.9)$$

**Definition 0.6.7.** We say that a map $(\mu_\epsilon^C)_\epsilon : \mathbb{R}^d \times B(\mathbb{R}^d) \to \tilde{\mathbb{C}}$ is a Colombeau complex transition kernel, if for every $\epsilon \in (0,1]$ and $x$, the map $A \mapsto \mu_\epsilon^C(x, A)$ is a finite complex Colombeau measure on $\mathbb{R}^d$, and for every $\epsilon \in (0,1]$ and $A \in \mathbf{B}(\mathbb{R}^d)$
**Definition 0.6.6.** Let $\Omega$ be a locally compact space. Set

$$E(\mathbf{M}(\Omega)) = (\mathbf{M}(\Omega))^I, I = (0,1]$$

$$E_M(\mathbf{M}(\Omega)) = \left\{ (\mu_\varepsilon)_\varepsilon \in E(\mathbf{M}(\Omega)) \middle| (\forall \alpha \in \mathbb{N}_0^n)(\exists p \in \mathbb{N}) \sup_{B \subseteq \Omega} |\mu_\varepsilon(B)| = O(\varepsilon^{-p}) \right\},$$

$$N_0(\mathbf{M}(\Omega)) = \left\{ (\mu_\varepsilon)_\varepsilon \in E(\mathbf{M}(\Omega)) \middle| (\forall \alpha \in \mathbb{N}_0^n)(\forall q \in \mathbb{N}) \sup_{B \subseteq \Omega} |\mu_\varepsilon(B)| = O(\varepsilon^q) \right\}, \quad (0.6.7)$$

$$G(E_M(\mathbf{M}(\Omega))) = E_M(\mathbf{M}(\Omega))/N_0(\mathbf{M}(\Omega)).$$

Elements of $E_M^0(\Omega)$ and $N_0(\Omega)$ are called moderate, resp. negligible. Colombeau-Borel measures on $\Omega$.

**Theorem 0.6.2. (Generalized Riesz-Markov theorem)** Let $\Omega$ be a locally compact space, then the set $E_M(\mathbf{M}(\Omega))$ of all finite complex regular Colombeau-Borel measures on $\Omega$, which coincides with the set of all continuous linear Colombeau generalized functionals on $E_M^0(\Omega)$.

Any complex $\sigma$-additive Colombeau measure $(\mu_\varepsilon)_\varepsilon$ on $\mathbb{R}^d$ has a representation of the form

$$(\mu_\varepsilon(dy))_\varepsilon = (f_\varepsilon(y)M_\varepsilon(dy))_\varepsilon \quad (0.6.8)$$

with a positive Colombeau measure $(M_\varepsilon(dy))_\varepsilon$ and a bounded for every $\varepsilon \in (0,1]$ complex-valued generalized function $(f_\varepsilon(y))_\varepsilon$. Moreover, the Colombeau measure $(M_\varepsilon(dy))_\varepsilon$ in (0.6.8) is uniquely defined under additional assumption that $|f_\varepsilon(y)|_\varepsilon = 1$ for all $y$. If this condition holds, the positive measure $(M_\varepsilon(dy))_\varepsilon$ is called the total variation measure of the complex measure $(\mu_\varepsilon^C(dy))_\varepsilon$ and is denoted by $(|\mu_\varepsilon^C(dy))_\varepsilon$. In general, if (0.6.8) holds, then

$$(\|\mu_\varepsilon^C\|)_\varepsilon = \left( \int |f_\varepsilon(y)| M_\varepsilon(dy) \right)_\varepsilon \quad (0.6.9)$$

**Definition 0.6.7.** We say that a map $(\mu_\varepsilon^C)_\varepsilon : \mathbb{R}^d \times B(\mathbb{R}^d) \to \tilde{\mathbb{C}}$ is a Colombeau complex transition kernel, if for every $\varepsilon \in (0,1]$ and $x$, the map $A \mapsto \mu_\varepsilon^C(x,A)$ is a finite complex Colombeau measure on $\mathbb{R}^d$, and for every $\varepsilon \in (0,1]$ and $A \in \mathbf{B}(\mathbb{R}^d)$
the map $x \to \mu_{\varepsilon}^{C}(x,A)$ is B-measurable.

**Definition 0.6.8.** A time homogeneous Colombeau complex transition function (abbreviated CCTF) on $\mathbb{R}^d$ is a family $\mu_{t,\varepsilon}^{C}, t \ge 0$ of Colombeau complex transition kernels such that for every $\varepsilon \in (0,1]$, $\mu_{0,\varepsilon}^{C}(x,dy) = \delta(x-y)$ for all $x$, where $\delta(x-y)$ is the Dirac measure in $x$, and such that for every non-negative $s, t$, the Colombeau-Chapman-Kolmogorov equation

$$ \left( \int \mu_{s,\varepsilon}^{C}(x,dy) \mu_{t,\varepsilon}^{C}(y,A) \right)_{\varepsilon} = (\mu_{s+t,\varepsilon}^{C}(x,A))_{\varepsilon} \quad (0.6.10) $$

is satisfied.

**Definition 0.6.9.** A CCTF will be called regular, if there exists a positive constant $K_{\varepsilon} = K(\varepsilon)$ such that $\forall \varepsilon \in (0,1]$ for all $x$ and $t > 0$, the norm $\|\mu_{t,\varepsilon}^{C}(x, \cdot)\|$ of the measure $A \mapsto \mu_{t,\varepsilon}^{C}(x,A)$ does not exceed $\exp(K_{\varepsilon}t)$.

**Remark 0.6.4.** We note that if $\forall \varepsilon \in (0,1]$, $T_{t,\varepsilon}$ is a strongly continuous semigroup of bounded linear operators in $E_M^0(\mathbb{R}^d)$, then there exists a timehomogeneous CCTF $\mu_{t,\varepsilon}^{C}(x,dy)$ such that

$$ (T_{t,\varepsilon}f_{\varepsilon}(x))_{\varepsilon} = \left( \int \mu_{t,\varepsilon}^{C}(x,dy)f_{\varepsilon}(y) \right)_{\varepsilon}. \quad (0.6.11) $$

Now we construct Colombeau measure on the path space corresponding to each regular CCTF, introducing first some (rather standard) notations. Let $\mathbb{R}^d$ denote the one point compactification of the Euclidean space $\mathbb{R}^d$ (i.e. $\dot{\mathbb{R}}^d = \mathbb{R}^d \cup \{\infty\}$) and is homeomorphic to the sphere $S^d$). Let $\dot{\mathbb{R}}^d_{[s,t]} = \prod_{s \le t \le t_s} \dot{\mathbb{R}}^d_t$ denote the infinite product of $[s,t]$, i.e. it is the set of all functions from $[s,t]$ to $\mathbb{R}^d$, the path space.

**Definition 0.6.10.** Let $\mathbf{B}(\dot{\mathbb{R}}^d)^m$ be the set of all bounded complex Borel functions on $(\dot{\mathbb{R}}^d)^m$. Set $E(\mathbf{B}(\dot{\mathbb{R}}^d)^m) = (\mathbf{B}(\dot{\mathbb{R}}^d)^m)^l, I = (0,1]$
the map $x \to \mu_{\varepsilon}^{C}(x, A)$ is B-measurable.

**Definition 0.6.8.** A time homogeneous Colombeau complex transition function (abbreviated CCTF) on $\mathbb{R}^d$ is a family $\mu_{t,\varepsilon}^{C}, t \ge 0$ of Colombeau complex transition kernels such that for every $\varepsilon \in (0,1]$, $\mu_{0,\varepsilon}^{C}(x,dy) = \delta(x-y)$ for all $x$, where $\delta(x-y)$ is the Dirac measure in $x$, and such that for every non-negative $s, t$, the Colombeau-Chapman-Kolmogorov equation

$$ \left( \int \mu_{s,\varepsilon}^{C}(x,dy) \mu_{t,\varepsilon}^{C}(y,A) \right)_{\varepsilon} = (\mu_{s+t,\varepsilon}^{C}(x,A))_{\varepsilon} \quad (0.6.10) $$

is satisfied.

**Definition 0.6.9.** A CCTF will be called regular, if there exists a positive constant $K_{\varepsilon} = K(\varepsilon)$ such that $\forall \varepsilon \in (0,1]$ for all $x$ and $t > 0$, the norm $\|\mu_{t,\varepsilon}^{C}(x, \cdot)\|$ of the measure $A \mapsto \mu_{t,\varepsilon}^{C}(x, A)$ does not exceed $\exp(K_{\varepsilon}t)$.

**Remark 0.6.4.** We note that if $\forall \varepsilon \in (0,1]$, $T_{t,\varepsilon}$ is a strongly continuous semigroup of bounded linear operators in $E_M^0(\mathbb{R}^d)$, then there exists a timehomogeneous CCTF $\mu_{t,\varepsilon}^{C}(x, dy)$ such that

$$ (T_{t,\varepsilon}f_{\varepsilon}(x))_{\varepsilon} = \left( \int \mu_{t,\varepsilon}^{C}(x,dy)f_{\varepsilon}(y) \right)_{\varepsilon}. \quad (0.6.11) $$

Now we construct Colombeau measure on the path space corresponding to each regular CCTF, introducing first some (rather standard) notations. Let $\mathbb{R}^d$ denote the one point compactification of the Euclidean space $\mathbb{R}^d$ (i.e. $\dot{\mathbb{R}}^d = \mathbb{R}^d \cup \{\infty\}$) and is homeomorphic to the sphere $S^d$). Let $\dot{\mathbb{R}}^d_{[s,t]} = \prod_{s \le t \le t'} \dot{\mathbb{R}}^d_t$ denote the infinite product of $[s,t]$, i.e. it is the set of all functions from $[s,t]$ to $\mathbb{R}^d$, the path space.

**Definition 0.6.10.** Let $\mathbf{B}(\dot{\mathbb{R}}^d)^m$ be the set of all bounded complex Borel functions on $(\dot{\mathbb{R}}^d)^m$. Set $E(\mathbf{B}(\dot{\mathbb{R}}^d)^m) = (\mathbf{B}(\dot{\mathbb{R}}^d)^m)^l, I = (0,1]$
$$E_M(\mathbf{B}((\dot{\mathbb{R}}^d)^m)) =$$

$$\{(f_{\varepsilon}(x))_{\varepsilon} \in E(\mathbf{B}((\dot{\mathbb{R}}^d)^m)) | (\exists p \in \mathbb{N}) = (\sup_{x \in \mathbb{R}^{d \times m}} |f_{\varepsilon}(x_1, \dots, x_n)| = O(\varepsilon^{-p}))\},$$

$$N(\mathbf{B}((\dot{\mathbb{R}}^d)^m)) = \qquad (0.6.12)$$

$$\{(f_{\varepsilon}(x))_{\varepsilon} \in E(\mathbf{B}((\dot{\mathbb{R}}^d)^m)) | (\forall p \in \mathbb{N}) = (\sup_{x \in \mathbb{R}^{d \times m}} |f_{\varepsilon}(x_1, \dots, x_n)| = O(\varepsilon^{-p}))\},$$

$$G(\mathbf{B}((\dot{\mathbb{R}}^d)^m)) = E_M(\mathbf{B}((\dot{\mathbb{R}}^d)^m))/N(\mathbf{B}((\dot{\mathbb{R}}^d)^m)).$$

**Definition 0.6.11.** Let $\mathfrak{R}_{y(\cdot)}^k(\dot{\mathbb{R}}_{[s,t]}^d)$ denote the set of a generalized functions on $\dot{\mathbb{R}}_{[s,t]}^d$ having the form

$$(\Phi_{t_0, \dots, t_{k+1, \varepsilon}}^{f}(y(\cdot)))_{\varepsilon} = (f_{\varepsilon}(y(t_0), \dots, y(t_{k+1})))_{\varepsilon}, \qquad (0.6.13)$$

where $(f_{\varepsilon})_{\varepsilon} \in G((\dot{\mathbb{R}}^d)^{k+2})$. The union $\mathfrak{R}_{y(\cdot)}^{\infty}(\dot{\mathbb{R}}_{[s,t]}^d) = \bigcup_{k \in \mathbb{N}} \mathfrak{R}_{y(\cdot)}^k(\dot{\mathbb{R}}_{[s,t]}^d)$ is called the set of cylindrical generalized functions (or generalized functionals) on $\dot{\mathbb{R}}_{[s,t]}^d$. Any **CCTF** $\mu_{t,\varepsilon}^{\mathfrak{C}}(x,dy)$ defines a family of linear generalized functionals $(\mu_{s,t,\varepsilon}^x, x \in \mathbb{R}^d$ on $\mathfrak{R}_{y(\cdot)}^{\infty}(\dot{\mathbb{R}}_{[s,t]}^d)$ by the formula

$$(\mu_{s,t,\varepsilon}^x[\Phi_{t_0, \dots, t_{k+1,\varepsilon}}^f])_{\varepsilon} = \qquad (0.6.14)$$

$$(\iint f_{\varepsilon}(x, y_1, \dots, y_{k+1}) \mu_{t_1-t_0, \varepsilon}^{\mathfrak{C}}(x, dy_1) \mu_{t_2-t_1, \varepsilon}^{\mathfrak{C}}(y_1, dy_2) \dots \mu_{t_{k+1}-t_{k-1}, \varepsilon}^{\mathfrak{C}}(y_k, dy_{k+1}))_{\varepsilon}.$$

**Theorem 0.6.3.** If the Colombeau semigroup $(T_{t,\varepsilon})_{\varepsilon}$ in $E_M^0(\mathbb{R}^d)$ is regular and $\mu_{t,\varepsilon}^{\mathfrak{C}}(x,dy)$ is its corresponding **CCTF**, then $\forall \varepsilon \in (0,1]$ the functional $\mu_{s,t,\varepsilon}^x[\Phi_{t_0, \dots, t_{k+1,\varepsilon}}^f]$ is bounded. Hence, it can be extended by continuity to a unique bounded linear functional $\tilde{\mu}_{s,t,\varepsilon}^x$ on $E_M^0(\dot{\mathbb{R}}_{[s,t]}^d)$, and consequently there exists a (regular) complex Colombeau-Borel measure $D_{s,t,\varepsilon}^x$ on the path space $\dot{\mathbb{R}}_{[s,t]}^d$ such that
$$E_M(\mathbf{B}((\dot{\mathbb{R}}^d)^m)) =$$

$$\{(f_{\varepsilon}(x))_{\varepsilon} \in E(\mathbf{B}((\dot{\mathbb{R}}^d)^m)) | (\exists p \in \mathbb{N}) = (\sup_{x \in \mathbb{R}^{d \times m}} |f_{\varepsilon}(x_1, \dots, x_n)| = O(\varepsilon^{-p}))\},$$

$$N(\mathbf{B}((\dot{\mathbb{R}}^d)^m)) = \qquad (0.6.12)$$

$$\{(f_{\varepsilon}(x))_{\varepsilon} \in E(\mathbf{B}((\dot{\mathbb{R}}^d)^m)) | (\forall p \in \mathbb{N}) = (\sup_{x \in \mathbb{R}^{d \times m}} |f_{\varepsilon}(x_1, \dots, x_n)| = O(\varepsilon^{-p}))\},$$

$$G(\mathbf{B}((\dot{\mathbb{R}}^d)^m)) = E_M(\mathbf{B}((\dot{\mathbb{R}}^d)^m))/N(\mathbf{B}((\dot{\mathbb{R}}^d)^m)).$$

**Definition 0.6.11.** Let $\mathfrak{R}_{y(\cdot)}^k(\dot{\mathbb{R}}_{[s,t]}^d)$ denote the set of a generalized functions on $\dot{\mathbb{R}}_{[s,t]}^d$ having the form

$$(\Phi_{t_0, \dots, t_{k+1, \varepsilon}}^{f}(y(\cdot)))_{\varepsilon} = (f_{\varepsilon}(y(t_0), \dots, y(t_{k+1})))_{\varepsilon}, \qquad (0.6.13)$$

where $(f_{\varepsilon})_{\varepsilon} \in G((\dot{\mathbb{R}}^d)^{k+2})$. The union $\mathfrak{R}_{y(\cdot)}^{\infty}(\dot{\mathbb{R}}_{[s,t]}^d) = \bigcup_{k \in \mathbb{N}} \mathfrak{R}_{y(\cdot)}^k(\dot{\mathbb{R}}_{[s,t]}^d)$ is called the set of cylindrical generalized functions (or generalized functionals) on $\dot{\mathbb{R}}_{[s,t]}^d$. Any **CCTF** $\mu_{t,\varepsilon}^{\mathfrak{C}}(x,dy)$ defines a family of linear generalized functionals $(\mu_{s,t,\varepsilon}^x, x \in \mathbb{R}^d$ on $\mathfrak{R}_{y(\cdot)}^{\infty}(\dot{\mathbb{R}}_{[s,t]}^d)$ by the formula

$$(\mu_{s,t,\varepsilon}^x[\Phi_{t_0, \dots, t_{k+1,\varepsilon}}^f])_{\varepsilon} = \qquad (0.6.14)$$

$$(\int f_{\varepsilon}(x, y_1, \dots, y_{k+1}) \mu_{t_1-t_0, \varepsilon}^{\mathfrak{C}}(x, dy_1) \mu_{t_2-t_1, \varepsilon}^{\mathfrak{C}}(y_1, dy_2) \dots \mu_{t_{k+1}-t_{k-1}, \varepsilon}^{\mathfrak{C}}(y_k, dy_{k+1}))_{\varepsilon}.$$

**Theorem 0.6.3.** If the Colombeau semigroup $(T_{t,\varepsilon})_{\varepsilon}$ in $E_M^0(\mathbb{R}^d)$ is regular and $\mu_{t,\varepsilon}^{\mathfrak{C}}(x,dy)$ is its corresponding **CCTF**, then $\forall \varepsilon \in (0,1]$ the functional $\mu_{s,t,\varepsilon}^x[\Phi_{t_0, \dots, t_{k+1,\varepsilon}}^f]$ is bounded. Hence, it can be extended by continuity to a unique bounded linear functional $\tilde{\mu}_{s,t,\varepsilon}^x$ on $E_M^0(\dot{\mathbb{R}}_{[s,t]}^d)$, and consequently there exists a (regular) complex Colombeau-Borel measure $D_{s,t,\varepsilon}^x$ on the path space $\dot{\mathbb{R}}_{[s,t]}^d$ such that
$$ (\tilde{\mu}_{s,t,\epsilon}^{x}[F_{\epsilon}])_{\epsilon} = \left( \int F_{\epsilon}(y(\tau)) \mathbf{D}_{s,t,\epsilon}^{x}[y(\tau)] \right)_{\epsilon} \quad (0.6.15) $$

for all $(F_{\epsilon})_{\epsilon} \in G_0(\dot{\mathbb{R}}_{[s,t]}^d)$. In particular,

$$ ((T_{t,\epsilon}f_{\epsilon})(x))_{\epsilon} = \left( \int f_{\epsilon}(y(t)) \mathbf{D}_{s,t,\epsilon}^{x}[y(\tau)] \right)_{\epsilon}. \quad (0.6.16) $$

**Theorem 0.6.4.** Let $(B_{\epsilon})_{\epsilon \in (0,1]}$ and $(A_{\epsilon})_{\epsilon \in (0,1]}$ be the nets of linear operators in $E_M^0(\dot{\mathbb{R}}_{[s,t]}^d)$

such that $\forall \epsilon \in (0,1]$, $A_{\epsilon}$ is bounded and $(B_{\epsilon})_{\epsilon}$ is the generator of a strongly continuous regular Colombeau semigroup $(T_{t,\epsilon})_{\epsilon}$. Then $(A_{\epsilon} + B_{\epsilon})_{\epsilon \in (0,1]}$ is also the generator of a regular semigroup, which we denote by $(\tilde{T}_{t,\epsilon})_{\epsilon}$.

**Definition 0.6.12.** Let $\mathcal{F}(\mathbb{R}^d)$ denote the Banach space of Fourier transforms of elements of $\mathbf{M}(\mathbb{R}^d)$, i.e. the space of continuous functions on $\mathbb{R}^d$ of the form

$$ V_{\mu}(x) = \int_{\mathbb{R}^d} \exp(ip \cdot x)\mu(dp) \quad (0.6.17) $$

for some $\mu \in \mathbf{M}(\mathbb{R}^d)$, with the induced norm $\|V_{\mu}\| = \|\mu\|$. Since $\mathbf{M}(\mathbb{R}^d)$ is a Banach algebra with convolution as the multiplication, it follows that $\mathcal{F}(\mathbb{R}^d)$ is also a Banach algebra with respect to the pointwise multiplication. We say that an element $f \in \mathcal{F}(\mathbb{R}^d)$

is infinitely divisible if there exists a family $f_i, t \ge 0$ of elements of $\mathcal{F}(\mathbb{R}^d)$ such that $f_0 = 1, f_1 = f$, and $f_{t+s} = f_i \cdot f_s$ for all positive $s, t$. Clearly if $f$ is infinitely divisible, then it has no zeros and a continuous function $g = \ln f$ is well defined (and is unique up to an imaginary shift). Moreover, the family $f_i$ has the form $f_i = \exp(tg)$

**Definition 0.6.13.** We say that a continuous function $g$ on $\mathbb{R}^d$ is a complex characteristic exponent (abbreviated CCE), if $\exp(g)$ is an infinitely divisible element of $\mathcal{F}(\mathbb{R}^d)$, or equivalently, if $\forall t, s \ge 0 : \exp(tg) \in \mathcal{F}(\mathbb{R}^d)$.

**Remark 0.6.5.** It follows from the definitions that the set of spatially homogeneous
$$ (\tilde{\mu}_{s,t,\epsilon}^{x}[F_{\epsilon}])_{\epsilon} = \left( \int f_{\epsilon}(y(\tau)) \mathbf{D}_{s,t,\epsilon}^{x}[y(\tau)] \right)_{\epsilon} \quad (0.6.15) $$

for all $(F_{\epsilon})_{\epsilon} \in G_0(\dot{\mathbb{R}}_{[s,t]}^d)$. In particular,

$$ ((T_{t,\epsilon}f_{\epsilon})(x))_{\epsilon} = \left( \int f_{\epsilon}(y(t)) \mathbf{D}_{s,t,\epsilon}^{x}[y(\tau)] \right)_{\epsilon}. \quad (0.6.16) $$

**Theorem 0.6.4.** Let $(B_{\epsilon})_{\epsilon \in (0,1]}$ and $(A_{\epsilon})_{\epsilon \in (0,1]}$ be the nets of linear operators in $E_M^0(\dot{\mathbb{R}}_{[s,t]}^d)$

such that $\forall \epsilon \in (0,1]$, $A_{\epsilon}$ is bounded and $(B_{\epsilon})_{\epsilon}$ is the generator of a strongly continuous regular Colombeau semigroup $(T_{t,\epsilon})_{\epsilon}$. Then $(A_{\epsilon} + B_{\epsilon})_{\epsilon \in (0,1]}$ is also the generator of a regular semigroup, which we denote by $(\tilde{T}_{t,\epsilon})_{\epsilon}$.

**Definition 0.6.12.** Let $\mathcal{F}(\mathbb{R}^d)$ denote the Banach space of Fourier transforms of elements of $\mathbf{M}(\mathbb{R}^d)$, i.e. the space of continuous functions on $\mathbb{R}^d$ of the form

$$ V_{\mu}(x) = \int_{\mathbb{R}^d} \exp(ip \cdot x)\mu(dp) \quad (0.6.17) $$

for some $\mu \in \mathbf{M}(\mathbb{R}^d)$, with the induced norm $\|V_{\mu}\| = \|\mu\|$. Since $\mathbf{M}(\mathbb{R}^d)$ is a Banach algebra with convolution as the multiplication, it follows that $\mathcal{F}(\mathbb{R}^d)$ is also a Banach algebra with respect to the pointwise multiplication. We say that an element $f \in \mathcal{F}(\mathbb{R}^d)$

is infinitely divisible if there exists a family $f_i, t \ge 0$ of elements of $\mathcal{F}(\mathbb{R}^d)$ such that $f_0 = 1, f_1 = f$, and $f_{t+s} = f_i \cdot f_s$ for all positive $s, t$. Clearly if $f$ is infinitely divisible, then it has no zeros and a continuous function $g = \ln f$ is well defined (and is unique up to an imaginary shift). Moreover, the family $f_i$ has the form $f_i = \exp(tg)$

**Definition 0.6.13.** We say that a continuous function $g$ on $\mathbb{R}^d$ is a complex characteristic exponent (abbreviated CCE), if $\exp(g)$ is an infinitely divisible element of $\mathcal{F}(\mathbb{R}^d)$, or equivalently, if $\forall t, s \ge 0 : \exp(tg) \in \mathcal{F}(\mathbb{R}^d)$.

**Remark 0.6.5.** It follows from the definitions that the set of spatially homogeneous
CTF's $\mu_t^C(dx)$ is in one-to-one correspondence with CCE g, in such a way that for any positive t the function $\exp(tg)$ is the Fourier transform of the transition measure $\mu_t^C(dx)$.

**Definition 0.6.14.**[54].**(Hardy spaces for the unit disk)**. The Hardy space $H^p(D)$ for $0 < p < \infty$ is the class of holomorphic functions $f(z)$ on the open unit disk satisfying

$$ \sup_{0<r<1} \left( \frac{1}{2\pi} \int_0^{2\pi} |f(re^{i\theta})|^p d\theta \right)^{\frac{1}{p}} < \infty. \qquad (0.6.18) $$

This class $H^p(D)$ is a vector space. The number on the left side of the above inequality is the Hardy space p-norm for $f$, denoted by $||f||_{H^p}$. It is a norm when $p \ge 1$, but not when $0 < p < 1$.

**Definition 0.6.15.**[54].**(Hardy spaces for the upper half plane).** The Hardy space on the upper half-plane $H^p(H)$ is defined to be the space of holomorphic functions $f(z)$ on $H$ with bounded (quasi-)norm, the norm being given by

$$ \|f\|_{H^p} = \sup_{0<y} \left( \int |f(x+iy)|^p dx \right)^{\frac{1}{p}} < \infty. \qquad (0.6.19) $$

Note that the unit disk $D$ and the upper half-plane $H$ can be mapped to one-another by means of Möbius transformations, they are not interchangeable as domains for Hardy spaces. Contributing to this difference is the fact that the unit circle has finite (one-dimensional) Lebesgue measure while the real line does not. However, for $H^2$, one may still state the following theorem: Given the Möbius transformation $\mathbf{m}: D \to H$

with $\mathbf{m}(z) = i(1+z)(1-z)^{-1}$ then there is an isometric isomorphism

$M: H^p(D) \to H^p(H)$

with $[Mf](x) = \sqrt{\pi} (1-z)^{-1}f(m(z))$.

**Definition 0.6.16.** A sequence of points $\{a_n\}_{n=1}^\infty$ inside the unit disk $D$ is said to satisfy the Blaschke condition when $\sum_{n=1}^\infty (1 - |a_n|) < \infty$. Given a sequence obeying the Blaschke condition, the Blaschke product $B(z)$ is defined as

$$ B(z) = \prod_n B_n(z), B_n(z) = - \frac{|a_n|}{a_n} \frac{z - a_n}{1 - \bar{a}_n \cdot z}. \qquad (0.6.20) $$

The Blaschke product $B(z)$ is analytic in the open unit disc, and is zero at the $a_n$
CTF's $\mu_t^C(dx)$ is in one-to-one correspondence with CCE g, in such a way that for any positive t the function $\exp(tg)$ is the Fourier transform of the transition measure $\mu_t^C(dx)$.

**Definition 0.6.14.** [54]. (Hardy spaces for the unit disk). The Hardy space $H^p(D)$ for $0 < p < \infty$ is the class of holomorphic functions $f(z)$ on the open unit disk satisfying

$$ \sup_{0<r<1} \left( \frac{1}{2\pi} \int_0^{2\pi} |f(re^{i\theta})|^p d\theta \right)^{\frac{1}{p}} < \infty. \qquad (0.6.18) $$

This class $H^p(D)$ is a vector space. The number on the left side of the above inequality is the Hardy space p-norm for $f$, denoted by $||f||_{H^p}$. It is a norm when $p \ge 1$, but not when $0 < p < 1$.

**Definition 0.6.15.** [54]. (Hardy spaces for the upper half plane). The Hardy space on the upper half-plane $H^p(H)$ is defined to be the space of holomorphic functions $f(z)$ on $H$ with bounded (quasi-)norm, the norm being given by

$$ ||f||_{H^p} = \sup_{0<y} \left( \int |f(x+iy)|^p dx \right)^{\frac{1}{p}} < \infty. \qquad (0.6.19) $$

Note that the unit disk $D$ and the upper half-plane $H$ can be mapped to one-another by means of Möbius transformations, they are not interchangeable as domains for Hardy spaces. Contributing to this difference is the fact that the unit circle has finite (one-dimensional) Lebesgue measure while the real line does not. However, for $H^2$, one may still state the following theorem: Given the Möbius transformation $\mathbf{m}: D \to H$

with $\mathbf{m}(z) = i(1+z)(1-z)^{-1}$ then there is an isometric isomorphism

$M: H^p(D) \to H^p(H)$

with $[Mf](x) = \sqrt{\pi} (1-z)^{-1}f(m(z))$.

**Definition 0.6.16.** A sequence of points $\{a_n\}_{n=1}^\infty$ inside the unit disk $D$ is said to satisfy the Blaschke condition when $\sum_{n=1}^\infty (1 - |a_n|) < \infty$. Given a sequence obeying the Blaschke condition, the Blaschke product $B(z)$ is defined as

$$ B(z) = \prod_n B_n(z), B_n(z) = - \frac{|a_n|}{a_n} \frac{z - a_n}{1 - \bar{a}_n \cdot z}. \qquad (0.6.20) $$

The Blaschke product $B(z)$ is analytic in the open unit disc, and is zero at the $a_n$
only.

**Definition 0.6.17.**[54].(i) One says that $G(z)$ is an **outer (exterior) function** if it takes the form

$$G(z) = c \exp \left[ \frac{1}{2\pi} \int_{-\pi}^{\pi} \frac{e^{i\theta} + z}{e^{i\theta} - z} \log \varphi(e^{i\theta}) d\theta \right] \quad (0.6.21)$$

for some complex number $c$ with $|c|= 1$, and some positive measurable function $\varphi$ on the unit circle such that $\log \varphi$ is integrable on the circle.

(ii) One says that $h(z)$ is an **inner (interior) function** if and only if $|h(z)|\le 1$ on the unit disk and the limit $\lim_{r\to 1^-} h(re^{i\theta})$ exists for almost all $\theta$ and its modulus is equal to 1.

Let $G$ be an outer function represented as above from a function $\varphi$ on the circle. Replacing $\varphi$ by $\varphi^\alpha, \alpha > 0$, a family $G_\alpha$ of outer functions is obtained, with the properties:

$$G_1 = G, G_{\alpha+\beta} = G_\alpha G_\beta, \qquad (0.6.22)$$

$$|G_\alpha| = |G|^\alpha.$$

**Remark 0.6.6.** For $0 < p \le \infty$, every non-zero function $f(z)$ in $H^p$ can be written as the product $f(z) = G(z)h(z)$ where $G(z)$ is an outer function and $h(z)$ is an inner function.

**Remark 0.6.7.** It is easy to show that if $f_1 \in \mathcal{F}(\mathbb{R})$ is infinite divisible and such that the measures corresponding to all functions $f_i, t > 0$, are concentrated on the half line $\mathbb{R}_+$ (complex generalisation of subordinators) and have densities from $\mathcal{L}_2(\mathbb{R}_+)$, then $f_1$ belongs to the Hardy space $H^2$ of analytic functions on the upper half plane, which have no Blaschke product in its canonical decomposition.

**Theorem 0.6.5.**[49]. If $V$ is a CCE, then the solution to the Cauchy problem

$$\frac{\partial u}{\partial t} = V \left( \frac{1}{i} \frac{\partial}{\partial y} \right) u \qquad (0.6.23)$$

defines a strongly continuous and spatially homogeneous semigroup $T_t$ of bounded linear operators in $C_0(\mathbb{R}^d)$ (i.e. $(T_t u_0)(y)$ is the solution to equation (0.6.23) with the initial function $u_0$). Conversely, each such semigroup is the solution to the Cauchy problem of an equation of type (0.6.23) with some CCE $g$.

**Definition 0.6.18.**[49]. We say that a CCE $V$ is regular, if equation (0.6.23) defines
only.

**Definition 0.6.17.**[54].(i) One says that $G(z)$ is an **outer (exterior) function** if it takes the form

$$G(z) = c \exp \left[ \frac{1}{2\pi} \int_{-\pi}^{\pi} \frac{e^{i\theta} + z}{e^{i\theta} - z} \log \varphi(e^{i\theta}) d\theta \right] \quad (0.6.21)$$

for some complex number $c$ with $|c|= 1$, and some positive measurable function $\varphi$ on the unit circle such that $\log \varphi$ is integrable on the circle.

(ii) One says that $h(z)$ is an **inner (interior) function** if and only if $|h(z)|\le 1$ on the unit disk and the limit $\lim_{r\to 1^-} h(re^{i\theta})$ exists for almost all $\theta$ and its modulus is equal to 1.

Let $G$ be an outer function represented as above from a function $\varphi$ on the circle. Replacing $\varphi$ by $\varphi^\alpha, \alpha > 0$, a family $G_\alpha$ of outer functions is obtained, with the properties:

$$G_1 = G, G_{\alpha+\beta} = G_\alpha G_\beta, \quad (0.6.22)$$

$$|G_\alpha| = |G|^\alpha.$$

**Remark 0.6.6.** For $0 < p \le \infty$, every non-zero function $f(z)$ in $H^p$ can be written as the product $f(z) = G(z)h(z)$ where $G(z)$ is an outer function and $h(z)$ is an inner function.

**Remark 0.6.7.** It is easy to show that if $f_1 \in \mathcal{F}(\mathbb{R})$ is infinite divisible and such that the measures corresponding to all functions $f_i, t > 0$, are concentrated on the half line $\mathbb{R}_+$ (complex generalisation of subordinators) and have densities from $\mathcal{L}_2(\mathbb{R}_+)$, then $f_1$ belongs to the Hardy space $H^2$ of analytic functions on the upper half plane, which have no Blaschke product in its canonical decomposition.

**Theorem 0.6.5.**[49]. If $V$ is a CCE, then the solution to the Cauchy problem

$$\frac{\partial u}{\partial t} = V \left( \frac{1}{i} \frac{\partial}{\partial y} \right) u \quad (0.6.23)$$

defines a strongly continuous and spatially homogeneous semigroup $T_t$ of bounded linear operators in $C_0(\mathbb{R}^d)$ (i.e. $(T_t u_0)(y)$ is the solution to equation (0.6.23) with the initial function $u_0$). Conversely, each such semigroup is the solution to the Cauchy problem of an equation of type (0.6.23) with some CCE $g$.

**Definition 0.6.18.**[49]. We say that a CCE $V$ is regular, if equation (0.6.23) defines
a regular semigroup.

**Theorem 0.6.6.**[49]. Let $V \in \mathcal{F}(\mathbb{R}^d)$, i.e. it is given by (0.6.17) with $\mu^C \in \mathbf{M}(\mathbb{R}^d)$. Then

$V$ is a regular CCE. Moreover, if the positive measure $M(dy)$ in the representation (0.6.2) for $\mu^C$ has no atom at the origin, i.e. $M(\{0\}) = 0$, then the corresponding measure $D_{0,t}^x$ on the path space is concentrated on the set of piecewise-constant paths in $\mathbb{R}^d$ with a finite number of jumps. In other words, $D_{0,t}^x$ is the measure of a jump-process.

Let us consider the pseudo-differential equation of the Schrödinger type

$$ \frac{\partial \tilde{u}}{\partial t} = -G[(-\Delta)^{\alpha}]\tilde{u} + (A, \frac{\partial}{\partial x})\tilde{u} + V(x)\tilde{u}, \quad (0.6.24) $$

where $G$ is a complex constant with a non-negative real part, $\alpha$ is any positive constant,

$A$ is a real-valued vector (if $\mathrm{Re}\,G > 0$, then $A$ can be also complex-valued), and $V(x)$ is a complex-valued function of form (0.6.17). The equation on the inverse Fourier transform

$$ u(y) = \int_{\mathbb{R}^d} dx \exp(-iyx) \tilde{u}(x) \quad (0.6.25) $$

of $\tilde{u}(x)$ has the form

$$ \frac{\partial u}{\partial t} = -G(y^2)^\alpha + i(A,y)u + V\left(\frac{1}{i}\frac{\partial}{\partial y}\right)u. \quad (0.6.26) $$

**Theorem 0.6.7.**[49]. The solution to the Cauchy problem of equation (0.6.26) can be

written in the form of a complex Feynman-Kac formula
a regular semigroup.

**Theorem 0.6.6.**[49]. Let $V \in \mathcal{F}(\mathbb{R}^d)$, i.e. it is given by (0.6.17) with $\mu^C \in \mathbf{M}(\mathbb{R}^d)$. Then

$V$ is a regular CCE. Moreover, if the positive measure $M(dy)$ in the representation (0.6.2) for $\mu^C$ has no atom at the origin, i.e. $M(\{0\}) = 0$, then the corresponding measure $D_{0,t}^x$ on the path space is concentrated on the set of piecewise-constant paths in $\mathbb{R}^d$ with a finite number of jumps. In other words, $D_{0,t}^x$ is the measure of a jump-process.

Let us consider the pseudo-differential equation of the Schrödinger type

$$ \frac{\partial \tilde{u}}{\partial t} = -G[(-\Delta)^{\alpha}]\tilde{u} + (A, \frac{\partial}{\partial x})\tilde{u} + V(x)\tilde{u}, \quad (0.6.24) $$

where $G$ is a complex constant with a non-negative real part, $\alpha$ is any positive constant,

$A$ is a real-valued vector (if $\mathrm{Re} G > 0$, then $A$ can be also complex-valued), and $V(x)$ is a complex-valued function of form (0.6.17). The equation on the inverse Fourier transform

$$ u(y) = \int_{\mathbb{R}^d} dx \exp(-iyx) \tilde{u}(x) \quad (0.6.25) $$

of $\tilde{u}(x)$ has the form

$$ \frac{\partial u}{\partial t} = -G(y^2)^\alpha + i(A, y)u + V\left(\frac{1}{i}\frac{\partial}{\partial y}\right)u. \quad (0.6.26) $$

**Theorem 0.6.7.**[49]. The solution to the Cauchy problem of equation (0.6.26) can be

written in the form of a complex Feynman-Kac formula
$$u(t, u) = \int \exp\left\{-\int_{0}^{t} [G(q^2(\tau))^a - (A, q(\tau))]\right\} u_0(q(t)) \mathbf{D}_{0,t}^x[q(\cdot)], \quad (0.6.27)$$

where $\mathbf{D}_{0,t}^x[q(\cdot)]$ is the measure of the jump process corresponding to equation (0.6.23).

As another example, let us consider the case of complex anharmonic oscillator, i.e. the equation

$$\frac{\partial \tilde{\psi}}{\partial t} = \frac{1}{2} (G\Delta - x^2 - iV(x)) \tilde{\psi}, \quad (0.6.28)$$

where $V = V_\mu \in \mathcal{F}(\mathbb{R}^d)$. The Fourier transform of this equation has the form

$$\frac{\partial \psi(t,p)}{\partial t} = \frac{1}{2} \left( \Delta - Gp^2 - iV \left( \frac{1}{i} \frac{\partial}{\partial p} \right) \right) \psi(t,p). \quad (0.6.29)$$

**Theorem 0.6.8.** [49]. If $\mathrm{Re}\,G \ge 0$, the Cauchy problem of equation (0.6.29) defines a regular semigroup of operators in $C_0(\mathbb{R}^d)$, and thus can be presented as the path integral from (Proposition 2.1.[49]).

**Remark 0.6.8.** [49]. The statement of the Proposition can be generalised easily to the following situation, which includes all Schrödinger equations, namely to the case of the equation

$$\frac{\partial \psi}{\partial t} = i(A - B)\psi. \quad (0.6.30)$$

where $A$ is selfadjoint operator, for which therefore exists (according to spectral theory) a unitary transformation $U$ such that $UAU^{-1}$ is the multiplication operator in some

$\mathcal{L}_2(X, d\mu)$, where $X$ is locally compact, and $B$ is such that $UBU^{-1}$ is a bounded operator in $C_0(X)$.

We shall consider now a formal Schrödinger operator with a magnetic field, namely the operator
$$u(t, u) = \int \exp\left\{-\int_{0}^{t} [G(q^2(\tau))^a - (A, q(\tau))]\right\} u_0(q(t)) \mathbf{D}_{0,t}^x[q(\cdot)], \quad (0.6.27)$$

where $\mathbf{D}_{0,t}^x[q(\cdot)]$ is the measure of the jump process corresponding to equation (0.6.23).

As another example, let us consider the case of complex anharmonic oscillator, i.e. the equation

$$\frac{\partial \tilde{\psi}}{\partial t} = \frac{1}{2} (G\Delta - x^2 - iV(x)) \tilde{\psi}, \quad (0.6.28)$$

where $V = V_\mu \in \mathcal{F}(\mathbb{R}^d)$. The Fourier transform of this equation has the form

$$\frac{\partial \psi(t,p)}{\partial t} = \frac{1}{2} \left( \Delta - Gp^2 - iV \left( \frac{1}{i} \frac{\partial}{\partial p} \right) \right) \psi(t,p). \quad (0.6.29)$$

**Theorem 0.6.8.** [49]. If $\mathrm{Re}\,G \ge 0$, the Cauchy problem of equation (0.6.29) defines a regular semigroup of operators in $C_0(\mathbb{R}^d)$, and thus can be presented as the path integral from (Proposition 2.1.[49]).

**Remark 0.6.8.** [49]. The statement of the Proposition can be generalised easily to the following situation, which includes all Schrödinger equations, namely to the case of the equation

$$\frac{\partial \psi}{\partial t} = i(A - B)\psi. \quad (0.6.30)$$

where $A$ is selfadjoint operator, for which therefore exists (according to spectral theory) a unitary transformation $U$ such that $UAU^{-1}$ is the multiplication operator in some

$\mathcal{L}_2(X, d\mu)$, where $X$ is locally compact, and $B$ is such that $UBU^{-1}$ is a bounded operator in $C_0(X)$.

We shall consider now a formal Schrödinger operator with a magnetic field, namely the operator
$$H = \left( i \frac{\partial}{\partial x} + A(x) \right)^2 + V(x) \qquad (0.6.31)$$

where $x = (x_1, \dots, x_d)$ and $\frac{\partial}{\partial x} = (\frac{\partial}{\partial x_1}, \dots, \frac{\partial}{\partial x_d})$ is the gradient operator in $\mathbb{R}^d$, under the following conditions [49]:

(C1) the magnetic vector-potential $A = (A^1, \dots, A^d)$ is a bounded measurable $\mathbb{R}^d$-valued function on $\mathbb{R}^d$,

(C2) the potential $V$ and the divergence $\mathrm{div} A = \sum_{j=1}^{d} \frac{\partial A^j}{\partial x_j}$ of $A$, defined in the sense of distributions, are both Borel measures,
(C3) if $d > 1$ there exist $\alpha > d-2$ and $C > 0$ such that for all $x \in \mathbb{R}^d$ and $r \in (0,1]$

$$|\mathrm{div} A|(B_r(x)) \leq Cr^\alpha, |V|(B_r(x)) \leq Cr^\alpha \qquad (0.6.32)$$

where $|\mathrm{div} A|(\cdot)$ and $|V|(\cdot)$ denote the total variations of the (possibly non-positive) measures $V$ and $\mathrm{div} A$ respectively, and $B_r(x)$ denotes the ball in $\mathbb{R}^d$ of the radius $r$ centered at $x$; if $d=1$, then the same holds for $\alpha = 0$.
Let us consider the following more general Cauchy problem

$$\frac{\partial u(t,x)}{\partial t} = -DHu(t,x), u(0,x) = u_0(x), \qquad (0.6.33)$$

where the (generalised) diffusion coefficient $D$ is an arbitrary complex number such that $\epsilon = \mathrm{Re} D \geq 0, |D| > 0$. In the interaction representation, equation (0.6.32) takes the form

$$u(t,x) = \exp(-Dt\Delta/2)u_0(x) - D \int_0^t \left[\exp(-D(t-s)\Delta/2)\right](W(x) - 2i(A(x),\nabla))u(s,x)ds. \qquad (0.6.34)$$

Here $W(x) = V(x) + |A(x)|^2 - i \mathrm{div} A(x)$. Equation (0.6.34) can be formally solved by iterations. In the case of the Green function $G^D(t,x,y)$ of equation (0.6.34), i.e. its
$$H = \left( i \frac{\partial}{\partial x} + A(x) \right)^2 + V(x) \qquad (0.6.31)$$

where $x = (x_1, \dots, x_d)$ and $\frac{\partial}{\partial x} = (\frac{\partial}{\partial x_1}, \dots, \frac{\partial}{\partial x_d})$ is the gradient operator in $\mathbb{R}^d$,

under the following conditions [49]:

(C1) the magnetic vector-potential $\mathbf{A} = (A^1, \dots, A^d)$ is a bounded measurable $\mathbb{R}^d$-valued function on $\mathbb{R}^d$,

(C2) the potential $V$ and the divergence $\mathrm{div}\mathbf{A} = \sum_{j=1}^{d} \frac{\partial A^j}{\partial x_j}$ of $\mathbf{A}$, defined in the sense

of distributions, are both Borel measures,

(C3) if $d > 1$ there exist $\alpha > d-2$ and $C > 0$ such that for all $x \in \mathbb{R}^d$ and $r \in (0,1]$

$$|\mathrm{div}\mathbf{A}|(B_r(x)) \leq Cr^\alpha, |V|(B_r(x)) \leq Cr^\alpha \qquad (0.6.32)$$

where $|\mathrm{div}\mathbf{A}|(\cdot)$ and $|V|(\cdot)$ denote the total variations of the (possibly non-positive) measures $V$ and $\mathrm{div}\mathbf{A}$ respectively, and $B_r(x)$ denotes the ball in $\mathbb{R}^d$ of the radius $r$ centered at $x$; if $d=1$, then the same holds for $\alpha = 0$.

Let us consider the following more general Cauchy problem

$$\frac{\partial u(t,x)}{\partial t} = -DHu(t,x), u(0,x) = u_0(x), \qquad (0.6.33)$$

where the (generalised) diffusion coefficient $D$ is an arbitrary complex number such that $\epsilon = \mathrm{Re}D \geq 0, |D| > 0$. In the interaction representation, equation (0.6.32) takes the form

$$u(t,x) = \\ \left[ \exp(-Dt\Delta/2) \right] u_0(x) - D \int_0^t \left[ \exp(-D(t-s)\Delta/2) \right] (W(x) - 2i(\mathbf{A}(x), \nabla)) u(s,x) ds. \qquad (0.6.34)$$

Here $W(x) = V(x) + |\mathbf{A}(x)|^2 - i\mathrm{div}\mathbf{A}(x)$. Equation (0.6.34) can be formally solved by iterations. In the case of the Green function $G^D(t,x,y)$ of equation (0.6.34), i.e. its
solution with the Dirac initial condition $u_0(x) = \delta(x - y)$, the iteration procedure leads to the following representation:

$$G^D(t,x,y) = \sum_{k=0}^{\infty} I_k^D(t,x,y),$$

$$I_k^D(t,x,y) =$$

$$-D \int_0^t \int_{\mathbb{R}^d} I_{k-1}^D(t-s,x,\xi)(W(d\xi) + 2iD^{-1}s^{-1}(\mathbf{A}, (\xi-y))d\xi)G_{\text{free}}^D(s, \xi-y)ds, \quad (0.6.35)$$

$$G_{\text{free}}^D(s, \xi - y) = (2\pi tD)^{-d/2} \exp\left[-\frac{(\xi - y)^2}{2Dt}\right].$$

**Theorem 0.6.9.** [49].(i) If $\epsilon = \text{Re}D > 0$, then all terms of series (0.6.35) are well defined

as absolutely convergent integrals, the series itself is absolutely convergent and its sum $G^D(t,x,y)$ is continuous in $x,y \in \mathbb{R}^d$, $t > 0$ (and $D$) and satisfies the following estimate

$$G^D(t,x,y) \leq K G_{\text{free}}^{\frac{|D|^2}{\epsilon}}(t,x-y) \exp(B|x-y|) \quad (0.6.36)$$

uniformly for $t \le t_0$ with any fixed $t_0$, where $B, K$ are constants.

(ii) The integral operators

$$[U^D(t)u_0](t,x) = \int u^D(t,x,y)u_0(y)dy \quad (0.6.37)$$

defining the solutions to equation (0.6.34) for $t \in [0, t_0]$ form a uniformly bounded family of operators $\mathcal{L}_2(\mathbb{R}^d) \to \mathcal{L}_2(\mathbb{R}^d)$.

(iii) If $D$ is real, i.e. $D = \epsilon > 0$, then there exists a constant $\omega > 0$ such that the Green function $G^\epsilon$ has the asymptotic representation
solution with the Dirac initial condition $u_0(x) = \delta(x - y)$, the iteration procedure leads to the following representation:

$$G^D(t,x,y) = \sum_{k=0}^{\infty} I_k^D(t,x,y),$$

$$I_k^D(t,x,y) =$$

$$-D \int_0^t \int_{\mathbb{R}^d} I_{k-1}^D(t-s,x,\xi)(W(d\xi) + 2iD^{-1}s^{-1}(\mathbf{A}, (\xi-y))d\xi)G_{\text{free}}^D(s, \xi-y)ds, \quad (0.6.35)$$

$$G_{\text{free}}^D(s, \xi - y) = (2\pi tD)^{-d/2} \exp\left[-\frac{(\xi - y)^2}{2Dt}\right].$$

**Theorem 0.6.9.** [49].(i) If $\epsilon = \text{Re}D > 0$, then all terms of series (0.6.35) are well defined

as absolutely convergent integrals, the series itself is absolutely convergent and its sum $G^D(t,x,y)$ is continuous in $x,y \in \mathbb{R}^d$, $t > 0$ (and $D$) and satisfies the following estimate

$$G^D(t,x,y) \leq K G_{\text{free}}^{\frac{|D|^2}{\epsilon}}(t,x-y) \exp(B|x-y|) \quad (0.6.36)$$

uniformly for $t \le t_0$ with any fixed $t_0$, where $B, K$ are constants.

(ii) The integral operators

$$[U^D(t)u_0](t,x) = \int u^D(t,x,y)u_0(y)dy \quad (0.6.37)$$

defining the solutions to equation (0.6.34) for $t \in [0, t_0]$ form a uniformly bounded family of operators $\mathcal{L}_2(\mathbb{R}^d) \to \mathcal{L}_2(\mathbb{R}^d)$.

(iii) If $D$ is real, i.e. $D = \epsilon > 0$, then there exists a constant $\omega > 0$ such that the Green function $G^\epsilon$ has the asymptotic representation
$$G^{\epsilon}(t,x,y) = G_{\text{free}}^{\epsilon}(t,x,y)(1 + O(t^{\omega}) + O(|x - y|)) \quad (0.6.38)$$

for small $t$ and $x-y$. In case of vanishing $A$, the multiplier $\exp(B|x-y|)$ in (0.6.36) can

be omitted, and the term $O(|x-y|)$ in (0.6.38) can be dropped. In this case, formula (0.6.38) gives global (uniform for all $x,y$) small time asymptotics for $G^{\epsilon}(t,x,y)$.

(iv) One can give rigorous meaning to formal expression (0.6.31) as a bounded below self adjoint operator in such a way that the family (0.6.37) of operators $U^D(t)$ (giving solutions to equation (0.6.34), which is formally equivalent to the evolutionary equation (0.6.33) with the formal generator $DH$) coincides with the semigroup $\exp(-tDH)$ defined by means of the functional calculus. Hence for the integral kernel of the operators $\exp(-tDH)$ the estimates (0.6.36) and (0.6.38) hold.

The rigorous construction a path integral representation for the solution $U^D(t)u_0$ of Eq.(0.6.33), by construction a measure on a path space that is supported on the set of continuous piecewise linear paths, was given in paper [49]. Denote this set by $\text{CPL}^{\mathbb{R}^d}$. Let $\text{CPL}^{x,y}(0,t)$ denote the class of paths $q : [0,t] \to \mathbb{R}^d$ from $\text{CPL}^{\mathbb{R}^d}$

joining $x$ and $y$ in time $t$, i.e. such that $q(0) = x, q(t) = y$. By $\text{CPL}_n^{x,y}(0,t)$ we denote its

subclass consisting of all paths from $\text{CPL}^{x,y}(0,t)$ that have exactly $n$ jumps of their derivative. Clearly, each $\text{CPL}_n^{x,y}(0,t)$ is parametrised by the simplex

$$\text{sim}_t^n = \{s_1, \dots, s_n | 0 < s_1 < s_2 < \dots < s_n < t\} \quad (0.6.39)$$

of the times of jumps $s_1, \dots, s_n$ of the derivatives of a path and by $n$ positions $q(s_j)$, $j = 1, \dots, n$, of this path at these points. In other words, an arbitrary path in $\text{CPL}_n^{x,y}(0,t)$ has the form

$$q_n(s) = q^{s_1, \dots, s_n} = \eta_j + (s - s_j) \frac{\eta_{j+1} - \eta_j}{s_{j+1} - s_j}, \quad (0.6.40)$$

$$s_j \in [s_j, s_{j+1}],$$

where $s_0 = 0, s_{n+1} = t, \eta_0 = x, \eta_{n+1} = y$. Therefore $\text{CPL}^{x,y}(0,t) = \bigcup_{n=0}^{\infty} \text{CPL}_n^{x,y}(0,t)$. To any $\mathbb{R}^{d+1}$-valued Borel measure $\tilde{M} = (\mu, \nu) = (\mu, \nu^1, \dots, \nu^n)$ on $\mathbb{R}^{d+1}$ there corresponds a $\sigma$-finite complex measure $\mathbf{D}^{\text{CPL}}[q(\tau)]$ on $\text{CPL}^{x,y}(0,t)$, which is
$$G^{\epsilon}(t,x,y) = G_{\text{free}}^{\epsilon}(t,x,y)(1 + O(t^{\omega}) + O(|x-y|)) \quad (0.6.38)$$

for small $t$ and $x-y$. In case of vanishing $A$, the multiplier $\exp(B|x-y|)$ in (0.6.36) can

be omitted, and the term $O(|x-y|)$ in (0.6.38) can be dropped. In this case, formula (0.6.38) gives global (uniform for all $x,y$) small time asymptotics for $G^{\epsilon}(t,x,y)$.

(iv) One can give rigorous meaning to formal expression (0.6.31) as a bounded below self adjoint operator in such a way that the family (0.6.37) of operators $U^D(t)$ (giving solutions to equation (0.6.34), which is formally equivalent to the evolutionary equation (0.6.33) with the formal generator $DH$) coincides with the semigroup $\exp(-tDH)$ defined by means of the functional calculus. Hence for the integral kernel of the operators $\exp(-tDH)$ the estimates (0.6.36) and (0.6.38) hold.

The rigorous construction a path integral representation for the solution $U^D(t)u_0$ of Eq.(0.6.33), by construction a measure on a path space that is supported on the set of continuous piecewise linear paths, was given in paper [49]. Denote this set by $\text{CPL}^{\mathbb{R}^d}$. Let $\text{CPL}^{x,y}(0,t)$ denote the class of paths $q : [0,t] \to \mathbb{R}^d$ from $\text{CPL}^{\mathbb{R}^d}$

joining $x$ and $y$ in time $t$, i.e. such that $q(0) = x, q(t) = y$. By $\text{CPL}_n^{x,y}(0,t)$ we denote its

subclass consisting of all paths from $\text{CPL}^{x,y}(0,t)$ that have exactly $n$ jumps of their derivative. Clearly, each $\text{CPL}_n^{x,y}(0,t)$ is parametrised by the simplex

$$\mathbf{sim}_t^n = \{s_1, \dots, s_n | 0 < s_1 < s_2 < \dots < s_n < t\} \quad (0.6.39)$$

of the times of jumps $s_1, \dots, s_n$ of the derivatives of a path and by $n$ positions $q(s_j)$, $j = 1, \dots, n$, of this path at these points. In other words, an arbitrary path in $\text{CPL}_n^{x,y}(0,t)$ has the form

$$q_n(s) = q^{s_1, \dots, s_n} = \eta_j + (s - s_j) \frac{\eta_{j+1} - \eta_j}{s_{j+1} - s_j}, \quad (0.6.40)$$

$$s_j \in [s_j, s_{j+1}],$$

where $s_0 = 0, s_{n+1} = t, \eta_0 = x, \eta_{n+1} = y$. Therefore $\text{CPL}^{x,y}(0,t) = \bigcup_{n=0}^{\infty} \text{CPL}_n^{x,y}(0,t)$. To any $\mathbb{R}^{d+1}$-valued Borel measure $\tilde{M} = (\mu, \nu) = (\mu, \nu^1, \dots, \nu^n)$ on $\mathbb{R}^{d+1}$ there corresponds a $\sigma$-finite complex measure $\mathbf{D}^{\text{CPL}}[q(\tau)]$ on $\text{CPL}^{x,y}(0,t)$, which is
defined as the sum of the measures $M_n^{CPL}$ $n \in \mathbb{N}$ on the finite-dimensional spaces $CPL_n^{x,y}(0,t)$ such that $M_0^{CPL}$ is

just the unit measure on the one-point set $CPL_0^{x,y}(0,t)$ and each $M_n^{CPL}, n > 0$, is defined

in the following way: if $\Phi(\cdot)$ is a functional on $CPL^{x,y}(0,t)$, then:

$$ \int_{CPL_n^{x,y}(0,t)} \Phi(q(\tau))D_n^{CPL}[q(\tau)] = \int_{sim_t^n} ds_1 \dots ds_n \times \\ \int_{\mathbb{R}^d} \dots \int_{\mathbb{R}^d} \left( \mu + 2i \frac{(\eta_2 - \eta_1, \nu)}{s_2 - s_1} \right) (d\eta_1) \dots \left( \mu + 2i \frac{(y - \eta_n, \nu)}{t - s_n} \right) (d\eta_n) \Phi(q(\tau)). \quad (0.6.41) $$

Now, we let $\tilde{M} = (W, D^{-1}A(x)dx)$.

**Theorem 0.6.10.** For any $D$ with $\epsilon = \text{Re}D > 0$, the Green function $G^D(t,x,y)$ of equation

(0.6.33) has the following path integral representation:

$$ G^D(t,x,y) = \int_{CPL^{x,y}(0,t)} \Phi_D(q(\tau)) \exp \left\{ \frac{1}{2D} \int_0^t \dot{q}^2(s) ds \right\} D^{CPL}[q(\tau)] \quad (0.6.42) $$

with $q(s)$ given by Eq.(0.6.40) and

$$ \Phi_D(q(\tau)) = D^n \prod_{j=1}^{n+1} (2\pi D(s_j - s_{j-1}))^{-d/2}. \quad (0.6.43) $$

For any $u_0 \in \mathcal{L}_2(\mathbb{R}^d)$ the solution $u(t,x)$ of the Cauchy problem (0.6.33) with $D = i$ has the form
defined as the sum of the measures $M_n^{CPL}$ $n \in \mathbb{N}$ on the finite-dimensional spaces $CPL_n^{x,y}(0,t)$ such that $M_0^{CPL}$ is

just the unit measure on the one-point set $CPL_0^{x,y}(0,t)$ and each $M_n^{CPL}, n > 0$, is defined

in the following way: if $\Phi(\cdot)$ is a functional on $CPL^{x,y}(0,t)$, then:

$$ \int_{CPL_n^{x,y}(0,t)} \Phi(q(\tau))D_n^{CPL}[q(\tau)] = \int_{\text{sim}_t^n} ds_1 \dots ds_n \times \qquad (0.6.41) $$

$$ \int_{\mathbb{R}^d} \dots \int_{\mathbb{R}^d} \left( \mu + 2i \frac{(\eta_2 - \eta_1, \nu)}{s_2 - s_1} \right) (d\eta_1) \dots \left( \mu + 2i \frac{(y - \eta_n, \nu)}{t - s_n} \right) (d\eta_n) \Phi(q(\tau)). $$

Now, we let $\tilde{M} = (W, D^{-1}A(x)dx)$.

**Theorem 0.6.10.** For any $D$ with $\epsilon = \text{Re}D > 0$, the Green function $G^D(t,x,y)$ of equation

(0.6.33) has the following path integral representation:

$$ G^D(t,x,y) = \int_{CPL^{x,y}(0,t)} \Phi_D(q(\tau)) \exp \left\{ \frac{1}{2D} \int_0^t \dot{q}^2(s) ds \right\} D^{CPL}[q(\tau)] \qquad (0.6.42) $$

with $q(s)$ given by Eq.(0.6.40) and

$$ \Phi_D(q(\tau)) = D^n \prod_{j=1}^{n+1} (2\pi D(s_j - s_{j-1}))^{-d/2}. \qquad (0.6.43) $$

For any $u_0 \in \mathcal{L}_2(\mathbb{R}^d)$ the solution $u(t,x)$ of the Cauchy problem (0.6.33) with $D = i$ has the form
$$u(t,x) = \lim_{\delta \to 0} \int_{CPL^{x,y}(0,t)} \int_{\mathbb{R}^d} u_0(x) \Phi_{i+\delta}(q(\tau)) \exp \left\{ \frac{1}{2(i+\delta)} \int_0^t \dot{q}^2(s) ds \right\} \mathbf{D}^{CPL}[q(\tau)]. \quad (0.6.44)$$

Here the limit is understood in $L_2(\mathbb{R}^d)$ sense.

**Theorem 0.6.11.** Suppose the vector potential A vanishes and V satisfies assumption (C3). Suppose additionally that $V$ has no atom at the origin and is a finite positive measure so that $\mu_V = V(\mathbb{R}^d) > 0$. Let the paths of CPL are parametrised by Eq. (0.6.40). Let $E_t$ denote the expectation with respect to the process of jumps $\eta_j$ which are identically independently distributed according to the probability measure $V/\mu_V$, and which occur at times $s_j$ from $[0, t]$ distributed according to Poisson process of intensity $\mu_V$. Then the integral (0.6.42) can be written in the form

$$G^D(t,x,y) = e^{\mu_V} \mathbf{E}_t \left[ \Phi_D(q(\tau)) \exp \left\{ -\frac{1}{2D} \int_0^t \dot{q}^2(s) ds \right\} \mathbf{D}^{CPL}[q(\tau)] \right]. \quad (0.6.45)$$

**Definition 0.6.19.** (Colombeau-Hardy spaces for the unit disk). Let $H^p(\mathbf{D})$ be the Hardy space for $0 < p < \infty$. Set $E(H^p(\mathbf{D})) = (H^p(\mathbf{D}))^I, I = (0,1]$,

$$E_M(H^p(\mathbf{D})) =$$

$$\left\{ (f_{\varepsilon})_{\varepsilon} \in E(H^p(\mathbf{D})) \middle| (\forall \alpha \in \mathbb{N})(\exists p \in \mathbb{N}) \left[ \sup_{z \in \mathbf{D}} \left| \frac{\partial^\alpha f_\varepsilon(z)}{\partial z^\alpha} \right| = O(\varepsilon^{-p}) \right] \right\},$$

$$N(H^p(\mathbf{D})) = \qquad (0.6.46)$$

$$\left\{ (f_{\varepsilon})_{\varepsilon} \in E(H^p(\mathbf{D})) \middle| (\forall \alpha \in \mathbb{N})(\forall q \in \mathbb{N}) \left[ \sup_{z \in \mathbf{D}} \left| \frac{\partial^\alpha f_\varepsilon(z)}{\partial z^\alpha} \right| = O(\varepsilon^q) \right] \right\},$$

$$G(H^p(\mathbf{D})) = E_M(H^p(\mathbf{D}))/N(H^p(\mathbf{D})).$$
$$
u(t,x) = \lim_{\delta \to 0} \int_{CPL^{x,y}(0,t)} \int_{\mathbb{R}^d} u_0(x) \Phi_{i+\delta}(q(\tau)) \exp \left\{ \frac{1}{2(i+\delta)} \int_0^t \dot{q}^2(s) ds \right\} \mathbf{D}^{\text{CPL}}[q(\tau)]. \quad (0.6.44)
$$

Here the limit is understood in $L_2(\mathbb{R}^d)$ sence.

**Theorem 0.6.11.** Suppose the vector potential A vanishes and V satisfies assumption (C3). Suppose additionally that $V$ has no atom at the origin and is a finite positive measure so that $\mu_V = V(\mathbb{R}^d) > 0$. Let the paths of CPL are parametrised by Eq.(0.6.40). Let $E_t$ denote the expectation with respect to the process of jumps $\eta_j$ which are identically independently distributed according to the probability measure $V/\mu_V$, and which occur at times $s_j$ from $[0,t]$ distributed according to Poisson process of intensity $\mu_V$. Then the integral (0.6.42) can be written in the form

$$
G^D(t,x,y) = e^{\mu V} \mathbf{E}_t \left[ \Phi_D(q(\tau)) \exp \left\{ -\frac{1}{2D} \int_0^t \dot{q}^2(s) ds \right\} \mathbf{D}^{\text{CPL}}[q(\tau)] \right]. \quad (0.6.45)
$$

**Definition 0.6.19. (Colombeau-Hardy spaces for the unit disk).** Let $H^p(\mathbf{D})$ be the Hardy space for $0 < p < \infty$. Set $E(H^p(\mathbf{D})) = (H^p(\mathbf{D}))^I, I = (0,1]$,

$$
\begin{align}
E_M(H^p(\mathbf{D})) &= \nonumber \\
&\quad \left\{ (f_\epsilon)_\epsilon \in E(H^p(\mathbf{D})) \middle| (\forall \alpha \in \mathbb{N})(\exists p \in \mathbb{N}) \left[ \sup_{z \in \mathbf{D}} \left| \frac{\partial^\alpha f_\epsilon(z)}{\partial z^\alpha} \right| = O(\epsilon^{-p}) \right] \right\}, \nonumber \\
N(H^p(\mathbf{D})) &= \tag{0.6.46}
\end{align}
$$

$$
\left\{ (f_{\epsilon})_{\epsilon} \in E(H^p(\mathbf{D})) \middle| (\forall \alpha \in \mathbb{N})(\forall q \in \mathbb{N}) \left[ \sup_{z \in \mathbf{D}} \left| \frac{\partial^\alpha f_\epsilon(z)}{\partial z^\alpha} \right| = O(\epsilon^q) \right] \right\},
$$

$$
G(H^p(\mathbf{D})) = E_M(H^p(\mathbf{D}))/N(H^p(\mathbf{D})).
$$
**Definition 0.6.20. (Colombeau-Hardy spaces for the upper half plane).** Let $H^p(\mathbf{H})$ be the Hardy space for $0 < p < \infty$. Set $E(H^p(\mathbf{H})) = (H^p(\mathbf{H}))^l, I = (0,1]$,

$$ E_M(H^p(\mathbf{H})) = \\ \left\{ (f_\varepsilon)_\varepsilon \in E(H^p(\mathbf{H})) \middle| (\forall \alpha \in \mathbb{N})(\exists p \in \mathbb{N}) \left[ \sup_{z \in \mathbf{D}} \left| \frac{\partial^\alpha f_\varepsilon(z)}{\partial z^\alpha} \right| = O(\varepsilon^{-p}) \right] \right\}, \\ \\ N(H^p(\mathbf{H})) = \tag{0.6.47} \\ \left\{ (f_\varepsilon)_\varepsilon \in E(H^p(\mathbf{H})) \middle| (\forall \alpha \in \mathbb{N})(\forall q \in \mathbb{N}) \left[ \sup_{z \in \mathbf{D}} \left| \frac{\partial^\alpha f_\varepsilon(z)}{\partial z^\alpha} \right| = O(\varepsilon^q) \right] \right\}, \\ \\ G(H^p(\mathbf{H})) = E_M(H^p(\mathbf{H}))/N(H^p(\mathbf{H})). $$

**Definition 0.6.21.** A sequence of points $\{(a_{n,\varepsilon})_\varepsilon\}_{n=1}^\infty$ inside the unit disk $\tilde{\mathbf{D}} \subset \tilde{\mathbb{C}}$ is said to satisfy the Blaschke condition when $(\sum_{n=1}^\infty (1 - |a_{n,\varepsilon}|)\varepsilon)^{-1} < \infty$. Given a sequence obeying the Blaschke condition, the Blaschke generalized product $(B_\varepsilon(z))_\varepsilon$ is defined as

$$ B_{\varepsilon}(z) = \prod_{n} (B_{n,\varepsilon}(z))_{\varepsilon}, \\ \\ B_{n,\varepsilon}(z) = - \frac{|a_{n,\varepsilon}|}{a_{n,\varepsilon}} \frac{z - a_{n,\varepsilon}}{1 - \bar{a}_{n,\varepsilon} \cdot z}. \tag{0.6.48} $$

The Blaschke generalized product $(B_\varepsilon(z))_\varepsilon$ is analytic in the open unit disc $\tilde{\mathbf{D}}$, and is zero at the $(a_{n,\varepsilon})_\varepsilon$ only.

**Definition 0.6.22. (i)** One says that $(G_\varepsilon(z))_\varepsilon$ is an **outer (exterior) generalized function** if it takes the form
**Definition 0.6.20. (Colombeau-Hardy spaces for the upper half plane).** Let $H^p(\mathbf{H})$ be the Hardy space for $0 < p < \infty$. Set $E(H^p(\mathbf{H})) = (H^p(\mathbf{H}))^l, I = (0, 1]$,

$$E_M(H^p(\mathbf{H})) =$$

$$\left\{ (f_{\varepsilon})_{\varepsilon} \in E(H^p(\mathbf{H})) \middle| (\forall \alpha \in \mathbb{N})(\exists p \in \mathbb{N}) \left[ \sup_{z \in \mathbf{D}} \left| \frac{\partial^\alpha f_\varepsilon(z)}{\partial z^\alpha} \right| = O(\varepsilon^{-p}) \right] \right\},$$

$$N(H^p(\mathbf{H})) = \tag{0.6.47}$$

$$\left\{ (f_{\varepsilon})_{\varepsilon} \in E(H^p(\mathbf{H})) \middle| (\forall \alpha \in \mathbb{N})(\forall q \in \mathbb{N}) \left[ \sup_{z \in \mathbf{D}} \left| \frac{\partial^\alpha f_\varepsilon(z)}{\partial z^\alpha} \right| = O(\varepsilon^q) \right] \right\},$$

$$G(H^p(\mathbf{H})) = E_M(H^p(\mathbf{H}))/N(H^p(\mathbf{H})).$$

**Definition 0.6.21.** A sequence of points $\{(a_{n,\varepsilon})_\varepsilon\}_{\varepsilon=1}^\infty$ inside the unit disk $\tilde{\mathbf{D}} \subset \tilde{\mathbb{C}}$ is said to satisfy the Blaschke condition when $(\sum_{n=1}^\infty (1 - |a_{n,\varepsilon}|))_\varepsilon < \infty$. Given a sequence obeying the Blaschke condition, the Blaschke generalized product $(B_\varepsilon(z))_\varepsilon$ is defined as

$$B_\varepsilon(z) = \prod_n (B_{n,\varepsilon}(z))_\varepsilon,$$

(0.6.48)

$$B_{n,\varepsilon}(z) = - \frac{|a_{n,\varepsilon}|}{a_{n,\varepsilon}} \frac{z - a_{n,\varepsilon}}{1 - \bar{a}_{n,\varepsilon} \cdot z}.$$

The Blaschke generalized product $(B_\varepsilon(z))_\varepsilon$ is analytic in the open unit disc $\tilde{\mathbf{D}}$, and is zero at the $(a_{n,\varepsilon})_\varepsilon$ only.

**Definition 0.6.22. (i)** One says that $(G_\varepsilon(z))_\varepsilon$ is an **outer (exterior) generalized function** if it takes the form
$$ (G_{\varepsilon}(z))_{\varepsilon} = ((c_{\varepsilon})_{\varepsilon}) \left( \exp \left[ \frac{1}{2\pi} \int_{-\pi}^{\pi} \frac{e^{i\theta} + z}{e^{i\theta} - z} \log \varphi_{\varepsilon}(e^{i\theta}) d\theta \right] \right)_{\varepsilon} \quad (0.6.49) $$

**Definition 0.6.23.** Let $\mathcal{F}^{\triangle}(\mathbb{R}^d) = G(\mathcal{F}(\mathbb{R}^d))$ denote the Colombeau algebra of Colombeau-Fourier transforms of elements of $\mathbf{M}^{\triangle}(\mathbb{R}^d) = G(\mathbf{M}(\mathbb{R}^d))$, i.e. the algebra of Colombeau generalized functions on $\mathbb{R}^d$ of the form

$$ (V_{\mu,\varepsilon}(x))_{\varepsilon} = \left( \int_{\mathbb{R}^d} \exp(ip \cdot x) \mu_{\varepsilon}(dp) \right)_{\varepsilon} \quad (0.6.50) $$

for some $(\mu_{\varepsilon})_{\varepsilon} \in \mathbf{M}^{\triangle}(\mathbb{R}^d)$, with the induced $\tilde{\mathbb{R}}$-valued norm $\|(V_{\mu,\varepsilon})_{\varepsilon}\| = (\|\mu_{\varepsilon}\|)_{\varepsilon}$. Since $\mathbf{M}^{\triangle}(\mathbb{R}^d)$ is a Colombeau-Banach algebra with convolution as the multiplication, it follows that $\mathcal{F}^{\triangle}(\mathbb{R}^d)$ is also a Colombeau-Banach algebra with respect to the pointwise multiplication. We say that an element $(f_{\varepsilon})_{\varepsilon} \in \mathcal{F}^{\triangle}(\mathbb{R}^d)$ is infinitely divisible if there exists a family $(f_{t,\varepsilon})_{\varepsilon}$, $t \ge 0$ of elements of $\mathcal{F}^{\triangle}(\mathbb{R}^d)$ such that $(f_{0,\varepsilon})_{\varepsilon} = 1, (f_{1,\varepsilon})_{\varepsilon} = (f_{\varepsilon})_{\varepsilon}$, and $(f_{t+s,\varepsilon})_{\varepsilon} = (f_{t,\varepsilon} \cdot f_{s,\varepsilon})_{\varepsilon}$ for all positive $s, t$. Clearly if $(f_{\varepsilon})_{\varepsilon}$ is infinitely divisible, then it has no zeros and a continuous generalized function $(g_{\varepsilon})_{\varepsilon} = (\ln f_{\varepsilon})_{\varepsilon}$ is well defined (and is unique up to an imaginary shift). Moreover, the family $(f_{t,\varepsilon})_{\varepsilon}$ has the form $(f_{t,\varepsilon})_{\varepsilon} = (\exp(tg_{\varepsilon}))_{\varepsilon}$.

**Definition 0.6.24.** We say that a Colombeau generalized function $(g_{\varepsilon})_{\varepsilon} \in G_0(\mathbb{R}^d)$ is a Colombeau complex characteristic exponent (abbreviated CCCE), if $\exp(g_{\varepsilon})_{\varepsilon}$ is an infinitely divisible element of $\mathcal{F}^{\triangle}(\mathbb{R}^d)$, or equivalently, if $\forall t, t \ge 0 : (\exp(tg_{\varepsilon}))_{\varepsilon} \in \mathcal{F}^{\triangle}(\mathbb{R}^d)$.

**Remark 0.6.8.** It follows from the definitions that the set of spatially homogeneous CCTF's $(\mu_{t,\varepsilon}(dx))_{\varepsilon}$ is in one-to-one correspondence with CCCE $(g_{\varepsilon})_{\varepsilon}$, in such a way that for any positive $t$ the generalized function $(\exp(tg_{\varepsilon}))_{\varepsilon}$ is the Colombeau-Fourier transform of the transition Colombeau measure $(\mu_{t,\varepsilon}(dx))_{\varepsilon}$.

**Remark 0.6.9.** It is easy to show that if $(f_{1,\varepsilon})_{\varepsilon} \in \mathcal{F}^{\triangle}(\mathbb{R})$ is infinite divisible and such that the measures corresponding to all functions $(f_{t,\varepsilon})_{\varepsilon}, t > 0$, are concentrated on the half line $\mathbb{R}_+$ (complex generalisation of subordinators) and have densities from Colombeau module $G_{\mathcal{L}_2(\mathbb{R}_+)}$, then $(f_{1,\varepsilon})_{\varepsilon}$ belongs to the Colombeau algebra $G(H^2)$ of Colombeau generalized analytic functions on the upper half plane, which have no Colombeau- Blaschke product in its canonical decomposition.

**Theorem 0.6.12.** If $(V_{\varepsilon})_{\varepsilon}$ is a CCCE, then the Colombeau solution to the
$$ (G_{\varepsilon}(z))_{\varepsilon} = ((c_{\varepsilon})_{\varepsilon}) \left( \exp \left[ \frac{1}{2\pi} \int_{-\pi}^{\pi} \frac{e^{i\theta} + z}{e^{i\theta} - z} \log \varphi_{\varepsilon}(e^{i\theta}) d\theta \right] \right)_{\varepsilon} \quad (0.6.49) $$

**Definition 0.6.23.** Let $\mathcal{F}^{\triangle}(\mathbb{R}^d) = G(\mathcal{F}(\mathbb{R}^d))$ denote the Colombeau algebra of Colombeau-Fourier transforms of elements of $\mathbf{M}^{\triangle}(\mathbb{R}^d) = G(\mathbf{M}(\mathbb{R}^d))$, i.e. the algebra of Colombeau generalized functions on $\mathbb{R}^d$ of the form

$$ (V_{\mu,\varepsilon}(x))_{\varepsilon} = \left( \int_{\mathbb{R}^d} \exp(ip \cdot x) \mu_{\varepsilon}(dp) \right)_{\varepsilon} \quad (0.6.50) $$

for some $(\mu_{\varepsilon})_{\varepsilon} \in \mathbf{M}^{\triangle}(\mathbb{R}^d)$, with the induced $\tilde{\mathbb{R}}$-valued norm $\|(V_{\mu,\varepsilon})_{\varepsilon}\| = (\|\mu_{\varepsilon}\|)_{\varepsilon}$. Since $\mathbf{M}^{\triangle}(\mathbb{R}^d)$ is a Colombeau-Banach algebra with convolution as the multiplication, it follows that $\mathcal{F}^{\triangle}(\mathbb{R}^d)$ is also a Colombeau-Banach algebra with respect to the pointwise multiplication. We say that an element $(f_{\varepsilon})_{\varepsilon} \in \mathcal{F}^{\triangle}(\mathbb{R}^d)$ is infinitely divisible if there exists a family $(f_{t,\varepsilon})_{\varepsilon}$, $t \ge 0$ of elements of $\mathcal{F}^{\triangle}(\mathbb{R}^d)$ such that $(f_{0,\varepsilon})_{\varepsilon} = 1, (f_{1,\varepsilon})_{\varepsilon} = (f_{\varepsilon})_{\varepsilon}$, and $(f_{t+s,\varepsilon})_{\varepsilon} = (f_{t,\varepsilon} \cdot f_{s,\varepsilon})_{\varepsilon}$ for all positive $s, t$. Clearly if $(f_{\varepsilon})_{\varepsilon}$ is infinitely divisible, then it has no zeros and a continuous generalized function $(g_{\varepsilon})_{\varepsilon} = (\ln f_{\varepsilon})_{\varepsilon}$ is well defined (and is unique up to an imaginary shift). Moreover, the family $(f_{t,\varepsilon})_{\varepsilon}$ has the form $(f_{t,\varepsilon})_{\varepsilon} = (\exp(tg_{\varepsilon}))_{\varepsilon}$.

**Definition 0.6.24.** We say that a Colombeau generalized function $(g_{\varepsilon})_{\varepsilon} \in G_0(\mathbb{R}^d)$ is a Colombeau complex characteristic exponent (abbreviated CCCE), if $\exp(g_{\varepsilon})_{\varepsilon}$ is an infinitely divisible element of $\mathcal{F}^{\triangle}(\mathbb{R}^d)$, or equivalently, if $\forall t, t \ge 0 : (\exp(tg_{\varepsilon}))_{\varepsilon} \in \mathcal{F}^{\triangle}(\mathbb{R}^d)$.

**Remark 0.6.8.** It follows from the definitions that the set of spatially homogeneous CCTF's $(\mu_{t,\varepsilon}(dx))_{\varepsilon}$ is in one-to-one correspondence with CCCE $(g_{\varepsilon})_{\varepsilon}$, in such a way that for any positive $t$ the generalized function $(\exp(tg_{\varepsilon}))_{\varepsilon}$ is the Colombeau-Fourier transform of the transition Colombeau measure $(\mu_{t,\varepsilon}(dx))_{\varepsilon}$.

**Remark 0.6.9.** It is easy to show that if $(f_{1,\varepsilon})_{\varepsilon} \in \mathcal{F}^{\triangle}(\mathbb{R})$ is infinite divisible and such that the measures corresponding to all functions $(f_{t,\varepsilon})_{\varepsilon}, t > 0$, are concentrated on the half line $\mathbb{R}_+$ (complex generalisation of subordinators) and have densities from Colombeau module $G_{\mathcal{L}_2(\mathbb{R}_+)}$, then $(f_{1,\varepsilon})_{\varepsilon}$ belongs to the Colombeau algebra $G(H^2)$ of Colombeau generalized analytic functions on the upper half plane, which have no Colombeau- Blaschke product in its canonical decomposition.

**Theorem 0.6.12.** If $(V_{\varepsilon})_{\varepsilon}$ is a CCCE, then the Colombeau solution to the
$$ \left( \frac{\partial u_{\varepsilon}}{\partial t} \right)_{\varepsilon} = \left( V_{\varepsilon} \left( \frac{1}{i} \frac{\partial}{\partial y} \right) u_{\varepsilon} \right)_{\varepsilon}, \qquad (0.6.51) $$

$$ (u_{\varepsilon}(0, y))_{\varepsilon} = (u_{0, \varepsilon}(y))_{\varepsilon} $$

defines a strongly continuous and spatially homogeneous Colombeau semigroup $(T_{t,\varepsilon})_{\varepsilon}$ of bounded linear operators in $C_0(\mathbb{R}^d)$ (i.e. $((T_{t,\varepsilon}u_{0,\varepsilon})(y))_{\varepsilon}$ is the Colombeau solution to equation (0.6.51) with the initial Colombeau generalized function $(u_{0,\varepsilon})_{\varepsilon}$). Conversely, each such semigroup is the solution to the Cauchy problem of an equation of type (0.6.51) with some **CCCE** $(g_{\varepsilon})_{\varepsilon}$.

**Definition 0.6.25.** We say that a **CCCE** $(V_{\varepsilon})_{\varepsilon}$ is regular, if equation (0.6.51) defines a regular Colombeau semigroup.

**Theorem 0.6.13.** Let $(V_{\varepsilon})_{\varepsilon} \in \mathcal{F}^{\Delta}(\mathbb{R}^d)$, i.e. it is given by (0.6.50) with $(\mu_{\varepsilon}^{\mathcal{C}})_{\varepsilon} \in \mathbf{M}^{\Delta}(\mathbb{R}^d)$. Then $(V_{\varepsilon})_{\varepsilon}$ is a regular **CCCE**. Moreover, if the positive Colombeau measure $(M_{\varepsilon}(dy))_{\varepsilon}$ in the representation (0.6.8) for $(\mu_{\varepsilon}^{\mathcal{C}})_{\varepsilon}$ has no atom at the origin, i.e. $(M_{\varepsilon}(\{0\}))_{\varepsilon} = 0$, then the corresponding Colombeau measure $(\mathbf{D}_{0,t,\varepsilon}^{x}[q(\cdot)])_{\varepsilon}$ on the path space is concentrated on the set of piecewise-constant paths in $\mathbb{R}^d_{[s,t]}$ with a finite number of jumps. In other words, $(\mathbf{D}_{0,t,\varepsilon}^{x}[q(\cdot)])_{\varepsilon}$ is the measure of a Colombeau generalizwd jump-process.

Let us consider now the pseudo-differential equation of the Colombeau-Schrödinger type

$$ \left( \frac{\partial \tilde{u}_{\varepsilon}}{\partial t} \right)_{\varepsilon} = -G[(-\Delta)^{a}](\tilde{u}_{\varepsilon})_{\varepsilon} + \left( \left( A(x)_{\varepsilon}, \frac{\partial}{\partial x} \right) \tilde{u}_{\varepsilon} \right)_{\varepsilon} + (V_{\varepsilon}(x)\tilde{u}_{\varepsilon})_{\varepsilon}, \qquad (0.6.52) $$

where $G$ is a complex constant with a non-negative real part, $\alpha$ is any positive constant,

$A$ is a real-valued vector (if $\mathrm{Re} G > 0$, then $(A_{\varepsilon}(x))_{\varepsilon}$ can be also $\tilde{\mathbb{C}}$-valued), and $(V_{\varepsilon}(x))_{\varepsilon}$ is a $\tilde{\mathbb{C}}$-valued function of form (0.6.50). The equation on the inverse Colombeau-Fourier transform

$$ (u_{\varepsilon}(y))_{\varepsilon} = \int_{\mathbb{R}^d} dx \exp(-iyx) (\tilde{u}_{\varepsilon}(x))_{\varepsilon} \qquad (0.6.53) $$
$$ \left( \frac{\partial u_{\varepsilon}}{\partial t} \right)_{\varepsilon} = \left( V_{\varepsilon} \left( \frac{1}{i} \frac{\partial}{\partial y} \right) u_{\varepsilon} \right)_{\varepsilon}, \qquad (0.6.51) $$

$$ (u_{\varepsilon}(0, y))_{\varepsilon} = (u_{0, \varepsilon}(y))_{\varepsilon} $$

defines a strongly continuous and spatially homogeneous Colombeau semigroup $(T_{t,\varepsilon})_{\varepsilon}$ of bounded linear operators in $C_0(\mathbb{R}^d)$ (i.e. $((T_{t,\varepsilon}u_{0,\varepsilon})(y))_{\varepsilon}$ is the Colombeau solution to equation (0.6.51) with the initial Colombeau generalized function $(u_{0,\varepsilon})_{\varepsilon}$). Conversely, each such semigroup is the solution to the Cauchy problem of an equation of type (0.6.51) with some **CCCE** $(g_{\varepsilon})_{\varepsilon}$.

**Definition 0.6.25.** We say that a **CCCE** $(V_{\varepsilon})_{\varepsilon}$ is regular, if equation (0.6.51) defines a regular Colombeau semigroup.

**Theorem 0.6.13.** Let $(V_{\varepsilon})_{\varepsilon} \in \mathcal{F}^{\Delta}(\mathbb{R}^d)$, i.e. it is given by (0.6.50) with $(\mu_{\varepsilon}^{\mathcal{C}})_{\varepsilon} \in \mathbf{M}^{\Delta}(\mathbb{R}^d)$. Then $(V_{\varepsilon})_{\varepsilon}$ is a regular **CCCE**. Moreover, if the positive Colombeau measure $(M_{\varepsilon}(dy))_{\varepsilon}$ in the representation (0.6.8) for $(\mu_{\varepsilon}^{\mathcal{C}})_{\varepsilon}$ has no atom at the origin, i.e. $(M_{\varepsilon}(\{0\}))_{\varepsilon} = 0$, then the corresponding Colombeau measure $(\mathbf{D}_{0,t,\varepsilon}^{x}[q(\cdot)])_{\varepsilon}$ on the path space is concentrated on the set of piecewise-constant paths in $\mathbb{R}^d_{[s,t]}$ with a finite number of jumps. In other words, $(\mathbf{D}_{0,t,\varepsilon}^{x}[q(\cdot)])_{\varepsilon}$ is the measure of a Colombeau generalizwd jump-process.

Let us consider now the pseudo-differential equation of the Colombeau-Schrödinger type

$$ \left( \frac{\partial \tilde{u}_{\varepsilon}}{\partial t} \right)_{\varepsilon} = -G[(-\Delta)^{a}](\tilde{u}_{\varepsilon})_{\varepsilon} + \left( \left( A(x)_{\varepsilon}, \frac{\partial}{\partial x} \right) \tilde{u}_{\varepsilon} \right)_{\varepsilon} + (V_{\varepsilon}(x)\tilde{u}_{\varepsilon})_{\varepsilon}, \qquad (0.6.52) $$

where $G$ is a complex constant with a non-negative real part, $\alpha$ is any positive constant,

$A$ is a real-valued vector (if $\mathrm{Re} G > 0$, then $(A_{\varepsilon}(x))_{\varepsilon}$ can be also $\tilde{\mathbb{C}}$-valued), and $(V_{\varepsilon}(x))_{\varepsilon}$ is a $\tilde{\mathbb{C}}$-valued function of form (0.6.50). The equation on the inverse Colombeau-Fourier transform

$$ (u_{\varepsilon}(y))_{\varepsilon} = \int_{\mathbb{R}^d} dx \exp(-iyx) (\tilde{u}_{\varepsilon}(x))_{\varepsilon} \qquad (0.6.53) $$
of $(\tilde{u}_{\varepsilon}(x))_{\varepsilon}$ has the form

$$ \left( \frac{\partial u_{\varepsilon}}{\partial t} \right)_{\varepsilon} = -G(y^2)^{\alpha} + i \left( (A_{\varepsilon}, y) u_{\varepsilon} \right)_{\varepsilon} + \left( V_{\varepsilon} \left( \frac{1}{i} \frac{\partial}{\partial y} \right) u_{\varepsilon} \right)_{\varepsilon}. \quad (0.6.54) $$

**Theorem 0.6.15.** The solution to the Colombeau-Cauchy problem of equation (0.6.54) can be written in the form of a complex Colombeau-Feynman-Kac formula

$$ (u_{\varepsilon}(t, u))_{\varepsilon} = \left( \int \exp \left\{ -\int_{0}^{t} [G(q^2(\tau))^{\alpha} - (A_{\varepsilon}, q(\tau))] \right\} u_{0,\varepsilon}(q(t)) \mathbf{D}_{0,t,\varepsilon}^{x}[q(\tau)] \right)_{\varepsilon}, \quad (0.6.55) $$

where $(\mathbf{D}_{0,t,\varepsilon}^{x}[q(\tau)])_{\varepsilon}$ is the measure of the Colombeau generalized jump process corresponding to equation (0.6.51).

As another example, let us consider the case of generalized complex anharmonic oscillator, i.e. the equation

$$ \left( \frac{\partial \tilde{\psi}_{\varepsilon}}{\partial t} \right)_{\varepsilon} = \frac{1}{2} \left( (G\Delta - x^2 - iV_{\varepsilon}(x)) \tilde{\psi}_{\varepsilon} \right)_{\varepsilon}, \quad (0.6.56) $$

where $(V_{\varepsilon})_{\varepsilon} = (V_{\mu,\varepsilon})_{\varepsilon} \in \mathcal{F}^{\Delta}(\mathbb{R}^d)$. The Colombeau-Fourier transform of this equation has the form

$$ \left( \frac{\partial \psi_{\varepsilon}(t, p)}{\partial t} \right) = \frac{1}{2} \left( \left( \Delta - Gp^2 - iV_{\varepsilon} \left( \frac{1}{i} \frac{\partial}{\partial p} \right) \right) \psi_{\varepsilon}(t, p) \right)_{\varepsilon}. \quad (0.6.57) $$

**Theorem 0.6.16.** If $\mathrm{Re}\,G \ge 0$, the Cauchy problem of equation (0.6.57) defines a Colombeau regular semigroup of operators in $C_0(\mathbb{R}^d)$, and thus can be presented as the Colombeau path integral from.

**Remark 0.6.10.** The statement of the Proposition can be generalised easily to the following situation, which includes all Schrödinger equations, namely to the case of the equation
of $(\tilde{u}_{\varepsilon}(x))_{\varepsilon}$ has the form

$$ \left( \frac{\partial u_{\varepsilon}}{\partial t} \right)_{\varepsilon} = -G(y^2)^{\alpha} + i \left( (A_{\varepsilon}, y) u_{\varepsilon} \right)_{\varepsilon} + \left( V_{\varepsilon} \left( \frac{1}{i} \frac{\partial}{\partial y} \right) u_{\varepsilon} \right)_{\varepsilon}. \quad (0.6.54) $$

**Theorem 0.6.15.** The solution to the Colombeau-Cauchy problem of equation (0.6.54) can be written in the form of a complex Colombeau-Feynman-Kac formula

$$ (u_{\varepsilon}(t, u))_{\varepsilon} = \left( \int \exp \left\{ -\int_{0}^{t} [G(q^2(\tau))^{\alpha} - (A_{\varepsilon}, q(\tau))] \right\} u_{0,\varepsilon}(q(t)) \mathbf{D}_{0,t,\varepsilon}^{x}[q(\tau)] \right)_{\varepsilon}, \quad (0.6.55) $$

where $(\mathbf{D}_{0,t,\varepsilon}^{x}[q(\tau)])_{\varepsilon}$ is the measure of the Colombeau generalized jump process corresponding to equation (0.6.51).

As another example, let us consider the case of generalized complex anharmonic oscillator, i.e. the equation

$$ \left( \frac{\partial \tilde{\psi}_{\varepsilon}}{\partial t} \right)_{\varepsilon} = \frac{1}{2} \left( (G\Delta - x^2 - iV_{\varepsilon}(x)) \tilde{\psi}_{\varepsilon} \right)_{\varepsilon}, \quad (0.6.56) $$

where $(V_{\varepsilon})_{\varepsilon} = (V_{\mu,\varepsilon})_{\varepsilon} \in \mathcal{F}^{\Delta}(\mathbb{R}^d)$. The Colombeau-Fourier transform of this equation has the form

$$ \left( \frac{\partial \psi_{\varepsilon}(t, p)}{\partial t} \right) = \frac{1}{2} \left( \left( \Delta - Gp^2 - iV_{\varepsilon} \left( \frac{1}{i} \frac{\partial}{\partial p} \right) \right) \psi_{\varepsilon}(t, p) \right)_{\varepsilon}. \quad (0.6.57) $$

**Theorem 0.6.16.** If $\mathrm{Re}\,G \ge 0$, the Cauchy problem of equation (0.6.57) defines a Colombeau regular semigroup of operators in $C_0(\mathbb{R}^d)$, and thus can be presented as the Colombeau path integral from.

**Remark 0.6.10.** The statement of the Proposition can be generalised easily to the following situation, which includes all Schrödinger equations, namely to the case of the equation
$$ \left( \frac{\partial \psi_{\varepsilon}}{\partial t} \right)_{\varepsilon} = i \left( (A_{\varepsilon} - B_{\varepsilon}) \psi_{\varepsilon} \right)_{\varepsilon}. \qquad (0.6.58) $$

where $(A_\varepsilon)_\varepsilon$ is Colombeau selfadjoint operator (i.e. $\forall \varepsilon \in (0,1]$, $A_\varepsilon$ is selfadjoint operator), for which therefore exists (according to spectral theory) a unitary transformations $U_\varepsilon$ such that $\forall \varepsilon \in (0,1]$, $U_\varepsilon A_\varepsilon U_\varepsilon^{-1}$ is the multiplication operator in some

$\mathcal{L}_2(X, d\mu)$, where $X$ is locally compact, and $(B_\varepsilon)_\varepsilon$ is such that $(U_\varepsilon B_\varepsilon U_\varepsilon^{-1})_\varepsilon$ is a bounded operator in $C_0(X)$.

We shall consider now Colombeau-Schrödinger type operator with a magnetic field, namely the operator

$$ (H_{\varepsilon})_{\varepsilon} = \left( \left( i \frac{\partial}{\partial x} + A_{\varepsilon}(x) \right)^2 + V_{\varepsilon}(x) \right)_{\varepsilon} \qquad (0.6.59) $$

where $x = (x_1, \dots, x_d)$ and $\frac{\partial}{\partial x} = \left( \frac{\partial}{\partial x_1}, \dots, \frac{\partial}{\partial x_d} \right)$ is the gradient operator in $\mathbb{R}^d$, under the following conditions:

(C\*1) the generalized magnetic vector-potential $A_\varepsilon = (A_\varepsilon^1, \dots, A_\varepsilon^d)$ is a bounded measurable $\mathbb{R}^d$-valued Colombeau generalized function on $\mathbb{R}^d$,

(C\*2) the potential $(V_\varepsilon)_\varepsilon$ and the divergence $\mathbf{div}(A_\varepsilon)_\varepsilon = \sum_{j=1}^d \left( \frac{\partial A_\varepsilon^j}{\partial x_j} \right)_\varepsilon$ of $(A_\varepsilon)_\varepsilon$,

defined in the sense of distributions, are both Colombeau-Borel measures,

(C\*3) if $d > 1$ there exist $\alpha > d-2$ and $C > 0$ such that for all $x \in \mathbb{R}^d$ and $r \in (0,1]$

$$ |\mathbf{div}(A_\varepsilon)_\varepsilon|(B_r(x)) \leq C(r+\varepsilon)^{-\alpha}, |(V_\varepsilon)_\varepsilon|(B_r(x)) \leq C(r+\varepsilon)^{-\alpha} \qquad (0.6.60) $$

where $|\mathbf{div}(A_\varepsilon)_\varepsilon|(\cdot)$ and $|V|(\cdot)$ denote the total variations of the (possibly non-positive) Colombeau-Borel measures $(V_\varepsilon)_\varepsilon$ and $\mathbf{div}A$ respectively, and $B_r(x)$ denotes the ball in $\mathbb{R}^d$ of the radius $r$ centered at $x$; if $d = 1$, then the same holds for $\alpha = 0$.

Let us consider the following more general Cauchy problem

$$ \left( \frac{\partial u_{\varepsilon}(t, x)}{\partial t} \right)_{\varepsilon} = -D(H_{\varepsilon} u_{\varepsilon}(t, x))_{\varepsilon}, \quad (u_{\varepsilon}(0, x))_{\varepsilon} = (u_{0, \varepsilon}(x))_{\varepsilon}, \qquad (0.6.61) $$
$$ \left( \frac{\partial \psi_{\varepsilon}}{\partial t} \right)_{\varepsilon} = i \left( (A_{\varepsilon} - B_{\varepsilon}) \psi_{\varepsilon} \right)_{\varepsilon}. \qquad (0.6.58) $$

where $(A_\varepsilon)_\varepsilon$ is Colombeau selfadjoint operator (i.e. $\forall \varepsilon \in (0,1]$, $A_\varepsilon$ is selfadjoint operator), for which therefore exists (according to spectral theory) a unitary transformations $U_\varepsilon$ such that $\forall \varepsilon \in (0,1]$, $U_\varepsilon A_\varepsilon U_\varepsilon^{-1}$ is the multiplication operator in some

$\mathcal{L}_2(X, d\mu)$, where $X$ is locally compact, and $(B_\varepsilon)_\varepsilon$ is such that $(U_\varepsilon B_\varepsilon U_\varepsilon^{-1})_\varepsilon$ is a bounded operator in $C_0(X)$.

We shall consider now Colombeau-Schrödinger type operator with a magnetic field, namely the operator

$$ (H_{\varepsilon})_{\varepsilon} = \left( \left( i \frac{\partial}{\partial x} + A_{\varepsilon}(x) \right)^2 + V_{\varepsilon}(x) \right)_{\varepsilon} \qquad (0.6.59) $$

where $x = (x_1, \dots, x_d)$ and $\frac{\partial}{\partial x} = \left( \frac{\partial}{\partial x_1}, \dots, \frac{\partial}{\partial x_d} \right)$ is the gradient operator in $\mathbb{R}^d$, under the following conditions:

(C\*1) the generalized magnetic vector-potential $A_\varepsilon = (A_\varepsilon^1, \dots, A_\varepsilon^d)$ is a bounded measurable $\mathbb{R}^d$-valued Colombeau generalized function on $\mathbb{R}^d$,

(C\*2) the potential $(V_\varepsilon)_\varepsilon$ and the divergence $\mathbf{div}(A_\varepsilon)_\varepsilon = \sum_{j=1}^d \left( \frac{\partial A_\varepsilon^j}{\partial x_j} \right)_\varepsilon$ of $(A_\varepsilon)_\varepsilon$,

defined in the sense of distributions, are both Colombeau-Borel measures,

(C\*3) if $d > 1$ there exist $\alpha > d-2$ and $C > 0$ such that for all $x \in \mathbb{R}^d$ and $r \in (0,1]$

$$ |\mathbf{div}(A_{\varepsilon})_{\varepsilon}|(B_r(x)) \leq C(r + \varepsilon)^{-\alpha}, |(V_{\varepsilon})_{\varepsilon}|(B_r(x)) \leq C(r + \varepsilon)^{-\alpha} \qquad (0.6.60) $$

where $|\mathbf{div}(A_\varepsilon)_\varepsilon|(\cdot)$ and $|V|(\cdot)$ denote the total variations of the (possibly non-positive) Colombeau-Borel measures $(V_\varepsilon)_\varepsilon$ and $\mathbf{div}A$ respectively, and $B_r(x)$ denotes the ball in $\mathbb{R}^d$ of the radius $r$ centered at $x$; if $d = 1$, then the same holds for $\alpha = 0$.

Let us consider the following more general Cauchy problem

$$ \left( \frac{\partial u_{\varepsilon}(t, x)}{\partial t} \right)_{\varepsilon} = -D(H_{\varepsilon} u_{\varepsilon}(t, x))_{\varepsilon}, \quad (u_{\varepsilon}(0, x))_{\varepsilon} = (u_{0, \varepsilon}(x))_{\varepsilon}, \qquad (0.6.61) $$
where the (generalised) diffusion coefficient $D$ is an arbitrary complex number such that $\epsilon = \text{Re}D \ge 0, |D| > 0$. In the interaction representation, equation (0.6.61) takes the form

$$ (u_{\epsilon}(t,x))_{\epsilon} = [\exp(-Dt\Delta/2)]((u_{0,\epsilon}(x))_{\epsilon}) - D \left( \int_{0}^{t} [\exp(-D(t-s)\Delta/2)](W_{\epsilon}(x) - 2i(\mathbf{A}_{\epsilon}(x),\nabla))u_{\epsilon}(s,x)ds \right)_{\epsilon}. \quad (0.6.62) $$

Here $(W_{\epsilon}(x))_{\epsilon} = (V_{\epsilon}(x))_{\epsilon} + (|\mathbf{A}_{\epsilon}(x)|^2)_{\epsilon} - i(\text{div}\mathbf{A}_{\epsilon}(x))_{\epsilon}$. Equation (0.6.62) can be formally solved by iterations. In the case of the Green function $(G_{\epsilon}^{D}(t,x,y))_{\epsilon}$ of equation (0.6.62), i.e. its solution with the Dirac initial condition $(u_{0,\epsilon}(x))_{\epsilon} = \delta(x-y)$, the iteration procedure leads to the following representation:

$$ \begin{gathered} (G_{\epsilon}^{D}(t,x,y))_{\epsilon} = \left( \sum_{k=0}^{\infty} I_{k,\epsilon}^{D}(t,x,y) \right)_{\epsilon}, \\ (I_{k,\epsilon}^{D}(t,x,y))_{\epsilon} = -D \int_{0}^{t} \int_{\mathbb{R}^{d}} (I_{k-1,\epsilon}^{D}(t-s,x,\xi)(W_{\epsilon}(d\xi) + 2iD^{-1}s^{-1}(\mathbf{A}_{\epsilon}(x), (\xi - y))d\xi))_{\epsilon} G_{\text{free}}^{D}(s, \xi - y)ds, \end{gathered} \quad (0.6.63) $$

$$ G_{\text{free}}^{D}(s, \xi - y) = (2\pi tD)^{-d/2} \exp\left[-\frac{(\xi - y)^2}{2Dt}\right]. $$

**Theorem 0.6.17.** (i) If $\epsilon = \text{Re}D > 0$, then all terms of series (0.6.35) are well defined as absolutely convergent Colombeau integrals, the series itself is absolutely convergent and its sum $(G_{\epsilon}^{D}(t,x,y))_{\epsilon}$ is continuous in $x,y \in \mathbb{R}^d, t > 0$ (and $D$) and satisfies the following estimate

$$ (G_{\epsilon}^{D}(t,x,y))_{\epsilon} \le ((K_{\epsilon})_{\epsilon}) G_{\text{free}}^{\frac{|D|^2}{\epsilon}}(t,x-y) \exp(((B_{\epsilon})_{\epsilon})|x-y|) \quad (0.6.64) $$
where the (generalised) diffusion coefficient $D$ is an arbitrary complex number such that $\epsilon = \text{Re}D \ge 0, |D| > 0$. In the interaction representation, equation (0.6.61) takes the form

$$ (u_{\epsilon}(t,x))_{\epsilon} = [\exp(-Dt\Delta/2)]((u_{0,\epsilon}(x))_{\epsilon}) - D \left( \int_{0}^{t} [\exp(-D(t-s)\Delta/2)](W_{\epsilon}(x) - 2i(\mathbf{A}_{\epsilon}(x),\nabla))u_{\epsilon}(s,x)ds \right)_{\epsilon} \quad (0.6.62) $$

Here $(W_{\epsilon}(x))_{\epsilon} = (V_{\epsilon}(x))_{\epsilon} + (|\mathbf{A}_{\epsilon}(x)|^2)_{\epsilon} - i(\text{div}\mathbf{A}_{\epsilon}(x))_{\epsilon}$. Equation (0.6.62) can be formally solved by iterations. In the case of the Green function $(G_{\epsilon}^{D}(t,x,y))_{\epsilon}$ of equation (0.6.62), i.e. its solution with the Dirac initial condition $(u_{0,\epsilon}(x))_{\epsilon} = \delta(x-y)$, the iteration procedure leads to the following representation:

$$ \begin{gathered} (G_{\epsilon}^{D}(t,x,y))_{\epsilon} = \left( \sum_{k=0}^{\infty} I_{k,\epsilon}^{D}(t,x,y) \right)_{\epsilon}, \\ (I_{k,\epsilon}^{D}(t,x,y))_{\epsilon} = -D \int_{0}^{t} \int_{\mathbb{R}^{d}} (I_{k-1,\epsilon}^{D}(t-s,x,\xi)(W_{\epsilon}(d\xi) + 2iD^{-1}s^{-1}(\mathbf{A}_{\epsilon}(x), (\xi-y))d\xi))_{\epsilon} G_{\text{free}}^{D}(s, \xi-y)ds, \end{gathered} \quad (0.6.63) $$

$$ G_{\text{free}}^{D}(s, \xi - y) = (2\pi tD)^{-d/2} \exp\left[-\frac{(\xi - y)^2}{2Dt}\right]. $$

**Theorem 0.6.17.** (i) If $\epsilon = \text{Re}D > 0$, then all terms of series (0.6.35) are well defined as absolutely convergent Colombeau integrals, the series itself is absolutely convergent and its sum $(G_{\epsilon}^{D}(t,x,y))_{\epsilon}$ is continuous in $x,y \in \mathbb{R}^d, t > 0$ (and $D$) and satisfies the following estimate

$$ (G_{\epsilon}^{D}(t,x,y))_{\epsilon} \le ((K_{\epsilon})_{\epsilon}) G_{\text{free}}^{\frac{|D|^2}{\epsilon}}(t,x-y) \exp(((B_{\epsilon})_{\epsilon})|x-y|) \quad (0.6.64) $$
uniformly for $t \le t_0$ with any fixed $t_0$, where $\mathbf{cl}[(B_\epsilon)_\epsilon]$, $\mathbf{cl}[(K_\epsilon)_\epsilon] \in \tilde{\mathbb{R}}$ are Colombeau constants.

(ii) The integral operators

$$([U_{\epsilon}^{D}(t)u_{0,\epsilon}](t,x))_{\epsilon} = \left(\int u_{\epsilon}^{D}(t,x,y)u_{0,\epsilon}(y)dy\right)_{\epsilon} \quad (0.6.65)$$

defining the solutions to equation (0.6.62) for $t \in [0, t_0]$ form a uniformly bounded family of Colombeau operators $\mathcal{L}_2(\mathbb{R}^d) \to \mathcal{L}_2(\mathbb{R}^d)$.

(iii) If $D$ is real, i.e. $D = \epsilon > 0$, then there exists a constant $\omega > 0$ such that the Green function $(G_{\epsilon}^{\epsilon}(t,x,y))_{\epsilon}$ has the asymptotic representation

$$(G_{\epsilon}^{\epsilon}(t,x,y))_{\epsilon} = G_{\text{free}}^{\epsilon}(t,x,y)(1 + (O_{\epsilon}(t^{\omega}))_{\epsilon} + (O_{\epsilon}(|x-y|))_{\epsilon}) \quad (0.6.66)$$

for small $t$ and $x-y$. In case of vanishing $A$, the multiplier $\exp(B|x-y|)$ in (0.6.64) can

be omitted, and the term $(O_{\epsilon}(|x-y|))_{\epsilon}$ in (0.6.66) can be dropped. In this case, formula (0.6.66) gives global (uniform for all $x,y$) small time asymptotics for $(G_{\epsilon}^{\epsilon}(t,x,y))_{\epsilon}$.

(iv) One can give rigorous meaning to formal expression (0.6.63) as a bounded below Colombeau self adjoint operator in such a way that the family (0.6.65) of operators $U^D(t)$ (giving solutions to equation (0.6.62), which is formally equivalent to the evolutionary equation (0.6.61) with the formal generator $D(H_\epsilon)_\epsilon$ coincides with the semigroup $(\exp(-tDH_\epsilon))_\epsilon$ defined by means of the functional calculus. Hence for the integral kernel of the operators $(\exp(-tDH_\epsilon))_\epsilon$ the estimates (0.6.65) and (0.6.66) hold.

The rigorous construction a path integral representation for the Colombeau solution $(U_{\epsilon}^{D}(t)u_{0,\epsilon})_{\epsilon}$ of Eq.(0.6.61), gives by construction an Colombeau-Borel measure on a path space that is supported on the set of continuous piecewise linear paths. Denote this set by $\text{CPL}^{\mathbb{R}^d}$. Let $\text{CPL}^{x,y}(0,t)$ denote the class of paths $q: [0,t] \to \mathbb{R}^d$

from $\text{CPL}^{\mathbb{R}^d}$ joining $x$ and $y$ in time $t$, i.e. such that $q(0) = x, q(t) = y$. By $\text{CPL}^{x,y}_n(0,t)$

we denote its subclass consisting of all paths from $\text{CPL}^{x,y}(0,t)$ that have exactly $n$ jumps of their derivative. Clearly, each $\text{CPL}^{x,y}_n(0,t)$ is parametrised by the simplex

$$\mathbf{sim}_t^n = \{s_1, \dots, s_n | 0 < s_1 < s_2 < \dots < s_n < t\} \quad (0.6.67)$$
uniformly for $t \le t_0$ with any fixed $t_0$, where $\mathrm{cl}[(B_\epsilon)_\epsilon]$, $\mathrm{cl}[(K_\epsilon)_\epsilon] \in \tilde{\mathbb{R}}$ are Colombeau constants.

(ii) The integral operators

$$([U_{\epsilon}^{D}(t)u_{0,\epsilon}](t,x))_{\epsilon} = \left(\int u_{\epsilon}^{D}(t,x,y)u_{0,\epsilon}(y)dy\right)_{\epsilon} \quad (0.6.65)$$

defining the solutions to equation (0.6.62) for $t \in [0, t_0]$ form a uniformly bounded family of Colombeau operators $\mathcal{L}_2(\mathbb{R}^d) \to \mathcal{L}_2(\mathbb{R}^d)$.

(iii) If $D$ is real, i.e. $D = \epsilon > 0$, then there exists a constant $\omega > 0$ such that the Green function $(G_{\epsilon}^{\epsilon}(t,x,y))_{\epsilon}$ has the asymptotic representation

$$(G_{\epsilon}^{\epsilon}(t,x,y))_{\epsilon} = G_{\mathrm{free}}^{\epsilon}(t,x,y)(1 + (O_{\epsilon}(t^{\omega}))_{\epsilon} + (O_{\epsilon}(|x - y|))_{\epsilon}) \quad (0.6.66)$$

for small $t$ and $x-y$. In case of vanishing $A$, the multiplier $\exp(B|x-y|)$ in (0.6.64) can

be omitted, and the term $(O_{\epsilon}(|x-y|))_{\epsilon}$ in (0.6.66) can be dropped. In this case, formula (0.6.66) gives global (uniform for all $x,y$) small time asymptotics for $(G_{\epsilon}^{\epsilon}(t,x,y))_{\epsilon}$.

(iv) One can give rigorous meaning to formal expression (0.6.63) as a bounded below Colombeau self adjoint operator in such a way that the family (0.6.65) of operators $U^D(t)$ (giving solutions to equation (0.6.62), which is formally equivalent to the evolutionary equation (0.6.61) with the formal generator $D(H_\epsilon)_\epsilon$ coincides with the semigroup $(\exp(-tDH_\epsilon))_\epsilon$ defined by means of the functional calculus. Hence for the integral kernel of the operators $(\exp(-tDH_\epsilon))_\epsilon$ the estimates (0.6.65) and (0.6.66) hold.

The rigorous construction a path integral representation for the Colombeau solution $(U_{\epsilon}^{D}(t)u_{0,\epsilon})_{\epsilon}$ of Eq.(0.6.61), gives by construction an Colombeau-Borel measure on a path space that is supported on the set of continuous piecewise linear paths. Denote this set by $\mathrm{CPL}^{\mathbb{R}^d}$. Let $\mathrm{CPL}^{x,y}(0,t)$ denote the class of paths $q: [0,t] \to \mathbb{R}^d$

from $\mathrm{CPL}^{\mathbb{R}^d}$ joining $x$ and $y$ in time $t$, i.e. such that $q(0) = x, q(t) = y$. By $\mathrm{CPL}^{x,y}_n(0,t)$

we denote its subclass consisting of all paths from $\mathrm{CPL}^{x,y}(0,t)$ that have exactly $n$ jumps of their derivative. Clearly, each $\mathrm{CPL}^{x,y}_n(0,t)$ is parametrised by the simplex

$$\mathrm{sim}_t^n = \{s_1, \dots, s_n | 0 < s_1 < s_2 < \dots < s_n < t\} \quad (0.6.67)$$
of the times of jumps $s_1, \dots, s_n$ of the derivatives of a path and by $n$ positions $q(s_j)$,
$j = 1, \dots, n$, of this path at these points. In other words, an arbitrary path in
$\text{CPL}_n^{x,y}(0,t)$
has the form

$$
\begin{equation} \tag{0.6.68}
\begin{aligned}
q_n(s) &= q^{s_1, \dots, s_n}_{\eta_1, \dots, \eta_n} = \eta_j + (s - s_j) \frac{\eta_{j+1} - \eta_j}{s_{j+1} - s_j}, \\
s_j &\in [s_j, s_{j+1}],
\end{aligned}
\end{equation}
$$

where $s_0 = 0, s_{n+1} = t, \eta_0 = x, \eta_{n+1} = y$. Therefore $\mathrm{CPL}^{x,y}(0,t) = \bigcup_{n=0}^{\infty} \mathrm{CPL}_n^{x,y}(0,t)$. To
any $\mathbb{R}^{d+1}$-valued Colombeau-Borel measure $(\tilde{M}_{\varepsilon})_{\varepsilon} = \{(\mu_{\varepsilon})_{\varepsilon}, (\nu_{\varepsilon})_{\varepsilon}\} =$
$\{(\mu_{\varepsilon})_{\varepsilon}, (\nu_{\varepsilon}^{1})_{\varepsilon}, \dots, (\nu_{\varepsilon}^{n})_{\varepsilon}\}$ on $\mathbb{R}^{d+1}$ there corresponds a $\sigma$-finite complex measure
$\mathbf{D}^{\mathrm{CPL}}[q(\tau)]$ on $\mathrm{CPL}^{x,y}(0,t)$, which is defined as the sum of the Colombeau-Borel
measures $(\mathbf{M}_{n,\varepsilon}^{\mathrm{CPL}})_{\varepsilon}$ $n \in \mathbb{N}$ on the finite-dimensional spaces $\mathrm{CPL}_n^{x,y}(0,t)$ such that
$\mathbf{M}_0^{\mathrm{CPL}}$ is just the unit measure on the one-point set $\mathrm{CPL}_0^{x,y}(0,t)$ and each
$\mathbf{M}_n^{\mathrm{CPL}}$, $n > 0$, is defined in the following way: if $\Phi(\cdot)$ is a functional on $\mathrm{CPL}^{x,y}(0,t)$,
then:

$$
\left( \int_{\mathrm{CPL}_n^{x,y}(0,t)} \Phi(q(\tau)) \mathrm{D}_{n,\varepsilon}^{\mathrm{CPL}}[q(\tau)] \right)_{\varepsilon} = \left( \int_{\mathrm{sim}_t^n} ds_1 \dots ds_n \times \right. \\
\left. \int_{\mathbb{R}^d} \dots \int_{\mathbb{R}^d} \left( \mu_{\varepsilon} + 2i \frac{(\eta_2 - \eta_1, \nu_{\varepsilon})}{s_2 - s_1} \right) (d\eta_1) \dots \left( \mu + 2i \frac{(y - \eta_n, \nu_{\varepsilon})}{t - s_n} \right) (d\eta_n) \Phi(q(\tau)) \right)_{\varepsilon}. \tag{0.6.69}
$$

Now, we let $(\tilde{M}_{\varepsilon})_{\varepsilon} = \{(W_{\varepsilon}(x))_{\varepsilon}, D^{-1}(A_{\varepsilon}(x)dx)_{\varepsilon}\}$.

**Theorem 0.6.10.** For any $D$ with $\epsilon = \mathrm{Re}D > 0$, the Green function $G_{\epsilon}^{D}(t,x,y)$ of equation (0.6.62) has the following path integral representation:
of the times of jumps $s_1, \dots, s_n$ of the derivatives of a path and by $n$ positions $q(s_j)$,
$j = 1, \dots, n$, of this path at these points. In other words, an arbitrary path in
$\text{CPL}_n^{x,y}(0,t)$
has the form

$$
\begin{equation} \tag{0.6.68}
\begin{aligned}
q_n(s) &= q^{s_1, \dots, s_n}_{\eta_1, \dots, \eta_n} = \eta_j + (s - s_j) \frac{\eta_{j+1} - \eta_j}{s_{j+1} - s_j}, \\
s_j &\in [s_j, s_{j+1}],
\end{aligned}
\end{equation}
$$

where $s_0 = 0, s_{n+1} = t, \eta_0 = x, \eta_{n+1} = y$. Therefore $\mathrm{CPL}^{x,y}(0,t) = \bigcup_{n=0}^{\infty} \mathrm{CPL}_n^{x,y}(0,t)$. To any $\mathbb{R}^{d+1}$-valued Colombeau-Borel measure $(\tilde{M}_{\varepsilon})_{\varepsilon} = \{(\mu_{\varepsilon})_{\varepsilon}, (v_{\varepsilon})_{\varepsilon}\} = \{(\mu_{\varepsilon})_{\varepsilon}, (v_{\varepsilon}^{1})_{\varepsilon}, \dots, (v_{\varepsilon}^{n})_{\varepsilon}\}$ on $\mathbb{R}^{d+1}$ there corresponds a $\sigma$-finite complex measure $\mathbf{D}^{\mathrm{CPL}}[q(\tau)]$ on $\mathrm{CPL}^{x,y}(0,t)$, which is defined as the sum of the Colombeau-Borel measures $(\mathbf{M}_{n,\varepsilon}^{\mathrm{CPL}})_{\varepsilon} \in \mathbb{N}$ on the finite-dimensional spaces $\mathrm{CPL}_n^{x,y}(0,t)$ such that $\mathbf{M}_0^{\mathrm{CPL}}$ is just the unit measure on the one-point set $\mathrm{CPL}_0^{x,y}(0,t)$ and each $\mathbf{M}_n^{\mathrm{CPL}}, n > 0$, is defined in the following way: if $\Phi(\cdot)$ is a functional on $\mathrm{CPL}^{x,y}(0,t)$, then:

$$
\left( \int_{\mathrm{CPL}_n^{x,y}(0,t)} \Phi(q(\tau)) \mathbf{D}_{n,\varepsilon}^{\mathrm{CPL}}[q(\tau)] \right)_{\varepsilon} = \left( \int_{\mathrm{sim}_t^n} ds_1 \dots ds_n \times \right. \\
\left. \int_{\mathbb{R}^d} \dots \int_{\mathbb{R}^d} \left( \mu_{\varepsilon} + 2i \frac{(\eta_2 - \eta_1, v_{\varepsilon})}{s_2 - s_1} \right) (d\eta_1) \dots \left( \mu + 2i \frac{(y - \eta_n, v_{\varepsilon})}{t - s_n} \right) (d\eta_n) \Phi(q(\tau)) \right)_{\varepsilon}. \tag{0.6.69}
$$

Now, we let $(\tilde{M}_{\varepsilon})_{\varepsilon} = \{(W_{\varepsilon}(x))_{\varepsilon}, D^{-1}(A_{\varepsilon}(x)dx)_{\varepsilon}\}$.

**Theorem 0.6.10.** For any $D$ with $\epsilon = \mathrm{Re}D > 0$, the Green function $G_{\epsilon}^{D}(t,x,y)$ of equation (0.6.62) has the following path integral representation:
$$G^D(t,x,y) = \int_{CPL^{x,y}(0,t)} \Phi_D(q(\tau)) \exp\left\{\frac{1}{2D} \int_0^t \dot{q}^2(s) ds\right\} \mathbf{D}^{CPL}[q(\tau)] \quad (0.6.70)$$

with $q(s)$ given by Eq.(0.6.68) and

$$\Phi_D(q(\tau)) = D^n \prod_{j=1}^{n+1} (2\pi D(s_j - s_{j-1}))^{-d/2}. \quad (0.6.71)$$

For any $(u_{0,\varepsilon})_{\varepsilon} \in G_{L_2(\mathbb{R}^d)}$ the Colombeau solution $(u_{\varepsilon}(t,x))_{\varepsilon}$ of the Colombeau-Cauchy problem (0.6.61) with $D = i$ has the form

$$(u_{\varepsilon}(t,x))_{\varepsilon} =$$

$$\left( \lim_{\delta \to 0} \int_{CPL^{x,y}(0,t)} \int_{\mathbb{R}^d} u_{0,\varepsilon}(x) \Phi_{i+\delta}(q(\tau)) \exp \left\{ \frac{1}{2(i+\delta)} \int_0^t \dot{q}^2(s) ds \right\} \mathbf{D}_{\varepsilon}^{CPL}[q(\tau)] \right)_{\varepsilon}. \quad (0.6.72)$$

Here the limit is understood in $L_2(\mathbb{R}^d)$ sense.

**Theorem 0.6.18.** Suppose the vector potential A vanishes and V satisfies assumption (C3). Suppose additionally that $(V_{\varepsilon})_{\varepsilon}$ has no atom at the origin and is a finite positive measure so that $(\mu_{V_{\varepsilon}})_{\varepsilon} = (V_{\varepsilon}(\mathbb{R}^d))_{\varepsilon} > 0$. Let the paths of CPL are parametrised by Eq.(0.6.40). Let $(E_{t,\varepsilon}(\cdot))_{\varepsilon}$ denote the expectation with respect to the generalized process of jumps $\eta_j$ which are identically independently distributed according to the probability Colombeau measure $(V_{\varepsilon}/\mu_{V_{\varepsilon}})_{\varepsilon}$, and which occur at times $s_j$ from $[0,t]$ distributed according to Colombeau generalized Poisson process of intensity $(\mu_{V_{\varepsilon}})_{\varepsilon}$. Then the integral (0.6.70) can be written in the form

$$(G_{\varepsilon}^{D}(t,x,y))_{\varepsilon} = \left( e^{\mu_{V_{\varepsilon}}} \mathbf{E}_{t,\varepsilon} \left[ \Phi_{D}(q(\tau)) \exp \left\{ -\frac{1}{2D} \int_{0}^{t} \dot{q}^{2}(s) ds \right\} \mathbf{D}_{\varepsilon}^{CPL}[q(\tau)] \right] \right)_{\varepsilon}. \quad (0.6.73)$$
$$G^D(t,x,y) = \int_{CPL^{x,y}(0,t)} \Phi_D(q(\tau)) \exp\left\{\frac{1}{2D} \int_0^t \dot{q}^2(s) ds\right\} \mathbf{D}^{CPL}[q(\tau)] \quad (0.6.70)$$

with $q(s)$ given by Eq.(0.6.68) and

$$\Phi_D(q(\tau)) = D^n \prod_{j=1}^{n+1} (2\pi D(s_j - s_{j-1}))^{-d/2}. \quad (0.6.71)$$

For any $(u_{0,\varepsilon})_{\varepsilon} \in G_{L_2(\mathbb{R}^d)}$ the Colombeau solution $(u_{\varepsilon}(t,x))_{\varepsilon}$ of the Colombeau-Cauchy problem (0.6.61) with $D = i$ has the form

$$(u_{\varepsilon}(t,x))_{\varepsilon} =$$

$$\left( \lim_{\delta \to 0} \int_{CPL^{x,y}(0,t)} \int_{\mathbb{R}^d} u_{0,\varepsilon}(x) \Phi_{i+\delta}(q(\tau)) \exp \left\{ \frac{1}{2(i+\delta)} \int_0^t \dot{q}^2(s) ds \right\} \mathbf{D}_{\varepsilon}^{CPL}[q(\tau)] \right)_{\varepsilon} \quad (0.6.72)$$

Here the limit is understood in $L_2(\mathbb{R}^d)$ sense.

**Theorem 0.6.18.** Suppose the vector potential A vanishes and V satisfies assumption (C3). Suppose additionally that $(V_{\varepsilon})_{\varepsilon}$ has no atom at the origin and is a finite positive measure so that $(\mu_{V_{\varepsilon}})_{\varepsilon} = (V_{\varepsilon}(\mathbb{R}^d))_{\varepsilon} > 0$. Let the paths of CPL are parametrised by Eq.(0.6.40). Let $(E_{t,\varepsilon}(\cdot))_{\varepsilon}$ denote the expectation with respect to the generalized process of jumps $\eta_j$ which are identically independently distributed according to the probability Colombeau measure $(V_{\varepsilon}/\mu_{V_{\varepsilon}})_{\varepsilon}$, and which occur at times $s_j$ from $[0,t]$ distributed according to Colombeau generalized Poisson process of intensity $(\mu_{V_{\varepsilon}})_{\varepsilon}$. Then the integral (0.6.70) can be written in the form

$$(G_{\varepsilon}^{D}(t,x,y))_{\varepsilon} = \left( e^{\mu_{V_{\varepsilon}}} \mathbf{E}_{t,\varepsilon} \left[ \Phi_{D}(q(\tau)) \exp \left\{ -\frac{1}{2D} \int_{0}^{t} \dot{q}^{2}(s) ds \right\} \mathbf{D}_{\varepsilon}^{CPL}[q(\tau)] \right] \right)_{\varepsilon} \quad (0.6.73)$$
(0.6.)

(0.6.)

### I.0.7. Colombeau-Wiener Path Integral and generalized Haba theorem.

Let us consider the heat equation with $D \ge 0$ and complex term

$$Z(x) = W(x) + iV(x).$$

$$\frac{\partial u(t,x)}{\partial t} = D\Delta u(t,x) + Z(x)u(t,x), u(0) = u_0(x), x \in \mathbb{R}^n. \quad (0.7.1)$$

Nelson [40] has shown how to solve such equations by means of Wiener integrals for the purely imaginary $Z(x) = iV(x)$ under following assumption (N) on the function V

(N): There is a closed set F in $\mathbb{R}^d$ of capacity 0 such that V is continuous and real on the complement of F.
(0.6.)

(0.6.)

### I.0.7. Colombeau-Wiener Path Integral and generalized Haba theorem.

Let us consider the heat equation with $D \ge 0$ and complex term

$$Z(x) = W(x) + iV(x).$$

$$\frac{\partial u(t,x)}{\partial t} = D\Delta u(t,x) + Z(x)u(t,x), u(0) = u_0(x), x \in \mathbb{R}^n. \quad (0.7.1)$$

Nelson [40] has shown how to solve such equations by means of Wiener integrals for the purely imaginary $Z(x) = iV(x)$ under following assumption (N) on the function V

(N): There is a closed set F in $\mathbb{R}^d$ of capacity 0 such that V is continuous and real on the complement of F.
For example, if $d > 1$ and $V$ is a function which is continuous except at one point, say the origin (with an arbitrary singularity at the origin and arbitrary growth at infinity), then $V$ satisfies (N).

Let $x$ be any point in the complement of $F$ and $Pr_x$ the corresponding Wiener measure, with diffusion constant $D > 0$, on the space $\Omega$ of all trajectories. Then for almost every $\omega \in \Omega$, $V(\omega(t))$ is a continuous function of $t$ for $0 \le t < \infty$, and so one may form the Wiener integral

$$[U_{m,v}^t u_0](x) = \int_{\Omega} \exp \left[ -i \int_0^t V(\omega(s)) ds \right] u_0(\omega(t)) Pr_x(d\omega) \quad (0.7.2)$$

for any $u_0 \in \mathcal{L}_2$ and any $t \ge 0$. In fact, the integrand is defined for almost every $\omega \in \Omega$, and it is bounded in absolute value by $|u_0(\omega(t))|$. By definition one obtain

$$\int_{\Omega} |u_0(\omega(t))| Pr_x(d\omega) = \Xi_m^t |u_0|(x) < \infty. \quad (0.7.3)$$

Since $V(\omega(t))$ is continuous for almost every $\omega \in \Omega$, one obtain

$$\int_0^t V(\omega(s)) ds = \lim_{n \to \infty} \sum_{j=1}^n \frac{t}{n} V\left(\omega\left(j \frac{t}{n}\right)\right). \quad (0.7.4)$$

Therefore by the Lebesgue dominated convergence theorem one obtain $\forall x \notin F$:

$$[U_{m,v}^t u_0](x) = \lim_{n \to \infty} \int_{\Omega} \exp \left[ -i \sum_{j=1}^n \frac{t}{n} V\left(\omega\left(j \frac{t}{n}\right)\right) \right] u_0(\omega(t)) Pr_x(d\omega) \quad (0.7.5)$$

and by definition of the Wiener integral Eq.(0.7.5) gives
For example, if $d > 1$ and $V$ is a function which is continuous except at one point, say the origin (with an arbitrary singularity at the origin and arbitrary growth at infinity), then $V$ satisfies (N).

Let $x$ be any point in the complement of $F$ and $\mathbf{Pr}_x$ the corresponding Wiener measure, with diffusion constant $D > 0$, on the space $\Omega$ of all trajectories. Then for almost every $\omega \in \Omega$, $V(\omega(t))$ is a continuous function of $t$ for $0 \le t < \infty$, and so one may form the Wiener integral

$$[U_{m,v}^t u_0](x) = \int_{\Omega} \exp \left[ -i \int_0^t V(\omega(s)) ds \right] u_0(\omega(t)) \mathbf{Pr}_x(d\omega) \quad (0.7.2)$$

for any $u_0 \in \mathcal{L}_2$ and any $t \ge 0$. In fact, the integrand is defined for almost every $\omega \in \Omega$, and it is bounded in absolute value by $|u_0(\omega(t))|$. By definition one obtain

$$\int_{\Omega} |u_0(\omega(t))| \mathbf{Pr}_x(d\omega) = \Xi_m^t |u_0|(x) < \infty. \quad (0.7.3)$$

Since $V(\omega(t))$ is continuous for almost every $\omega \in \Omega$, one obtain

$$\int_0^t V(\omega(s)) ds = \lim_{n \to \infty} \sum_{j=1}^n \frac{t}{n} V\left(\omega\left(j \frac{t}{n}\right)\right). \quad (0.7.4)$$

Therefore by the Lebesgue dominated convergence theorem one obtain $\forall x \notin F$:

$$[U_{m,v}^t u_0](x) = \lim_{n \to \infty} \int_{\Omega} \exp \left[ -i \sum_{j=1}^n \frac{t}{n} V\left(\omega\left(j \frac{t}{n}\right)\right) \right] u_0(\omega(t)) \mathbf{Pr}_x(d\omega) \quad (0.7.5)$$

and by definition of the Wiener integral Eq.(0.7.5) gives
$$[U_{m,V}^t u_0](x) = \left(\Xi_m^{\frac{t}{n}} M_V^{\frac{t}{n}}\right)^n u_0(x). \qquad (0.7.6)$$

**Theorem 0.7.1.** (Nelson [41]). Let $D > 0$, $Z$ be purely imaginary and let $V$ satisfy condition (N). Then for all $u_0(x) \in \mathcal{L}_2(\mathbb{R}^n)$ and $t \ge 0$, the limit (0.4.6) exists, where $\Xi_m^t$ and $M_V^t$ are defined by (0.3.4) and (0.3.5), respectively. The operators $U_{m,V}^t$ defined (0.7.6) have the semigroup property

$$U_{m,V}^t U_{m,V}^s = U_{m,V}^{t+s}, \quad 0 \le t, s < \infty \qquad (0.7.7)$$

and for all $u_0(x) \in \mathcal{L}_2(\mathbb{R}^n)$, $t \to U_{m,V}^t$ is continuous from $[0, \infty)$ to $\mathcal{L}_2(\mathbb{R}^n)$. Let us consider now the heat equation with $D \ge 0$ and generalized Colombeau complex

term $(Z_\epsilon(x))_\epsilon = (W_\epsilon(x))_\epsilon + i(V_\epsilon(x))_\epsilon$ such that $\mathbf{cl}[(W_\epsilon(x))_\epsilon], \mathbf{cl}[(V_\epsilon(x))_\epsilon], \mathbf{cl}[(u_{0,\epsilon}(x))_\epsilon] \in G(\mathbb{R}^n)$

$$\frac{\partial}{\partial t}((u_\epsilon(t,x))_\epsilon) = D\Delta((u_\epsilon(t,x))_\epsilon) + ((Z_\epsilon(x))_\epsilon)((u_\epsilon(t,x))_\epsilon), \qquad (0.7.8)$$

$$(u_\epsilon(0))_\epsilon = (u_{0,\epsilon}(x))_\epsilon, x \in \mathbb{R}^n.$$

One can solve such equations by means of Colombeau-Wiener integrals for the $D > 0$ under following assumption $(\tilde{\mathbb{N}})$ on the Colombeau generalized function $(Z_\epsilon)_\epsilon = (W_\epsilon(x))_\epsilon + i(V_\epsilon(x))_\epsilon$

$(\tilde{\mathbb{N}}): \mathbf{cl}[(W_\epsilon(x))_\epsilon], \mathbf{cl}[(V_\epsilon(x))_\epsilon] \in G(\mathbb{R}^n)$. There is a closed set $F$ in $\mathbb{R}^d$ of capacity 0 such that $\forall \epsilon \in (0, 1]$, $Z_\epsilon(x)$ is continuous on the complement of $F$.

Let $x$ be any point in the complement of $F$ and $\mathbf{Pr}_x$ the corresponding Wiener measure, with diffusion constant $D > 0$, on the space $\Omega$ of all trajectories. Then for almost every $\omega \in \Omega$, $Z_\epsilon(\omega(t)), \epsilon \in (0, 1]$ is a continuous function of $t$ for $0 \le t < \infty$, and so one may form the Colombeau-Wiener integral:
$$[U_{m,V}^t u_0](x) = \left(\Xi_m^{\frac{t}{n}} M_V^{\frac{t}{n}}\right)^n u_0(x). \qquad (0.7.6)$$

**Theorem 0.7.1.** (Nelson [41]). Let $D > 0$, $Z$ be purely imaginary and let $V$ satisfy condition (N). Then for all $u_0(x) \in \mathcal{L}_2(\mathbb{R}^n)$ and $t \ge 0$, the limit (0.4.6) exists, where $\Xi_m^t$ and $M_V^t$ are defined by (0.3.4) and (0.3.5), respectively. The operators $U_{m,V}^t$ defined (0.7.6) have the semigroup property

$$U_{m,V}^t U_{m,V}^s = U_{m,V}^{t+s}, \quad 0 \le t, s < \infty \qquad (0.7.7)$$

and for all $u_0(x) \in \mathcal{L}_2(\mathbb{R}^n)$, $t \to U_{m,V}^t$ is continuous from $[0, \infty)$ to $\mathcal{L}_2(\mathbb{R}^n)$. Let us consider now the heat equation with $D \ge 0$ and generalized Colombeau complex

term $(Z_\epsilon(x))_\epsilon = (W_\epsilon(x))_\epsilon + i(V_\epsilon(x))_\epsilon$ such that $\mathbf{cl}[(W_\epsilon(x))_\epsilon], \mathbf{cl}[(V_\epsilon(x))_\epsilon], \mathbf{cl}[(u_{0,\epsilon}(x))_\epsilon] \in G(\mathbb{R}^n)$

$$\frac{\partial}{\partial t}((u_{\epsilon}(t,x))_{\epsilon}) = D\Delta((u_{\epsilon}(t,x))_{\epsilon}) + ((Z_{\epsilon}(x))_{\epsilon})((u_{\epsilon}(t,x))_{\epsilon}), \qquad (0.7.8)$$

$$(u_{\epsilon}(0))_{\epsilon} = (u_{0,\epsilon}(x))_{\epsilon}, x \in \mathbb{R}^n.$$

One can solve such equations by means of Colombeau-Wiener integrals for the $D > 0$ under following assumption $(\tilde{\mathbb{N}})$ on the Colombeau generalized function $(Z_\epsilon)_\epsilon = (W_\epsilon(x))_\epsilon + i(V_\epsilon(x))_\epsilon$

$(\tilde{\mathbb{N}}): \mathbf{cl}[(W_\epsilon(x))_\epsilon], \mathbf{cl}[(V_\epsilon(x))_\epsilon] \in G(\mathbb{R}^n)$. There is a closed set $F$ in $\mathbb{R}^d$ of capacity 0 such that $\forall \epsilon \in (0, 1]$, $Z_\epsilon(x)$ is continuous on the complement of $F$.

Let $x$ be any point in the complement of $F$ and $\mathbf{Pr}_x$ the corresponding Wiener measure, with diffusion constant $D > 0$, on the space $\Omega$ of all trajectories. Then for almost every $\omega \in \Omega$, $Z_\epsilon(\omega(t)), \epsilon \in (0, 1]$ is a continuous function of $t$ for $0 \le t < \infty$, and so one may form the Colombeau-Wiener integral:
$$([U_{m,Z_{\varepsilon}}^{t} u_0](x))_{\varepsilon} =$$

$$\left( \int_{\Omega} \exp \left[ -i \int_{0}^{t} Z_{\varepsilon}(\omega(s)) ds \right] u_{0,\varepsilon}(\omega(t)) \mathbf{Pr}_{x}(d\omega) \right)_{\varepsilon} = \qquad (0.7.9)$$

$$\int_{\Omega} \exp \left[ -i \int_{0}^{t} ((Z_{\varepsilon}(\omega(s)))_{\varepsilon}) ds \right] ((u_{0,\varepsilon}(\omega(t))))_{\varepsilon} \mathbf{Pr}_{x}(d\omega)$$

for any $(u_{0,\varepsilon})_{\varepsilon}$ such that $\forall \varepsilon \in (0,1]$, $u_{0,\varepsilon} \in \mathcal{L}_2$ and any $t \ge 0$. In fact, the integrand is defined for almost every $\omega \in \Omega$, and it is bounded in absolute value by $|u_{0,\varepsilon}(\omega(t))|\exp(\varepsilon^{-N})$ for some $N \in \mathbb{N}$. By definition one obtain $\forall \varepsilon \in (0,1]$:

$$\int_{\Omega} |u_{0,\varepsilon}(\omega(t))| \mathbf{Pr}_{x}(d\omega) = \Xi_{m}^{t} |u_{0,\varepsilon}|(x) < \infty. \qquad (0.7.10)$$

Since $\forall \varepsilon \in (0,1]$, $Z_{\varepsilon}(\omega(t))$ is continuous for almost every $\omega \in \Omega$, one obtain

$$\left( \int_{0}^{t} Z_{\varepsilon}(\omega(s)) ds \right)_{\varepsilon} = \left( \lim_{n \to \infty} \sum_{j=1}^{n} \frac{t}{n} Z_{\varepsilon} \left( \omega \left( j \frac{t}{n} \right) \right) \right)_{\varepsilon}. \qquad (0.7.11)$$

Therefore by the Lebesgue dominated convergence theorem one obtain $\forall \varepsilon \in (0,1]$, $\forall x \notin F$:

$$([U_{m,Z_{\varepsilon}}^{t} u_0](x))_{\varepsilon} =$$

$$\left( \lim_{n \to \infty} \int_{\Omega} \exp \left[ -i \sum_{j=1}^{n} \frac{t}{n} Z_{\varepsilon} \left( \omega \left( j \frac{t}{n} \right) \right) \right] u_{0,\varepsilon}(\omega(t)) \mathbf{Pr}_{x}(d\omega) \right)_{\varepsilon} \qquad (0.7.12)$$
$$([U_{m,Z_{\varepsilon}}^{t} u_0](x))_{\varepsilon} =$$

$$\left( \int_{\Omega} \exp \left[ -i \int_{0}^{t} Z_{\varepsilon}(\omega(s)) ds \right] u_{0,\varepsilon}(\omega(t)) \mathbf{Pr}_{x}(d\omega) \right)_{\varepsilon} = \qquad (0.7.9)$$

$$\int_{\Omega} \exp \left[ -i \int_{0}^{t} ((Z_{\varepsilon}(\omega(s)))_{\varepsilon}) ds \right] ((u_{0,\varepsilon}(\omega(t))))_{\varepsilon} \mathbf{Pr}_{x}(d\omega)$$

for any $(u_{0,\varepsilon})_{\varepsilon}$ such that $\forall \varepsilon \in (0,1]$, $u_{0,\varepsilon} \in \mathcal{L}_2$ and any $t \ge 0$. In fact, the integrand is defined for almost every $\omega \in \Omega$, and it is bounded in absolute value by $|u_{0,\varepsilon}(\omega(t))|\exp(\varepsilon^{-N})$ for some $N \in \mathbb{N}$. By definition one obtain $\forall \varepsilon \in (0,1]$:

$$\int_{\Omega} |u_{0,\varepsilon}(\omega(t))| \mathbf{Pr}_{x}(d\omega) = \Xi_{m}^{t} |u_{0,\varepsilon}|(x) < \infty. \qquad (0.7.10)$$

Since $\forall \varepsilon \in (0,1]$, $Z_{\varepsilon}(\omega(t))$ is continuous for almost every $\omega \in \Omega$, one obtain

$$\left( \int_{0}^{t} Z_{\varepsilon}(\omega(s)) ds \right)_{\varepsilon} = \left( \lim_{n \to \infty} \sum_{j=1}^{n} \frac{t}{n} Z_{\varepsilon} \left( \omega \left( j \frac{t}{n} \right) \right) \right)_{\varepsilon}. \qquad (0.7.11)$$

Therefore by the Lebesgue dominated convergence theorem one obtain $\forall \varepsilon \in (0,1]$, $\forall x \notin F$:

$$([U_{m,Z_{\varepsilon}}^{t} u_0](x))_{\varepsilon} =$$

$$\left( \lim_{n \to \infty} \int_{\Omega} \exp \left[ -i \sum_{j=1}^{n} \frac{t}{n} Z_{\varepsilon} \left( \omega \left( j \frac{t}{n} \right) \right) \right] u_{0,\varepsilon}(\omega(t)) \mathbf{Pr}_{x}(d\omega) \right)_{\varepsilon} \qquad (0.7.12)$$
and by definition of the Wiener integral Eq.(0.7.13) gives

$$ ([U_{m,Z_{\varepsilon}}^t u_{0,\varepsilon}](x))_{\varepsilon} = \left( \left( \Xi_m^{\frac{t}{n}} M_{Z_{\varepsilon}}^{\frac{t}{n}} \right)^n u_{0,\varepsilon}(x) \right)_{\varepsilon}. \quad (0.7.13) $$

Here

$$ ([\Xi_m^t u_{0,\varepsilon}](x))_{\varepsilon} = \left(\frac{2\pi it}{m}\right)^{-d/2} \left( \int_{\mathbb{R}^n} d^d y u_{0,\varepsilon}(y) \exp\left[i \frac{m}{2} \left(\frac{|x-y|^2}{t}\right)\right] \right)_{\varepsilon}, \quad (0.7.14) $$

$$ (M_{Z_{\varepsilon}}^t)_{\varepsilon} = (\exp\{-tZ_{\varepsilon}\})_{\varepsilon}. $$

Haba [44] has found conditions on $u_0(x)$ and $Z(x)$ under which Eq.(0.7.2) valid for complex $D \in \mathbb{C}$.

**Definition 0.7.1.** Let $\mathcal{F}_d$ denote the complex Banach algebra of functions which can be represented of the form

$$ g(x) = \int_{\mathbb{R}^d} \exp(ipx) d\mu_g(p). \quad (0.7.15) $$

where $\mu_g(p)$ is a complex measure on $\mathbb{R}^d$ with a bounded variation $|\mu_g|(p)$.

**Definition 0.7.2.** Let $\mathcal{F}_d^\triangle$ denote the set of functions which can be represented of the form (0.7.15) with $\mu_g(p)$ such that

$$ \forall \epsilon(\epsilon > 0) \exists K(\epsilon) \forall a(a \in \mathbb{R}) \left[ \int_{\mathbb{R}^d} \exp(a|p|) d|\mu_g|(p) \le K(\epsilon) \exp(\epsilon a^2) \right]. \quad (0.7.16) $$

Note that $\mathcal{F}_d$ and $\mathcal{F}_d^\triangle$ are dense subsets in $\mathcal{L}_2(\mathbb{R}^d)$.

Let us consider now a complex extension of the Schrödinger equation (0.1.1) in the form
and by definition of the Wiener integral Eq.(0.7.13) gives

$$ ([U_{m,Z_\varepsilon}^t u_{0,\varepsilon}](x))_{\varepsilon} = \left( \left( \Xi_m^{\frac{t}{n}} M_{Z_\varepsilon}^{\frac{t}{n}} \right)^n u_{0,\varepsilon}(x) \right)_{\varepsilon}. \quad (0.7.13) $$

Here

$$ ([\Xi_m^t u_{0,\varepsilon}](x))_{\varepsilon} = \left( \frac{2\pi it}{m} \right)^{-d/2} \left( \int_{\mathbb{R}^n} d^d y u_{0,\varepsilon}(y) \exp \left[ i \frac{m}{2} \left( \frac{|x-y|^2}{t} \right) \right] \right)_{\varepsilon}, \quad (0.7.14) $$

$$ (M_{Z_\varepsilon}^t)_{\varepsilon} = (\exp\{-tZ_\varepsilon\})_{\varepsilon}. $$

Haba [44] has found conditions on $u_0(x)$ and $Z(x)$ under which Eq.(0.7.2) valid for complex $D \in \mathbb{C}$.

**Definition 0.7.1.** Let $\mathcal{F}_d$ denote the complex Banach algebra of functions which can be represented of the form

$$ g(x) = \int_{\mathbb{R}^d} \exp(ipx) d\mu_g(p). \quad (0.7.15) $$

where $\mu_g(p)$ is a complex measure on $\mathbb{R}^d$ with a bounded variation $|\mu_g|(p)$.

**Definition 0.7.2.** Let $\mathcal{F}_d^\triangle$ denote the set of functions which can be represented of the form (0.7.15) with $\mu_g(p)$ such that

$$ \forall \epsilon(\epsilon > 0) \exists K(\epsilon) \forall a(a \in \mathbb{R}) \left[ \int_{\mathbb{R}^d} \exp(a|p|) d|\mu_g|(p) \le K(\epsilon) \exp(\epsilon a^2) \right]. \quad (0.7.16) $$

Note that $\mathcal{F}_d$ and $\mathcal{F}_d^\triangle$ are dense subsets in $\mathcal{L}_2(\mathbb{R}^d)$.

Let us consider now a complex extension of the Schrödinger equation (0.1.1) in the form
$$ \frac{\partial \psi(t, x)}{\partial t} = \frac{1}{2} \sigma^2 \lambda^2 \Delta \psi(t, x) + g \hbar^{-1} \lambda^{-2} Z(x) \psi(t, x), \psi(0, x) = \psi_0(x), x \in \mathbb{R}^d. \quad (0.7.17) $$

Here $\lambda \in \mathbb{C}, \sigma = \sqrt{\hbar m^{-1}}$. Schrödinger equation corresponds to $\lambda = \sqrt{i} = (1+i)/\sqrt{2}$.

Haba [44] has express the Feynman path integral corresponding to Eq.(0.7.17) using Wiener measure corresponding to Wiener process $b(\tau) \triangleq b_{\tau} \in \mathbb{R}^d$, (Brownian motion). Here $b_{\tau} \in \mathbb{R}^d$ is the Gaussian process with the covariance

$$ \mathbb{E}[b^k(\tau_1)b^r(\tau_2)] = \delta_{kr} \min(\tau_1, \tau_2). \quad (0.7.18) \\ k, r = 1, \dots, d. $$

We use the notation

$$ \mathbb{E}[F] \triangleq \int_{\Omega_t} d\mu_t^W(\omega)F(\omega), \quad (0.7.19) $$

where $\mu_t^W(\omega)$ is the Wiener measure and the integral is over the Wiener space $\Omega_t = C([0,t])$ which consists of continuous functions defined on the interval $[0,t]$ with $\omega(0) = 0$.

**Lemma 0.7.1.** [44]. Let $\mu$ be a Gaussian measure on the Wiener space $\Omega_t = C([0,t])$.

Assume that $Q$ is a non-negative quadratic form on $\Omega_t$ such that $Q = \lim_{m \to \infty} Q_m$ with probability 1, where $Q_m$ is a continuous non-negative quadratic form. Let $\mathcal{L}(\omega) = \sum_{k=1}^r a_k \omega(\tau_k)$, where $a_k \in \mathbb{R}$ ($k = 1, 2, \dots, r$) and $0 < \tau_k < t$ are arbitrary. Then

$$ \left| \int d\mu(\omega) \exp[-Q(\omega) - i\lambda \mathcal{L}(\omega)] \right| \le 1 \quad (0.7.20) $$

if $\mathrm{Re}(\lambda^2) \ge 0$.

Let us denote
$$ \frac{\partial \psi(t, x)}{\partial t} = \frac{1}{2} \sigma^2 \lambda^2 \Delta \psi(t, x) + g \hbar^{-1} \lambda^{-2} Z(x) \psi(t, x), \psi(0, x) = \psi_0(x), x \in \mathbb{R}^d. \quad (0.7.17) $$

Here $\lambda \in \mathbb{C}, \sigma = \sqrt{\hbar m^{-1}}$. Schrödinger equation corresponds to $\lambda = \sqrt{i} = (1+i)/\sqrt{2}$.

Haba [44] has express the Feynman path integral corresponding to Eq.(0.7.17) using Wiener measure corresponding to Wiener process $b(\tau) \triangleq b_{\tau} \in \mathbb{R}^d$, (Brownian motion). Here $b_{\tau} \in \mathbb{R}^d$ is the Gaussian process with the covariance

$$ \mathbb{E}[b^k(\tau_1)b^r(\tau_2)] = \delta_{kr} \min(\tau_1, \tau_2). \quad (0.7.18) \\ k, r = 1, \dots, d. $$

We use the notation

$$ \mathbb{E}[F] \triangleq \int_{\Omega_t} d\mu_t^W(\omega)F(\omega), \quad (0.7.19) $$

where $\mu_t^W(\omega)$ is the Wiener measure and the integral is over the Wiener space $\Omega_t = C([0,t])$ which consists of continuous functions defined on the interval $[0,t]$ with $\omega(0) = 0$.

**Lemma 0.7.1.** [44]. Let $\mu$ be a Gaussian measure on the Wiener space $\Omega_t = C([0,t])$.

Assume that $Q$ is a non-negative quadratic form on $\Omega_t$ such that $Q = \lim_{m \to \infty} Q_m$ with probability 1, where $Q_m$ is a continuous non-negative quadratic form. Let $\mathcal{L}(\omega) = \sum_{k=1}^r a_k \omega(\tau_k)$, where $a_k \in \mathbb{R}$ ($k = 1, 2, \dots, r$) and $0 < \tau_k < t$ are arbitrary. Then

$$ \left| \int d\mu(\omega) \exp[-Q(\omega) - i\lambda \mathcal{L}(\omega)] \right| \le 1 \quad (0.7.20) $$

if $\mathrm{Re}(\lambda^2) \ge 0$.

Let us denote
$$ \theta_R(x) = \exp\left(\frac{|x^2|}{2R}\right), $$

$$ \Theta_t[g\theta_R(\omega(\tau)); Z(x + \lambda\sigma\omega(\tau))] = \qquad (0.7.21) $$

$$ \exp\left\{ \frac{g}{\hbar\lambda^2} \int_0^t d\tau [\theta_R(\omega(\tau))Z(x + \lambda\sigma\omega(\tau))] \right\}. $$

We shall also use a shorthand notation $\Theta_t[g\theta_R; Z]$ for the quantity $\Theta_t[g\theta_R(\omega(\tau)); Z(x + \lambda\sigma\omega(\tau))]$.

**Theorem 0.7.2.** (Haba [44]). Assume that $\psi_0, Z \in \mathcal{F}_d^{\uparrow}$. Then, $\Theta_t[g\theta_R; Z]$ is an analytic function of $g \in \mathbb{C}$ and $\lambda \neq 0$ as long as $\mathrm{Re}\,\lambda^2 \geq 0$. Define

$$ \psi^R(t,x) = \mathbb{E}[\Theta_t[g\theta_R(\omega(\tau)); Z(x + \lambda\sigma\omega(\tau))]\psi_0(x + \lambda\sigma\omega(\tau))] \quad (0.7.22) $$

Then, limit

$$ \psi(t,x) = \lim_{R \to \infty} \psi^R(t,x) \qquad (0.7.23) $$

exists uniformly in $t$ and $x$. Function $\psi(t,x)$ solves the Schrödinger equation (0.7.17). The solution is an analytic function of $g$ and $\lambda$ in the same region where $\mathbb{E}[\Theta_t]$ is analytic.

Let $Z(t,x)$ be the function

$$ Z(t,x) = \frac{1}{2}\omega(t)x^2 + \vartheta(t)x + gZ(x), \qquad (0.7.24) $$

where $Z(x) \in \mathcal{F}_d^{\uparrow}$ and $\omega(t), \vartheta(t)$ are continuous functions of time.

**Theorem 0.7.3.** (Haba [44]). Assume that $\omega(t) \leq \omega_0$ for each $t$. If $t < \pi/\omega_0$ then theorem 0.7.2 holds true for potential given by Eq.(0.7.24).

Let us consider a decomposition of $\mathbb{R}^{2n}$ into $\mathbb{R}^n \oplus \mathbb{R}^n$ where
$$ \theta_R(x) = \exp\left(\frac{|x^2|}{2R}\right), $$

$$ \Theta_t[g\theta_R(\omega(\tau)); Z(x + \lambda\sigma\omega(\tau))] = \qquad (0.7.21) $$

$$ \exp\left\{ \frac{g}{\hbar\lambda^2} \int_0^t d\tau [\theta_R(\omega(\tau))Z(x + \lambda\sigma\omega(\tau))] \right\}. $$

We shall also use a shorthand notation $\Theta_t[g\theta_R; Z]$ for the quantity $\Theta_t[g\theta_R(\omega(\tau)); Z(x + \lambda\sigma\omega(\tau))]$.

**Theorem 0.7.2.** (Haba [44]). Assume that $\psi_0, Z \in \mathcal{F}_d^{\uparrow}$. Then, $\Theta_t[g\theta_R; Z]$ is an analytic function of $g \in \mathbb{C}$ and $\lambda \neq 0$ as long as $\mathrm{Re}\,\lambda^2 \geq 0$. Define

$$ \psi^R(t,x) = \mathbb{E}[\Theta_t[g\theta_R(\omega(\tau)); Z(x + \lambda\sigma\omega(\tau))]\psi_0(x + \lambda\sigma\omega(\tau))] \qquad (0.7.22) $$

Then, limit

$$ \psi(t,x) = \lim_{R \to \infty} \psi^R(t,x) \qquad (0.7.23) $$

exists uniformly in $t$ and $x$. Function $\psi(t,x)$ solves the Schrödinger equation (0.7.17). The solution is an analytic function of $g$ and $\lambda$ in the same region where $\mathbb{E}[\Theta_t]$ is analytic.

Let $Z(t,x)$ be the function

$$ Z(t,x) = \frac{1}{2}\omega(t)x^2 + \vartheta(t)x + gZ(x), \qquad (0.7.24) $$

where $Z(x) \in \mathcal{F}_d^{\uparrow}$ and $\omega(t), \vartheta(t)$ are continuous functions of time.

**Theorem 0.7.3.** (Haba [44]). Assume that $\omega(t) \leq \omega_0$ for each $t$. If $t < \pi/\omega_0$ then theorem 0.7.2 holds true for potential given by Eq. (0.7.24).

Let us consider a decomposition of $\mathbb{R}^{2n}$ into $\mathbb{R}^n \oplus \mathbb{R}^n$ where
$z = (x_1, x_2), x_1, x_2 \in \mathbb{R}^n$. Let $G_z$ be a subset $G_z \subset \mathbb{R}^{2n}$

$$G_z = \{w \in \mathbb{R}^{2n} | w = (x_1 + y, x_2 + y), y \in \mathbb{R}^n\} \quad (0.7.25)$$

and let $Z(x_1, x_2)$ be a complex-valued potential defined on $\mathbb{R}^{2n}$ such that on $G_z$

$$|\operatorname{Im} Z(x_1 + y, x_2 + y)| \leq A(x_1, x_2)(1 + |y|) \quad (0.7.26)$$

with a certain continuous non-negative function $A$. We define a semigroup $S_t$ on the set of functions defined on $\mathbb{R}^{2n}$ such that on $G_z$

$$|\Phi(x_1 + y, x_2 + y)| \leq M(x_1, x_2) \exp(|y|C(x_1, x_2)) \quad (0.7.27)$$

with certain non-negative continuous functions $C(x_1, x_2)$ and $M(x_1, x_2)$. Let $S_t$ is defined by the Feynman-Kac formula:
$z = (x_1,x_2), x_1,x_2 \in \mathbb{R}^n$. Let
$G_z$ be a subset $G_z \subset \mathbb{R}^{2n}$

$$
G_z = \{w \in \mathbb{R}^{2n} | w = (x_1 + y, x_2 + y), y \in \mathbb{R}^n\} \quad (0.7.25)
$$

and let $Z(x_1,x_2)$ be a complex-valued potential defined on $\mathbb{R}^{2n}$ such that on $G_z$

$$
|\operatorname{Im} Z(x_1 + y, x_2 + y)| \leq A(x_1, x_2)(1 + |y|) \quad (0.7.26)
$$

with a certain continuous non-negative function $A$. We define a semigroup $S_t$ on the
set of functions defined on $\mathbb{R}^{2n}$ such that on $G_z$

$$
|\Phi(x_1 + y, x_2 + y)| \leq M(x_1, x_2) \exp(|y|C(x_1, x_2)) \quad (0.7.27)
$$

with certain non-negative continuous functions $C(x_1,x_2)$ and $M(x_1,x_2)$. Let $S_t$ is
defined by the Feynman-Kac formula:
$$ (S_t \Phi)(x_1, x_2) = $$

$$ \mathbb{E} \left[ \exp \left( -\frac{i}{\hbar} \int_0^t d\tau Z \left( x_1 + \frac{\sigma\omega(\tau)}{\sqrt{2}}, x_2 + \frac{\sigma\omega(\tau)}{\sqrt{2}} \right) \times \Phi \left( x_1 + \frac{\sigma\omega(t)}{\sqrt{2}}, x_2 + \frac{\sigma\omega(t)}{\sqrt{2}} \right) \right) \right] = $$

$$ \int_{\mathbb{R}^n} \int_{\mathbb{R}^n} \mathbb{E} \left[ \delta \left( y_1 - x_1 - \frac{\sigma\omega(\tau)}{\sqrt{2}} \right) \delta \left( y_2 - x_2 - \frac{\sigma\omega(\tau)}{\sqrt{2}} \right) \times \right. \quad (0.7.28) $$

$$ \exp\left(-\frac{i}{\hbar} \int_0^t d\tau \tilde{Z}\left(x_1 + \frac{\sigma\omega(\tau)}{\sqrt{2}}, x_2 + \frac{\sigma\omega(\tau)}{\sqrt{2}}\right)\right) \times $$

$$ \Phi(y_1, y_2) dy_1 dy_2 = $$

$$ \int_{\mathbb{R}^n} \int_{\mathbb{R}^n} S_t(x_1, x_2; y_1, y_2) \Phi(y_1, y_2) dy_1 dy_2, $$

where $\sigma = \sqrt{\hbar/m}$. Using Eq.(0.7.28) one can express the kernel $S_t(x_1, x_2; y_1, y_2)$ of $S_t$, by the Brownian bridge $\alpha(s)$. The Brownian bridge is the Gaussian process defined on the interval $[0, 1]$ with the covariance

$$ \mathbb{E}[\alpha_k(s)\alpha_r(s')] = \delta_{kr}(1-s') \quad \text{iff } s \le s' \quad (0.7.29) $$

and the boundary conditions $\alpha(0) = \alpha(1) = 0$. Then the expectation value in the second line of Eq.(0.7.28) is equal to the expectation value
$$ (S_t \Phi)(x_1, x_2) = $$

$$ \mathbb{E} \left[ \exp \left( -\frac{i}{\hbar} \int_0^t d\tau Z \left( x_1 + \frac{\sigma\omega(\tau)}{\sqrt{2}}, x_2 + \frac{\sigma\omega(\tau)}{\sqrt{2}} \right) \times \Phi \left( x_1 + \frac{\sigma\omega(t)}{\sqrt{2}}, x_2 + \frac{\sigma\omega(t)}{\sqrt{2}} \right) \right) \right] = $$

$$ \int_{\mathbb{R}^n} \int_{\mathbb{R}^n} \mathbb{E} \left[ \delta \left( y_1 - x_1 - \frac{\sigma\omega(\tau)}{\sqrt{2}} \right) \delta \left( y_2 - x_2 - \frac{\sigma\omega(\tau)}{\sqrt{2}} \right) \times \right. \quad (0.7.28) $$

$$ \left. \exp \left( -\frac{i}{\hbar} \int_0^t d\tau \tilde{Z} \left( x_1 + \frac{\sigma\omega(\tau)}{\sqrt{2}}, x_2 + \frac{\sigma\omega(\tau)}{\sqrt{2}} \right) \right) \right] \times $$

$$ \Phi(y_1, y_2) dy_1 dy_2 = $$

$$ \int_{\mathbb{R}^n} \int_{\mathbb{R}^n} S_t(x_1, x_2; y_1, y_2) \Phi(y_1, y_2) dy_1 dy_2, $$

where $\sigma = \sqrt{\hbar/m}$. Using Eq.(0.7.28) one can express the kernel $S_t(x_1, x_2; y_1, y_2)$ of $S_t$, by the Brownian bridge $\alpha(s)$. The Brownian bridge is the Gaussian process defined on the interval $[0, 1]$ with the covariance

$$ \mathbb{E}[\alpha_k(s)\alpha_r(s')] = \delta_{kr}(1-s') \quad \text{iff } s \le s' \quad (0.7.29) $$

and the boundary conditions $\alpha(0) = \alpha(1) = 0$. Then the expectation value in the second line of Eq.(0.7.28) is equal to the expectation value
$$ 
\begin{gathered}
\mathbf{E} \left[ \delta \left( y_1 - x_1 - \frac{\sigma \omega(\tau)}{\sqrt{2}} \right) \delta \left( y_2 - x_2 - \frac{\sigma \omega(\tau)}{\sqrt{2}} \right) \right] = \\
(4\pi t \sigma^2)^{-n/2} \exp \left( - \frac{(x_1 - x_2 - y_1 + y_2)^2}{4t\sigma^2} \right) \delta(x_1 - x_2 - y_1 + y_2)
\end{gathered}
\qquad (0.7.30)
$$

times the expectation value of $\exp(-(i/\hbar) \int_0^t Z)$, where the Brownian motion is constrained to end at $y$. A solution of this constraint gives a simple formula for the kernel

$$ 
\begin{gathered}
S_t(x_1, x_2; y_1, y_2) = \\
(4\pi t \sigma^2)^{-n/2} \exp\left(-\frac{(x_1 - x_2 - y_1 + y_2)^2}{4t\sigma^2}\right) \delta(x_1 - x_2 - y_1 + y_2) \times \\
\mathbf{E} \left[ \exp\left(-\frac{i}{\hbar} \int_0^t d\tau Z \left(x_1\left(1-\frac{s}{t}\right) + y_1\left(\frac{s}{t}\right) + \sqrt{\frac{t}{2}} \sigma \mathbf{a}\left(\frac{s}{t}\right)\right) \right), \right. \\
\left. x_2\left(1-\frac{s}{t}\right) + y_2\left(\frac{s}{t}\right) + \sqrt{\frac{t}{2}} \sigma \mathbf{a}\left(\frac{s}{t}\right) \right) ds \right]
\end{gathered}
\qquad (0.7.31)
$$

Now we restrict our variables to functions of the form

$$ Z(x_1, x_2) = \tilde{Z}(x_1 + ix_2), \Phi(x_1, x_2) = \varphi(x_1 + ix_2), \qquad (0.7.32) $$

where $Z$ and $\varphi$ are holomorphic functions of their arguments. Substituting equation (0.7.32) into equation (0.7.28) we obtain
$$ 
\begin{gathered}
\mathbf{E} \left[ \delta \left( y_1 - x_1 - \frac{\sigma \omega(\tau)}{\sqrt{2}} \right) \delta \left( y_2 - x_2 - \frac{\sigma \omega(\tau)}{\sqrt{2}} \right) \right] = \\
(4\pi t \sigma^2)^{-n/2} \exp \left( - \frac{(x_1 - x_2 - y_1 + y_2)^2}{4t\sigma^2} \right) \delta(x_1 - x_2 - y_1 + y_2)
\end{gathered}
\qquad (0.7.30)
$$

times the expectation value of $\exp(-(i/\hbar) \int_0^t Z)$, where the Brownian motion is constrained to end at $y$. A solution of this constraint gives a simple formula for the kernel

$$ 
\begin{gathered}
S_t(x_1, x_2; y_1, y_2) = \\
(4\pi t \sigma^2)^{-n/2} \exp\left(-\frac{(x_1 - x_2 - y_1 + y_2)^2}{4t\sigma^2}\right) \delta(x_1 - x_2 - y_1 + y_2) \times \\
\mathbf{E} \left[ \exp\left(-\frac{i}{\hbar} \int_0^t d\tau Z\left(x_1\left(1 - \frac{s}{t}\right) + y_1\left(\frac{s}{t}\right) + \sqrt{\frac{t}{2}} \sigma \mathbf{a}\left(\frac{s}{t}\right)\right)\right), \right. \\
\left. x_2\left(1 - \frac{s}{t}\right) + y_2\left(\frac{s}{t}\right) + \sqrt{\frac{t}{2}} \sigma \mathbf{a}\left(\frac{s}{t}\right) \right) ds \right]
\end{gathered}
\qquad (0.7.31)
$$

Now we restrict our variables to functions of the form

$$ Z(x_1, x_2) = \tilde{Z}(x_1 + ix_2), \Phi(x_1, x_2) = \varphi(x_1 + ix_2), \qquad (0.7.32) $$

where $Z$ and $\varphi$ are holomorphic functions of their arguments. Substituting equation (0.7.32) into equation (0.7.28) we obtain
$$ (S_t \varphi)(x_1 + ix_2) = $$

$$ (4\pi t\sigma^2)^{-n/2} \int_{\mathbb{R}^n} dy E_t(y, x_1, x_2) \varphi\left(\frac{1}{2}(y+x_1+x_2) + \frac{i}{2}(y-x_1+x_2)\right) \times \quad (0.7.33) $$

$$ \exp\left(-\frac{(y - x_1 - x_2)^2}{4t\sigma^2}\right) $$

where by $E_t$ we have denoted $\mathbb{E}[\dots]$. We note that

$$ (4\pi t\sigma^2)^{-n/2} \int_{\mathbb{R}^n} dy \chi\left(\frac{1}{2}(y+x) + \frac{i}{2}(y-x)\right) \exp\left(-\frac{(x-y)^2}{4t\sigma^2}\right) = \quad (0.7.34) $$

$$ (4\pi t\sigma^2)^{-n/2} \int_{\mathbb{R}^n} dy \chi \exp\left(-\frac{(x-y)^2}{2it\sigma^2}\right) $$

(where $i^{n/2} = \exp(\frac{1}{2}in\pi)$) for any analytic function $\chi \in \mathcal{L}_1(\mathbb{R}^n)$ of the form
$\chi(y) = \int dp \chi^-(p) \exp(ipy)$. Finally, we set $x_2 = 0$ in equation (0.7.33) and apply the identity (0.7.34). We obtain the final formula for the Feynman propagator:

$$ K_t(x,y) = $$

$$ (4\pi t\sigma^2)^{-n/2} \exp\left(-\frac{(x-y)^2}{2it\sigma^2}\right) \times \quad (0.7.35) $$

$$ \mathbb{E} \left[ \exp \left( -\frac{i}{\hbar} \int_0^t d\tau \tilde{Z} \left( x \left( 1 - \frac{s}{t} \right) + y \left( \frac{s}{t} \right) + \lambda \sqrt{t} \sigma \alpha \left( \frac{s}{t} \right) \right) \right) \right] = $$

$$ K_t^0(x,y)\Re(t,x,y). $$

Here $\lambda = \sqrt{i}$ and we have denoted the free propagator by $K_t^0(x,y)$.
$$(S_t \varphi)(x_1 + ix_2) =$$

$$(4\pi t\sigma^2)^{-n/2} \int_{\mathbb{R}^n} dy E_t(y, x_1, x_2) \varphi\left(\frac{1}{2}(y+x_1+x_2) + \frac{i}{2}(y-x_1+x_2)\right) \times \quad (0.7.33)$$

$$\exp\left(-\frac{(y - x_1 - x_2)^2}{4t\sigma^2}\right)$$

where by $E_t$ we have denoted $E[\dots]$. We note that

$$(4\pi t\sigma^2)^{-n/2} \int_{\mathbb{R}^n} dy \chi\left(\frac{1}{2}(y+x) + \frac{i}{2}(y-x)\right) \exp\left(-\frac{(x-y)^2}{4t\sigma^2}\right) = \quad (0.7.34)$$

$$(4\pi t\sigma^2)^{-n/2} \int_{\mathbb{R}^n} dy \chi \exp\left(-\frac{(x-y)^2}{2it\sigma^2}\right)$$

(where $i^{n/2} = \exp(\frac{1}{2}in\pi)$) for any analytic function $\chi \in \mathcal{L}_1(\mathbb{R}^n)$ of the form
$\chi(y) = \int dp \chi^-(p) \exp(ipy)$. Finally, we set $x_2 = 0$ in equation (0.7.33) and apply the identity (0.7.34). We obtain the final formula for the Feynman propagator:

$$K_t(x,y) =$$

$$(4\pi t\sigma^2)^{-n/2} \exp\left(-\frac{(x-y)^2}{2it\sigma^2}\right) \times \quad (0.7.35)$$

$$\mathbf{E} \left[ \exp \left( -\frac{i}{\hbar} \int_0^t d\tau \tilde{Z} \left( x \left( 1 - \frac{s}{t} \right) + y \left( \frac{s}{t} \right) + \lambda \sqrt{t} \sigma \alpha \left( \frac{s}{t} \right) \right) \right) \right] =$$

$$K_t^0(x,y)\Re(t,x,y).$$

Here $\lambda = \sqrt{i}$ and we have denoted the free propagator by $K_t^0(x,y)$.
**Theorem 0.7.4.** (Haba [44]). Define

$$ \mathfrak{R}_t^R(t,x,y) = \mathbf{E} \left[ \exp \left\{ \frac{g}{\hbar \lambda^2} \int_0^t ds \theta_R \left( \alpha \left( \frac{s}{t} \right) \right) Z \left( x \left( 1 - \frac{s}{t} \right) + y \frac{s}{t} + \lambda \sigma \right) \right\} \right], \quad (0.7.36) $$

where $Z \in \mathcal{F}_n^\triangle$. The limit $R \to \infty$ exists uniformly in $t,x,y$. $\mathfrak{R}(t,x,y)$, is an analytic function

of $g$ and $\lambda$ if $\mathrm{Re}\,\lambda^2 \ge 0$. Formula $K_t(x,y) = K_t^0(x,y) \lim_{R \to \infty} \mathfrak{R}_t^R(t,x,y)$ gives the Feynman propagator (0.7.35) for any $Z \in \mathcal{F}_n^\triangle$.

**Definition 0.7.3.** Let $G_{\mathcal{F}_d^\triangle}(\mathbb{R}^d)$ denote the Colombeau module over $\tilde{\mathbb{C}}$ of Colombeau generalized functions which can be represented of the form

$$ (g_{\varepsilon}(x))_{\varepsilon} = \left( \int_{\mathbb{R}^d} \exp(ipx) d\mu_{\varepsilon}(p; \mathbf{cl}[(g_{\varepsilon}(x))_{\varepsilon}]) \right)_{\varepsilon} = \int_{\mathbb{R}^d} \exp(ipx) (d\mu_{\varepsilon}(p; \mathbf{cl}[(g_{\varepsilon}(x))_{\varepsilon}]))_{\varepsilon}. \quad (0.7.37) $$

where $(\mu_{\varepsilon}(p; \mathbf{cl}[(g_{\varepsilon}(x))_{\varepsilon}]))_{\varepsilon}$ is a complex Colombeau measure on $\mathbb{R}^d$ with a variation
$$ (|\mu_{\varepsilon}|(p; \mathbf{cl}[(g_{\varepsilon}(x))_{\varepsilon}]))_{\varepsilon}. $$

**Definition 0.7.4.** Let $G_{\mathcal{F}_d^\triangle}(\mathbb{R}^n)$ denote Colombeau module over $\tilde{\mathbb{C}}$ of Colombeau generalized functions which can be represented of the form (0.7.37) with $\mu_{g,\varepsilon}(p) = (\mu_{\varepsilon}(p; \mathbf{cl}[(g_{\varepsilon}(x))_{\varepsilon}]))_{\varepsilon}, g = \mathbf{cl}[(g_{\varepsilon}(x))_{\varepsilon}]$ such that

$$ \begin{gathered} \forall \epsilon(\epsilon > 0) \exists K(\epsilon) \exists N(N \in \mathbb{N}) \forall a(a \in \mathbb{R}) \\ \forall \epsilon(\epsilon \in (0,1]) \left[ \int_{\mathbb{R}^d} \exp(a|p|) d|\mu_{g,\epsilon}|(p) \le \epsilon^{-N} K(\epsilon) \exp(\epsilon a^2) \right]. \end{gathered} \quad (0.7.38) $$
**Theorem 0.7.4.** (Haba [44]). Define

$$ \Re_t^R(t, x, y) = \mathbf{E} \left[ \exp \left\{ \frac{g}{\hbar \lambda^2} \int_0^t ds \theta_R \left( \alpha \left( \frac{s}{t} \right) \right) Z \left( x \left( 1 - \frac{s}{t} \right) + y \frac{s}{t} + \lambda \sigma \right) \right\} \right], \quad (0.7.36) $$

where $Z \in \mathcal{F}_n^\triangle$. The limit $R \to \infty$ exists uniformly in $t, x, y$. $\Re(t, x, y)$, is an analytic function

of $g$ and $\lambda$ if $\mathrm{Re}\,\lambda^2 \ge 0$. Formula $K_t(x, y) = K_t^0(x, y) \lim_{R \to \infty} \Re_t^R(t, x, y)$ gives the Feynman propagator (0.7.35) for any $Z \in \mathcal{F}_n^\triangle$.

**Definition 0.7.3.** Let $G_{\mathcal{F}_d^\triangle}(\mathbb{R}^d)$ denote the Colombeau module over $\tilde{\mathbb{C}}$ of Colombeau generalized functions which can be represented of the form

$$ (g_{\varepsilon}(x))_{\varepsilon} = \left( \int_{\mathbb{R}^d} \exp(ipx) d\mu_{\varepsilon}(p; \mathbf{cl}[(g_{\varepsilon}(x))_{\varepsilon}]) \right)_{\varepsilon} = \int_{\mathbb{R}^d} \exp(ipx) (d\mu_{\varepsilon}(p; \mathbf{cl}[(g_{\varepsilon}(x))_{\varepsilon}]))_{\varepsilon}. \quad (0.7.37) $$

where $(\mu_{\varepsilon}(p; \mathbf{cl}[(g_{\varepsilon}(x))_{\varepsilon}]))_{\varepsilon}$ is a complex Colombeau measure on $\mathbb{R}^d$ with a variation
$$ (|\mu_{\varepsilon}|(p; \mathbf{cl}[(g_{\varepsilon}(x))_{\varepsilon}]))_{\varepsilon}. $$

**Definition 0.7.4.** Let $G_{\mathcal{F}_d^\triangle}(\mathbb{R}^n)$ denote Colombeau module over $\tilde{\mathbb{C}}$ of Colombeau generalized functions which can be represented of the form (0.7.37) with $\mu_{g,\varepsilon}(p) = (\mu_{\varepsilon}(p; \mathbf{cl}[(g_{\varepsilon}(x))_{\varepsilon}]))_{\varepsilon}, g = \mathbf{cl}[(g_{\varepsilon}(x))_{\varepsilon}]$ such that

$$ \begin{gathered} \forall \epsilon(\epsilon > 0) \exists K(\epsilon) \exists N(N \in \mathbb{N}) \forall a(a \in \mathbb{R}) \\ \forall \epsilon(\epsilon \in (0,1]) \left[ \int_{\mathbb{R}^d} \exp(a|p|) d|\mu_{g,\epsilon}|(p) \le \epsilon^{-N} K(\epsilon) \exp(\epsilon a^2) \right]. \end{gathered} \quad (0.7.38) $$
Let us consider now a complex extension of the Colombeau-Schrödinger equation in the form

$$ \left( \frac{\partial \psi_{\varepsilon}(t, x)}{\partial t} \right)_{\varepsilon} = \frac{1}{2} \sigma^2 \lambda^2 (\Delta \psi_{\varepsilon}(t, x))_{\varepsilon} + g \hbar^{-1} \lambda^{-2} ((Z_{\varepsilon}(x))_{\varepsilon}) ((\psi_{\varepsilon}(t, x))_{\varepsilon}), \quad (0.7.39) $$

$$ (\psi_{\varepsilon}(0, x))_{\varepsilon} = (\psi_{0, \varepsilon}(x))_{\varepsilon}, x \in \mathbb{R}^{d}. $$

Here $\lambda \in \mathbb{C}, \sigma = \sqrt{\hbar m^{-1}}$. Schrödinger equation corresponds to $\lambda = \sqrt{i} = (1 + i)/\sqrt{2}$.

We wish to express the Colombeau-Feynman path integral corresponding to Eq.(0.7.39) using Wiener measure corresponding to Wiener process

$\mathbf{b}(\tau) \triangleq \mathbf{b}_{\tau} \in \mathbb{R}^d$, Here $\mathbf{b}_{\tau} \in \mathbb{R}^d$ is the Gaussian process with the covariance given by

Eq.(0.7.18).We use the notation

$$ (\mathbf{E}_{\varepsilon}[F_{\varepsilon}])_{\varepsilon} \triangleq \left( \int_{\Omega_{\varepsilon}} d\mu_{t, \varepsilon}^{W}(\omega) F_{\varepsilon}(\omega) \right)_{\varepsilon}, \quad (0.7.40) $$

where $\mu_{t, \varepsilon}^{W}(\omega)$ is the Colombeau-Wiener measure and the Colombeau integral is over the Wiener space $\Omega_{\varepsilon} = \mathbb{C}([0, t])$ which consists of continuous functions defined on the interval $[0, t]$ with $\omega(0) = 0$.

Let us denote
Let us consider now a complex extension of the Colombeau-Schrödinger equation in the form

$$ \left( \frac{\partial \psi_{\varepsilon}(t, x)}{\partial t} \right)_{\varepsilon} = \frac{1}{2} \sigma^2 \lambda^2 (\Delta \psi_{\varepsilon}(t, x))_{\varepsilon} + g \hbar^{-1} \lambda^{-2} ((Z_{\varepsilon}(x))_{\varepsilon}) ((\psi_{\varepsilon}(t, x))_{\varepsilon}), \quad (0.7.39) $$

$$ (\psi_{\varepsilon}(0, x))_{\varepsilon} = (\psi_{0, \varepsilon}(x))_{\varepsilon}, x \in \mathbb{R}^{d}. $$

Here $\lambda \in \mathbb{C}, \sigma = \sqrt{\hbar m^{-1}}$. Schrödinger equation corresponds to $\lambda = \sqrt{i} = (1 + i)/\sqrt{2}$.

We wish to express the Colombeau-Feynman path integral corresponding to Eq.(0.7.39) using Wiener measure corresponding to Wiener process $b(\tau) \triangleq b_{\tau} \in \mathbb{R}^d$, Here $b_{\tau} \in \mathbb{R}^d$ is the Gaussian process with the covariance given by

Eq.(0.7.18).We use the notation

$$ (\mathbf{E}_{\varepsilon}[F_{\varepsilon}])_{\varepsilon} \triangleq \left( \int_{\Omega_t} d\mu_{t,\varepsilon}^{W}(\omega) F_{\varepsilon}(\omega) \right)_{\varepsilon}, \quad (0.7.40) $$

where $\mu_{t,\varepsilon}^W(\omega)$ is the Colombeau-Wiener measure and the Colombeau integral is over the Wiener space $\Omega_t = C([0,t])$ which consists of continuous functions defined on the interval $[0,t]$ with $\omega(0) = 0$.

Let us denote
$$ \theta_R(x) = \exp\left(\frac{|x^2|}{2R}\right), $$

$$ (\Theta_{t,\varepsilon}[g\theta_R(\omega(\tau)); Z_{\varepsilon}(x + \lambda\sigma\omega(\tau))])_{\varepsilon} = \qquad (0.7.41) $$

$$ \exp\left\{ \frac{g}{\hbar\lambda^2} \left( \int_0^t d\tau [\theta_R(\omega(\tau))Z_{\varepsilon}(x + \lambda\sigma\omega(\tau))] \right)_{\varepsilon} \right\}. $$

We shall also use a shorthand notation $\Theta_{t,\varepsilon}[g\theta_R; Z]$ for the Colombeau quantity $(\Theta_{t,\varepsilon}[g\theta_R(\omega(\tau)); Z_{\varepsilon}(x + \lambda\sigma\omega(\tau))])_{\varepsilon}$.

**Theorem 0.7.5.** Assume that $cl[(\psi_{0,\varepsilon})_{\varepsilon}, cl[(Z_{\varepsilon})_{\varepsilon}] \in G_{\mathcal{F}_d}(\mathbb{R}^d)$. Then, $\forall \varepsilon \in (0,1]$, $\Theta_{t,\varepsilon}[g\theta_R; Z_{\varepsilon}]$ is an analytic function of $g \in \mathbb{C}$ and $\lambda \neq 0$ as long as $\text{Re}\,\lambda^2 \geq 0$. Define

$$ (\psi_{\varepsilon}^{R}(t,x))_{\varepsilon} = (E_{\varepsilon}[\Theta_{t,\varepsilon}[g\theta_{R}(\omega(\tau)); Z_{\varepsilon}(x + \lambda\sigma\omega(\tau))]\psi_{0,\varepsilon}(x + \lambda\sigma\omega(\tau))])_{\varepsilon} \quad (0.7.42) $$

Then, limit

$$ \psi_{\varepsilon}(t,x) = \lim_{R \to \infty} \psi_{\varepsilon}^{R}(t,x) \qquad (0.7.43) $$

exists uniformly in $t$ and $x$. Function $(\psi_{\varepsilon}(t,x))_{\varepsilon}$ solves the Colombeau-Schrödinger equation (0.7.39). The solution is an analytic function of $g$ and $\lambda$ in the same region where $E[\Theta_t]$ is analytic. Let $(Z_{\varepsilon}(t,x))_{\varepsilon}$ be Colombeau generalized function

$$ (Z_{\varepsilon}(t,x))_{\varepsilon} = \frac{1}{2}\omega_{\varepsilon}(t)x^2 + \vartheta_{\varepsilon}(t)x + g(Z_{\varepsilon}(x)), \qquad (0.7.44) $$

where $cl[(Z_{\varepsilon}(x))_{\varepsilon}] \in G_{\mathcal{F}_d}(\mathbb{R}^d)$ and $\forall \varepsilon \in (0,1]$, $\omega_{\varepsilon}(t), \vartheta_{\varepsilon}(t)$ are continuous functions of time.

**Theorem 0.7.6.** Assume that $\omega_{\varepsilon}(t) \leq \omega_{0,\varepsilon}$ for each $t$. If $t < \pi/\omega_{0,\varepsilon}$ then theorem 0.7.5 holds true for potential given by Eq.(0.7.44).

Let us consider a decomposition of $\mathbb{R}^{2n}$ into $\mathbb{R}^n \oplus \mathbb{R}^n$ where
$$ \theta_R(x) = \exp\left(\frac{|x^2|}{2R}\right), $$

$$ (\Theta_{t,\varepsilon}[g\theta_R(\omega(\tau)); Z_{\varepsilon}(x + \lambda\sigma\omega(\tau))])_{\varepsilon} = \qquad (0.7.41) $$

$$ \exp\left\{ \frac{g}{\hbar\lambda^2} \left( \int_0^t d\tau [\theta_R(\omega(\tau))Z_{\varepsilon}(x + \lambda\sigma\omega(\tau))] \right)_{\varepsilon} \right\}. $$

We shall also use a shorthand notation $\Theta_{t,\varepsilon}[g\theta_R; Z]$ for the Colombeau quantity $(\Theta_{t,\varepsilon}[g\theta_R(\omega(\tau)); Z_{\varepsilon}(x + \lambda\sigma\omega(\tau))])_{\varepsilon}$.

**Theorem 0.7.5.** Assume that $\mathrm{cl}[(\psi_{0,\varepsilon})_{\varepsilon}, \mathrm{cl}[(Z_{\varepsilon})_{\varepsilon}] \in G_{\mathcal{F}\triangle}(\mathbb{R}^d)$. Then, $\forall \varepsilon \in (0,1]$, $\Theta_{t,\varepsilon}[g\theta_R; Z_{\varepsilon}]$ is an analytic function of $g \in \mathbb{C}$ and $\lambda \neq 0$ as long as $\mathrm{Re}\,\lambda^2 \geq 0$. Define

$$ (\psi_{\varepsilon}^{R}(t,x))_{\varepsilon} = (\mathbf{E}_{\varepsilon}[\Theta_{t,\varepsilon}[g\theta_{R}(\omega(\tau)); Z_{\varepsilon}(x + \lambda\sigma\omega(\tau))]\psi_{0,\varepsilon}(x + \lambda\sigma\omega(\tau))])_{\varepsilon} \quad (0.7.42) $$

Then, limit

$$ \psi_{\varepsilon}(t,x) = \lim_{R \to \infty} \psi_{\varepsilon}^{R}(t,x) \qquad (0.7.43) $$

exists uniformly in $t$ and $x$. Function $(\psi_{\varepsilon}(t,x))_{\varepsilon}$ solves the Colombeau-Schrödinger equation (0.7.39). The solution is an analytic function of $g$ and $\lambda$ in the same region where $\mathrm{E}[\Theta_t]$ is analytic. Let $(Z_{\varepsilon}(t,x))_{\varepsilon}$ be Colombeau generalized function

$$ (Z_{\varepsilon}(t,x))_{\varepsilon} = \frac{1}{2}\omega_{\varepsilon}(t)x^2 + \vartheta_{\varepsilon}(t)x + g(Z_{\varepsilon}(x)), \qquad (0.7.44) $$

where $\mathrm{cl}[(Z_{\varepsilon}(x))_{\varepsilon}] \in G_{\mathcal{F}\triangle}(\mathbb{R}^d)$ and $\forall \varepsilon \in (0,1]$, $\omega_{\varepsilon}(t), \vartheta_{\varepsilon}(t)$ are continuous functions of time.

**Theorem 0.7.6.** Assume that $\omega_{\varepsilon}(t) \le \omega_{0,\varepsilon}$ for each $t$. If $t < \pi/\omega_{0,\varepsilon}$ then theorem 0.7.5 holds true for potential given by Eq.(0.7.44).

Let us consider a decomposition of $\mathbb{R}^{2n}$ into $\mathbb{R}^n \oplus \mathbb{R}^n$ where
$z = (x_1,x_2), x_1,x_2 \in \mathbb{R}^n$. Let
$G_z$ be a subset $G_z \subset \mathbb{R}^{2n}$

$$
G_z = \{w \in \mathbb{R}^{2n} | w = (x_1 + y, x_2 + y), y \in \mathbb{R}^n\} \quad (0.7.45)
$$

and let $Z_{\varepsilon}(x_1,x_2)$ be a complex-valued potential defined on $\mathbb{R}^{2n}$ such that $\forall \varepsilon \in (0,1]$ on $G_z$

$$
|\operatorname{Im} Z_{\varepsilon}(x_1 + y, x_2 + y)| \leq A_{\varepsilon}(x_1, x_2)(1 + |y|) \quad (0.7.46)
$$

with a certain continuous non-negative function $A_\varepsilon$. We define Colombeau
semigroup $S_{\ell,\varepsilon}$ on the set of functions defined on $\mathbb{R}^{2n}$ such that on $G_z$

$$
|\Phi_{\varepsilon}(x_1 + y, x_2 + y)| \leq M_{\varepsilon}(x_1, x_2) \exp(|y|C_{\varepsilon}(x_1, x_2)) \quad (0.7.47)
$$

with certain non-negative continuous functions $C_\epsilon(x_1,x_2)$ and $M_\epsilon(x_1,x_2)$. Let $(S_{\ell,\epsilon})_\epsilon$, is defined by the Feynman-Kac formula:
$z = (x_1,x_2), x_1,x_2 \in \mathbb{R}^n$. Let
$G_z$ be a subset $G_z \subset \mathbb{R}^{2n}$

$$
G_z = \{w \in \mathbb{R}^{2n} | w = (x_1 + y, x_2 + y), y \in \mathbb{R}^n\} \quad (0.7.45)
$$

and let $Z_{\varepsilon}(x_1,x_2)$ be a complex-valued potential defined on $\mathbb{R}^{2n}$ such that $\forall \varepsilon \in (0,1]$ on $G_z$

$$
|\operatorname{Im} Z_{\varepsilon}(x_1 + y, x_2 + y)| \leq A_{\varepsilon}(x_1, x_2)(1 + |y|) \quad (0.7.46)
$$

with a certain continuous non-negative function $A_\varepsilon$. We define Colombeau
semigroup $S_{\ell,\varepsilon}$ on the set of functions defined on $\mathbb{R}^{2n}$ such that on $G_z$

$$
|\Phi_{\varepsilon}(x_1 + y, x_2 + y)| \leq M_{\varepsilon}(x_1, x_2) \exp(|y|C_{\varepsilon}(x_1, x_2)) \quad (0.7.47)
$$

with certain non-negative continuous functions $C_{\varepsilon}(x_1,x_2)$ and $M_{\varepsilon}(x_1,x_2)$. Let $(S_{\ell,\varepsilon})_{\varepsilon}$, is defined by the Feynman-Kac formula:
$$((\mathbf{S}_{t,\varepsilon}\Phi_{\varepsilon})(x_1,x_2))_{\varepsilon} =$$

$$\left( \mathbf{E}_{\varepsilon} \left[ \exp \left( -\frac{i}{\hbar} \int_{0}^{t} d\tau Z_{\varepsilon} \left( x_1 + \frac{\sigma\omega(\tau)}{\sqrt{2}}, x_2 + \frac{\sigma\omega(\tau)}{\sqrt{2}} \right) \times \Phi_{\varepsilon} \left( x_1 + \frac{\sigma\omega(t)}{\sqrt{2}}, x_2 + \frac{\sigma\omega(t)}{\sqrt{2}} \right) \right) \right] \right)_{\varepsilon} =$$

$$\left( \int_{\mathbb{R}^n} \int_{\mathbb{R}^n} \mathbf{E}_{\varepsilon} \left[ \delta \left( y_1 - x_1 - \frac{\sigma\omega(\tau)}{\sqrt{2}} \right) \delta \left( y_2 - x_2 - \frac{\sigma\omega(\tau)}{\sqrt{2}} \right) \times \right. \right. \quad (0.7.48)$$

$$\exp\left(-\frac{i}{\hbar} \int_{0}^{t} d\tau \tilde{Z}\left(x_{1}+\frac{\sigma \omega(\tau)}{\sqrt{2}}, x_{2}+\frac{\sigma \omega(\tau)}{\sqrt{2}}\right)\right] \Phi_{\varepsilon}(y_{1}, y_{2}) d y_{1} d y_{2})_{\varepsilon} =$$

$$\left( \int_{\mathbb{R}^n} \int_{\mathbb{R}^n} S_{t,\varepsilon}(x_1,x_2;y_1,y_2) \Phi_{\varepsilon}(y_1,y_2) dy_1 dy_2 \right)_{\varepsilon},$$

where $\sigma = \sqrt{\hbar/m}$. Using Eq.(0.7.48) one can express the kernel $(S_{t,\varepsilon}(x_1,x_2;y_1,y_2))_{\varepsilon}$ of $(S_{t,\varepsilon})_{\varepsilon}$, by the Brownian bridge $\alpha(s)$. The Brownian bridge is the Gaussian process defined on the interval $[0, 1]$ with the covariance

$$\mathbf{E}[\alpha_k(s)\alpha_r(s')] = \delta_{kr}(1-s') \quad \text{iff } s \le s' \qquad (0.7.49)$$

and the boundary conditions $\alpha(0) = \alpha(1) = 0$. Then the expectation value in the second line of Eq.(0.7.48) is equal to the expectation value
$$((S_{t,\varepsilon}\Phi_{\varepsilon})(x_1,x_2))_{\varepsilon} =$$

$$\left( \mathbf{E}_{\varepsilon} \left[ \exp \left( -\frac{i}{\hbar} \int_{0}^{t} d\tau Z_{\varepsilon} \left( x_1 + \frac{\sigma\omega(\tau)}{\sqrt{2}}, x_2 + \frac{\sigma\omega(\tau)}{\sqrt{2}} \right) \times \Phi_{\varepsilon} \left( x_1 + \frac{\sigma\omega(t)}{\sqrt{2}}, x_2 + \frac{\sigma\omega(t)}{\sqrt{2}} \right) \right) \right] \right)_{\varepsilon} =$$

$$\left( \int_{\mathbb{R}^n} \int_{\mathbb{R}^n} \mathbf{E}_{\varepsilon} \left[ \delta \left( y_1 - x_1 - \frac{\sigma\omega(\tau)}{\sqrt{2}} \right) \delta \left( y_2 - x_2 - \frac{\sigma\omega(\tau)}{\sqrt{2}} \right) \times \exp \left( -\frac{i}{\hbar} \int_{0}^{t} d\tau \tilde{Z} \left( x_1 + \frac{\sigma\omega(\tau)}{\sqrt{2}}, x_2 + \frac{\sigma\omega(\tau)}{\sqrt{2}} \right) \right) \right] \Phi_{\varepsilon}(y_1, y_2) dy_1 dy_2 \right)_{\varepsilon} = \quad (0.7.48)$$

$$\left( \int_{\mathbb{R}^n} \int_{\mathbb{R}^n} S_{t,\varepsilon}(x_1,x_2;y_1,y_2) \Phi_{\varepsilon}(y_1,y_2) dy_1 dy_2 \right)_{\varepsilon},$$

where $\sigma = \sqrt{\hbar/m}$. Using Eq.(0.7.48) one can express the kernel $(S_{t,\varepsilon}(x_1,x_2;y_1,y_2))_{\varepsilon}$ of $(S_{t,\varepsilon})_{\varepsilon}$, by the Brownian bridge $\alpha(s)$. The Brownian bridge is the Gaussian process defined on the interval $[0,1]$ with the covariance

$$\mathbf{E}[\alpha_k(s)\alpha_r(s')] = \delta_{kr}(1-s') \quad \text{iff } s \le s' \qquad (0.7.49)$$

and the boundary conditions $\alpha(0) = \alpha(1) = 0$. Then the expectation value in the second line of Eq.(0.7.48) is equal to the expectation value
$$ 
\begin{aligned}
& \mathbf{E}_{\varepsilon} \left[ \delta \left( y_1 - x_1 - \frac{\sigma \omega(\tau)}{\sqrt{2}} \right) \delta \left( y_2 - x_2 - \frac{\sigma \omega(\tau)}{\sqrt{2}} \right) \right] = \\
& \qquad (4\pi t \sigma^2)^{-n/2} \exp \left( - \frac{(x_1 - x_2 - y_1 + y_2)^2}{4t\sigma^2} \right) \delta(x_1 - x_2 - y_1 + y_2)
\end{aligned} 
\qquad (0.7.50) 
$$

times the expectation value of $\exp(-(i/h) \int_0^t d\tau (Z_\varepsilon(\tau))_\varepsilon)$, where the Brownian motion is constrained to end at y. A solution of this constraint gives a simple formula for the kernel

$$ 
\begin{aligned}
(S_{t,\varepsilon}(x_1,x_2;y_1,y_2))_{\varepsilon} = & \\
& (4\pi t \sigma^2)^{-n/2} \exp\left(-\frac{(x_1-x_2-y_1+y_2)^2}{4t\sigma^2}\right) \delta(x_1-x_2-y_1+y_2) \times \\
& \left( \mathbf{E} \left[ \exp\left(-\frac{i}{\hbar} \int_0^t d\tau Z_\varepsilon \left(x_1\left(1-\frac{s}{t}\right)\right) + y_1\left(\frac{s}{t}\right) + \sqrt{\frac{t}{2}} \sigma \mathbf{a}\left(\frac{s}{t}\right) \right), \right. \\
& \left. x_2\left(1-\frac{s}{t}\right) + y_2\left(\frac{s}{t}\right) + \sqrt{\frac{t}{2}} \sigma \mathbf{a}\left(\frac{s}{t}\right) \right) ds \right] \right)_{\varepsilon}
\end{aligned} 
\qquad (0.7.51) 
$$

Now we restrict our sevles to functions of the form

$$ (Z_{\varepsilon}(x_1, x_2))_{\varepsilon} = (\tilde{Z}_{\varepsilon}(x_1 + ix_2))_{\varepsilon}, (\Phi_{\varepsilon}(x_1, x_2))_{\varepsilon} = (\varphi_{\varepsilon}(x_1 + ix_2))_{\varepsilon}, \qquad (0.7.52) $$

where $\forall \varepsilon \in (0,1]$, $Z_\varepsilon$ and $\varphi_\varepsilon$ are holomorphic functions of their arguments. Substituting equation (0.7.52) into equation (0.7.48) we obtain
$$ 
\begin{aligned}
& \mathbf{E}_{\varepsilon} \left[ \delta \left( y_1 - x_1 - \frac{\sigma \omega(\tau)}{\sqrt{2}} \right) \delta \left( y_2 - x_2 - \frac{\sigma \omega(\tau)}{\sqrt{2}} \right) \right] = \\
& \qquad (4\pi t \sigma^2)^{-n/2} \exp \left( - \frac{(x_1 - x_2 - y_1 + y_2)^2}{4t\sigma^2} \right) \delta(x_1 - x_2 - y_1 + y_2)
\end{aligned}
\qquad (0.7.50)
$$

times the expectation value of $\exp(-(i/\hbar) \int_0^t d\tau (Z_\varepsilon(\tau))_\varepsilon)$, where the Brownian motion is constrained to end at $y$. A solution of this constraint gives a simple formula for the kernel

$$ 
\begin{aligned}
(S_{t,\varepsilon}(x_1,x_2;y_1,y_2))_\varepsilon = & \\
& (4\pi t \sigma^2)^{-n/2} \exp\left(-\frac{(x_1 - x_2 - y_1 + y_2)^2}{4t\sigma^2}\right) \delta(x_1 - x_2 - y_1 + y_2) \times \\
& \left( \mathbf{E} \left[ \exp\left(-\frac{i}{\hbar} \int_0^t d\tau Z_\varepsilon \left(x_1\left(1-\frac{s}{t}\right)\right) + y_1\left(\frac{s}{t}\right) + \sqrt{\frac{t}{2}} \sigma \mathbf{a}\left(\frac{s}{t}\right)\right), \right. \\
& \left. x_2\left(1-\frac{s}{t}\right) + y_2\left(\frac{s}{t}\right) + \sqrt{\frac{t}{2}} \sigma \mathbf{a}\left(\frac{s}{t}\right) \right) ds \right] \right)_\varepsilon
\end{aligned}
\qquad (0.7.51)
$$

Now we restrict our sevles to functions of the form

$$ (Z_{\varepsilon}(x_1, x_2))_{\varepsilon} = (\tilde{Z}_{\varepsilon}(x_1 + ix_2))_{\varepsilon}, \quad (\Phi_{\varepsilon}(x_1, x_2))_{\varepsilon} = (\varphi_{\varepsilon}(x_1 + ix_2))_{\varepsilon}, \qquad (0.7.52) $$

where $\forall \varepsilon \in (0,1]$, $Z_\varepsilon$ and $\varphi_\varepsilon$ are holomorphic functions of their arguments. Substituting equation (0.7.52) into equation (0.7.48) we obtain
$$((\mathbf{S}_{t,\varepsilon}\varphi_{\varepsilon})(x_1 + ix_2))_{\varepsilon} =$$

$$(4\pi t\sigma^2)^{-n/2} \left( \int_{\mathbb{R}^n} dy E_{t,\varepsilon}(y, x_1, x_2) \varphi_{\varepsilon} \left( \frac{1}{2}(y+x_1+x_2) + \frac{i}{2}(y-x_1+x_2) \right) \times \right. \quad (0.7.53)$$

$$\left. \exp\left(-\frac{(y-x_1-x_2)^2}{4t\sigma^2}\right) \right)_{\varepsilon}$$

where by $E_{t,\varepsilon}$ we have denoted $E[\dots]$. We note that

$$(4\pi t\sigma^2)^{-n/2} \int_{\mathbb{R}^n} dy \chi \left( \frac{1}{2}(y+x) + \frac{i}{2}(y-x) \right) \exp \left( -\frac{(x-y)^2}{4t\sigma^2} \right) = \quad (0.7.54)$$

$$(4\pi t\sigma^2)^{-n/2} \int_{\mathbb{R}^n} dy \chi \exp\left(-\frac{(x-y)^2}{2it\sigma^2}\right)$$

(where $i^{n/2} = \exp(\frac{1}{2}in\pi)$) for any analytic function $\chi_{\varepsilon} \in \mathcal{L}_1(\mathbb{R}^n)$ of the form

$\forall \varepsilon \in (0,1], \chi_{\varepsilon}(y) = \int dp \chi_{\varepsilon}(p) \exp(ipy)$. Finally, we set $x_2 = 0$ in equation (0.7.53) and apply the identity (0.7.54). We obtain the final formula for the generalized Feynman propagator:
$$((\mathbf{S}_{t,\epsilon}\varphi_{\epsilon})(x_1 + ix_2))_{\epsilon} =$$

$$(4\pi t\sigma^2)^{-n/2} \left( \int_{\mathbb{R}^n} dy E_{t,\epsilon}(y, x_1, x_2) \varphi_{\epsilon} \left( \frac{1}{2}(y+x_1+x_2) + \frac{i}{2}(y-x_1+x_2) \right) \times \right. \quad (0.7.53)$$

$$\left. \exp\left(-\frac{(y-x_1-x_2)^2}{4t\sigma^2}\right) \right)_{\epsilon}$$

where by $E_{t,\epsilon}$ we have denoted $E[\dots]$. We note that

$$(4\pi t\sigma^2)^{-n/2} \int_{\mathbb{R}^n} dy \chi \left( \frac{1}{2}(y+x) + \frac{i}{2}(y-x) \right) \exp \left( -\frac{(x-y)^2}{4t\sigma^2} \right) = \quad (0.7.54)$$

$$(4\pi t\sigma^2)^{-n/2} \int_{\mathbb{R}^n} dy \chi \exp\left(-\frac{(x-y)^2}{2it\sigma^2}\right)$$

(where $i^{n/2} = \exp(\frac{1}{2}in\pi)$) for any analytic function $\chi_{\epsilon} \in \mathcal{L}_1(\mathbb{R}^n)$ of the form

$\forall \epsilon \in (0,1], \chi_{\epsilon}(y) = \int dp \chi_{\epsilon}(p) \exp(ipy)$. Finally, we set $x_2 = 0$ in equation (0.7.53) and apply the identity (0.7.54). We obtain the final formula for the generalized Feynman propagator:
$$ (K_{t,\varepsilon}(x,y))_{\varepsilon} = $$

$$ (4\pi t\sigma^2)^{-n/2} \exp\left(-\frac{(x-y)^2}{2it\sigma^2}\right) \times $$

$$ \left( \mathbf{E} \left[ \exp \left( -\frac{i}{\hbar} \int_0^t d\tau \tilde{Z}_\varepsilon \left( x\left(1-\frac{s}{t}\right) + y\left(\frac{s}{t}\right) + \lambda \sqrt{t} \sigma \mathbf{a}\left(\frac{s}{t}\right) \right) \right) \right] \right)_\varepsilon = \quad (0.7.55) $$

$$ K_t^0(x,y)(\mathfrak{R}_\varepsilon(t,x,y))_\varepsilon. $$

Here $\lambda = \sqrt{i}$ and we have denoted the free propagator by $K_t^0(x,y)$. **Theorem 0.7.7.** Define

$$ (\mathfrak{R}_{t,\varepsilon}^R(t,x,y))_\varepsilon = $$

$$ \mathbf{E} \left[ \exp \left\{ \frac{g}{\hbar \lambda^2} \int_0^t ds \theta_R \left( \alpha \left( \frac{s}{t} \right) \right) Z_\varepsilon \left( x \left( 1 - \frac{s}{t} \right) + y \frac{s}{t} + \lambda \sigma \right) \right\} \right], \quad (0.7.56) $$

where $(Z_\varepsilon)_\varepsilon \in G_{\mathcal{F}_n^\uparrow}(\mathbb{R}^d)$. The limit $R \to \infty$ exists $\forall \varepsilon \in (0,1]$ uniformly in $t,x,y$. $\forall \varepsilon \in (0,1]$, $\mathfrak{R}_\varepsilon(t,x,y)$, is an analytic function of $g$ and $\lambda$ if $\mathrm{Re}\,\lambda^2 \ge 0$. Formula

$$ (K_{t,\varepsilon}(x,y))_{\varepsilon} = K_t^0(x,y)\left(\lim_{R\to\infty} \mathfrak{R}_{t,\varepsilon}^R(t,x,y)\right)_{\varepsilon} \quad (0.7.57) $$

gives the Feynman propagator (0.7.55) for any $(Z_\varepsilon)_\varepsilon \in G_{\mathcal{F}_n^\uparrow}(\mathbb{R}^d)$.

### I.0.8. Colombeau-Feynman Path Integral using Fujiwara approach.
$$ (K_{t,\epsilon}(x,y))_{\epsilon} = $$

$$ (4\pi t\sigma^2)^{-n/2} \exp\left(-\frac{(x-y)^2}{2it\sigma^2}\right) \times $$

$$ \left( \mathbf{E} \left[ \exp \left( -\frac{i}{\hbar} \int_0^t d\tau \tilde{Z}_\epsilon \left( x\left(1-\frac{s}{t}\right) + y\left(\frac{s}{t}\right) + \lambda \sqrt{t} \sigma \mathbf{a}\left(\frac{s}{t}\right) \right) \right] \right)_\epsilon = \quad (0.7.55) $$

$$ K_t^0(x,y)(\mathfrak{R}_\epsilon(t,x,y))_\epsilon. $$

Here $\lambda = \sqrt{i}$ and we have denoted the free propagator by $K_t^0(x,y)$. **Theorem 0.7.7.** Define

$$ (\mathfrak{R}_{t,\epsilon}^R(t,x,y))_\epsilon = $$

$$ \mathbf{E} \left[ \exp \left\{ \frac{g}{\hbar \lambda^2} \int_0^t ds \theta_R \left( \alpha \left( \frac{s}{t} \right) \right) Z_\epsilon \left( x \left( 1 - \frac{s}{t} \right) + y \frac{s}{t} + \lambda \sigma \right) \right\} \right], \quad (0.7.56) $$

where $(Z_\epsilon)_\epsilon \in G_{\mathcal{F}_n^\Delta}(\mathbb{R}^d)$. The limit $R \to \infty$ exists $\forall \epsilon \in (0,1]$ uniformly in $t,x,y$. $\forall \epsilon \in (0,1]$, $\mathfrak{R}_\epsilon(t,x,y)$, is an analytic function of $g$ and $\lambda$ if $\mathrm{Re}\,\lambda^2 \ge 0$. Formula

$$ (K_{t,\epsilon}(x,y))_{\epsilon} = K_{t}^{0}(x,y)\left(\lim_{R \to \infty} \mathfrak{R}_{t,\epsilon}^{R}(t,x,y)\right)_{\epsilon} \quad (0.7.57) $$

gives the Feynman propagator (0.7.55) for any $(Z_\epsilon)_\epsilon \in G_{\mathcal{F}_n^\Delta}(\mathbb{R}^d)$.

### I.0.8. Colombeau-Feynman Path Integral using Fujiwara approach.
We shall treat now the quantum dynamical system using Colombeau-Feynman Path Integral. Let $\mathcal{L}(\dot{x}, x, t)$ be the Lagrangian of the form:

$$ \mathcal{L}(\dot{x}, x, t) = \frac{1}{2} |\dot{x}|^2 - V(t, x). \qquad (0.8.1) $$

Here potential $V(t,x)$ is assumed to satisfy the following assumptions:

(i) $V(t,x)$ is a real valued function of $(t,x) \in \mathbb{R}_+ \times \mathbb{R}^n$. For any fixed $t \in \mathbb{R}_+$, $V(t,x)$ is a function of $x$ of class $C^\infty$. $V(t,x)$ is a measurable function of $(t,x) \in \mathbb{R}_+ \times \mathbb{R}^n$.

(ii) For any multi-index $\alpha$, with its length $|\alpha| \ge 2$, the non-negative measurable function $\mu_\alpha(t)$ defined by

$$ \mu_{\alpha}(t) = \sup_{x \in \mathbb{R}^n} \left| \left( \frac{\partial}{\partial x} \right)^{\alpha} V(t,x) \right| + \sup_{|x| \le 1} |V(t,x)| \qquad (0.8.2) $$

is essentially bounded on every bounded interval of $\mathbb{R}$.

(iii) We fix a sufficiently large $K$, and we let $T = \infty$ if

$$ \mu^* = \sum_{2 \le a \le K} \operatorname{ess. sup}_{t \in \mathbb{R}} \mu_a(t) < \infty \qquad (0.8.3) $$

Otherwise, we let $T$ be an arbitrarily fixed finite positive number. We shall discuss everything in the time interval $(-T, T)$.

Let $S(t,s,x,y)$ be the classical action along the classical orbit starting from $y$ at time $s$ and reaching $x$ at time $t$. It well known that there exists a positive constant $\delta(T)$ such that $S(t,s,x,y)$ is uniquely defined for any $x$ and $y$ in $\mathbb{R}^n$ if $|t-s| < \delta(T)$ [40]. Using this, we define the following oscillatory integral transformations used in [40]. For any $N = 0, 1, 2, \dots$

$$ E^{(N)}(\tilde{\lambda}, t, s)f(x) = \left(\frac{-\tilde{\lambda}}{2\pi(t-s)}\right)^{n/2} \int_{\mathbb{R}^n} d^nyf(y)a^{(N)}(\tilde{\lambda}, t, s, x, y) \exp[\lambda S(t, s, x, y)]. \qquad (0.8.4) $$

Here $\tilde{\lambda} = i\hbar^{-1}, \hbar$ being a small positive parameter. In Eq.(0.8.4), the amplitude
We shall treat now the quantum dynamical system using Colombeau-Feynman Path Integral. Let $\mathcal{L}(\dot{x}, x, t)$ be the Lagrangian of the form:

$$ \mathcal{L}(\dot{x}, x, t) = \frac{1}{2} |\dot{x}|^2 - V(t, x). \qquad (0.8.1) $$

Here potential $V(t,x)$ is assumed to satisfy the following assumptions:

(i) $V(t,x)$ is a real valued function of $(t,x) \in \mathbb{R}_+ \times \mathbb{R}^n$. For any fixed $t \in \mathbb{R}_+$, $V(t,x)$ is a function of $x$ of class $C^\infty$. $V(t,x)$ is a measurable function of $(t,x) \in \mathbb{R}_+ \times \mathbb{R}^n$.

(ii) For any multi-index $\alpha$, with its length $|\alpha| \ge 2$, the non-negative measurable function $\mu_\alpha(t)$ defined by

$$ \mu_{\alpha}(t) = \sup_{x \in \mathbb{R}^n} \left| \left( \frac{\partial}{\partial x} \right)^{\alpha} V(t,x) \right| + \sup_{|x| \le 1} |V(t,x)| \qquad (0.8.2) $$

is essentially bounded on every bounded interval of $\mathbb{R}$.

(iii) We fix a sufficiently large $K$, and we let $T = \infty$ if

$$ \mu^* = \sum_{2 \le a \le K} \sup_{t \in \mathbb{R}} \mu_a(t) < \infty \qquad (0.8.3) $$

Otherwise, we let $T$ be an arbitrarily fixed finite positive number. We shall discuss everything in the time interval $(-T, T)$.

Let $S(t,s,x,y)$ be the classical action along the classical orbit starting from $y$ at time $s$ and reaching $x$ at time $t$. It well known that there exists a positive constant $\delta(T)$ such that $S(t,s,x,y)$ is uniquely defined for any $x$ and $y$ in $\mathbb{R}^n$ if $|t-s| < \delta(T)$ [40]. Using this, we define the following oscillatory integral transformations used in [40]. For any $N = 0, 1, 2, \dots$

$$ E^{(N)}(\tilde{\lambda}, t, s)f(x) = \left(\frac{-\tilde{\lambda}}{2\pi(t-s)}\right)^{n/2} \int_{\mathbb{R}^n} d^nyf(y)a^{(N)}(\tilde{\lambda}, t, s, x, y) \exp[\lambda S(t, s, x, y)]. \qquad (0.8.4) $$

Here $\tilde{\lambda} = i\hbar^{-1}, \hbar$ being a small positive parameter. In Eq.(0.8.4), the amplitude
function $a^{(N)}(\tilde{\lambda}, t, s, x, y)$ is defined by

$$a^{(N)}(\tilde{\lambda}, t, s, x, y) = 1, \qquad (0.8.5)$$

$$a^{(N)}(\tilde{\lambda}, t, s, x, y) = \sum_{j=1}^{N} \tilde{\lambda}^{1-j} a_j(t, s, x, y),$$

where $a_j(t,s,x,y)$ will be defined by transport equation [40]. Let $[s,t]$ be an arbitrary time interval in $(-T, T)$ and let $\Delta = \{s = t_0 < t_1 < \dots < t_L = t\}$ be an arbitrary subdivision of the interval $[s,t]$. We put $\delta(\Delta) = \max_{1 \le j \le L} (|t_j - t_{j-1}|)$. We introduce the iterated integral operators $E^{(N)}(\Delta|\tilde{\lambda}, t, s)$ and $E^{(N)}(\Delta|\tilde{\lambda}, s, t)$ associated with the subdivision $\Delta$ by the formulae

$$E^{(N)}(\Delta|\tilde{\lambda}, t, s) = E^{(N)}(\tilde{\lambda}, t, t_{L-1}) E^{(N)}(\tilde{\lambda}, t_{L-1}, t_{L-2}) \cdots E^{(N)}(\tilde{\lambda}, t_{L-1}, t), \qquad (0.8.6)$$

$$E^{(N)}(\Delta|\tilde{\lambda}, s, t) = E^{(N)}(\tilde{\lambda}, s, t_1) E^{(N)}(\tilde{\lambda}, t_1, t_2) \cdots E^{(N)}(\tilde{\lambda}, t_1, t)$$

for $N = 0, 1, 2, \dots$. Let $I^{(N)}(\Delta|\tilde{\lambda}, t, s, x, y)$ and $I^{(N)}(\Delta|\tilde{\lambda}, s, t, x, y)$ be the kernel functions of $E^{(N)}(\Delta|\tilde{\lambda}, t, s)$ and $E^{(N)}(\Delta|\tilde{\lambda}, s, t)$, respectively. Then, we have

$$E^{(N)}(\Delta|\tilde{\lambda}, t, s)f(x) = \int_{\mathbb{R}^n} d^nyf(y)I^{(N)}(\Delta|\tilde{\lambda}, t, s, x, y),$$

$$I^{(N)}(\Delta|\tilde{\lambda}, t, s, x, y) = \prod_{j=1}^{L} \left( \frac{-\tilde{\lambda}}{2\pi(t_j - t_{j-1})} \right)^{n/2} \int_{\mathbb{R}^n} \cdots \int_{\mathbb{R}^n} \prod_{j=1}^{L} a^{(N)}(\tilde{\lambda}, t_j, t_{j-1}, x^j, y^{j-1}) \times \qquad (0.8.7)$$

$$\prod_{j=1}^{L} \left( \frac{-\tilde{\lambda}}{2\pi(t_j - t_{j-1})} \right)^{n/2} \int_{\mathbb{R}^n} \cdots \int_{\mathbb{R}^n} \prod_{j=1}^{L} a^{(N)}(\tilde{\lambda}, t_j, t_{j-1}, x^j, y^{j-1}) \times \\ \exp\left\{ \tilde{\lambda} \left[ \sum_{j=1}^{L} S(t_j, t_{j-1}, x^j, y^{j-1}) \right] \right\} \prod_{j=1}^{L-1} dx^j,$$

where $x^0 = y$ and $x^L = x$. Similar formulae are valid for $E^{(N)}(\Delta|\tilde{\lambda}, s, t)$. Feynman formally "stated" in [13] that limit
function $a^{(N)}(\tilde{\lambda}, t, s, x, y)$ is defined by

$$ 
\begin{gathered}
a^{(N)}(\tilde{\lambda}, t, s, x, y) = 1, \\
a^{(N)}(\tilde{\lambda}, t, s, x, y) = \sum_{j=1}^{N} \tilde{\lambda}^{1-j} a_j(t, s, x, y),
\end{gathered}
\qquad (0.8.5) 
$$

where $a_j(t,s,x,y)$ will be defined by transport equation [40]. Let $[s,t]$ be an arbitrary time interval in $(-T, T)$ and let $\Delta = \{s = t_0 < t_1 < \dots < t_L = t\}$ be an arbitrary subdivision of the interval $[s,t]$. We put $\delta(\Delta) = \max_{1 \le j \le L} (|t_j - t_{j-1}|)$. We introduce the iterated integral operators $E^{(N)}(\Delta|\tilde{\lambda}, t, s)$ and $E^{(N)}(\Delta|\tilde{\lambda}, s, t)$ associated with the subdivision $\Delta$ by the formulae

$$ 
\begin{aligned}
E^{(N)}(\Delta|\tilde{\lambda}, t, s) &= E^{(N)}(\tilde{\lambda}, t, t_{L-1}) E^{(N)}(\tilde{\lambda}, t_{L-1}, t_{L-2}) \cdots E^{(N)}(\tilde{\lambda}, t_{L-1}, t), \\
E^{(N)}(\Delta|\tilde{\lambda}, s, t) &= E^{(N)}(\tilde{\lambda}, s, t_1) E^{(N)}(\tilde{\lambda}, t_1, t_2) \cdots E^{(N)}(\tilde{\lambda}, t_1, t)
\end{aligned}
\qquad (0.8.6) 
$$

for $N = 0, 1, 2, \dots$. Let $\mathbf{I}^{(N)}(\Delta|\tilde{\lambda}, t, s, x, y)$ and $\mathbf{I}^{(N)}(\Delta|\tilde{\lambda}, s, t, x, y)$ be the kernel functions of $E^{(N)}(\Delta|\tilde{\lambda}, t, s)$ and $E^{(N)}(\Delta|\tilde{\lambda}, s, t)$, respectively. Then, we have

$$ 
\begin{aligned}
E^{(N)}(\Delta|\tilde{\lambda}, t, s)f(x) &= \int_{\mathbb{R}^n} d^nyf(y)\mathbf{I}^{(N)}(\Delta|\tilde{\lambda}, t, s, x, y), \\
\mathbf{I}^{(N)}(\Delta|\tilde{\lambda}, t, s, x, y) &= \\
&\quad \prod_{j=1}^{L} \left( \frac{-\tilde{\lambda}}{2\pi(t_j - t_{j-1})} \right)^{n/2} \int_{\mathbb{R}^n} \cdots \int_{\mathbb{R}^n} \prod_{j=1}^{L} a^{(N)}(\tilde{\lambda}, t_j, t_{j-1}, x^j, y^{j-1}) \times \\
&\quad \exp\left\{ \tilde{\lambda} \left[ \sum_{j=1}^{L} S(t_j, t_{j-1}, x^j, y^{j-1}) \right] \right\} \prod_{j=1}^{L-1} dx^j,
\end{aligned}
\qquad (0.8.7) 
$$

where $x^0 = y$ and $x^L = x$. Similar formulae are valid for $E^{(N)}(\Delta|\tilde{\lambda}, s, t)$. Feynman formally "stated" in [13] that limit
$$ \lim_{\delta(\Delta) \to 0} \mathbf{I}^{(0)}(\Delta|\tilde{\lambda}, t, s, x, y) \qquad (0.8.8) $$

exists and equals the kernel function of the fundamental solution (Green's function) for the Schrödinger equation (0.1.1) or

$$ \frac{\partial}{\partial \tilde{\lambda}} u(t,x) + \frac{1}{2} \sum_{j=1}^{n} \left( \frac{\partial}{\partial \tilde{\lambda}} \partial_{x_j} \right)^2 u(t,x) + V(t,x) u(t,x) = 0, \qquad (0.8.9) $$

$$ u(0,x) = u_0(x). $$

Note that

$$ \mathbf{I}^{(N)}(\Delta|\tilde{\lambda}, t, s, x, y) = \left( \frac{-\tilde{\lambda}}{2\pi(t-s)} \right)^{n/2} a^{(N)}(\tilde{\lambda}, t, s, x, y) \exp[\tilde{\lambda}S(t, s, x, y)] \qquad (0.8.10) $$

and

$$ \mathbf{I}^{(N)}(\Delta|\tilde{\lambda}, s, t, x, y) = \left( \frac{-\tilde{\lambda}}{2\pi(s-t)} \right)^{n/2} a^{(N)}(\tilde{\lambda}, s, t, x, y) \exp[\tilde{\lambda}S(s, t, x, y)] \qquad (0.8.11) $$

if $|t-s| < \delta(T)$. These amplitude functions $a^{(N)}(\tilde{\lambda}, t, s, x, y)$ and $a^{(N)}(\tilde{\lambda}, s, t, x, y)$ belong to the function space $D(\mathbb{R}_x^n \times \mathbb{R}_y^n)$ of Schwartz [47] with parameters $\Delta, \tilde{\lambda}, t, s$. As usual we shall employ the norm $\|f\|_m$ defined by

$$ \|f\|_m = \sum_{|a+\beta| \le m} \sup_{(x,y) \in \mathbb{R}_x^n \times \mathbb{R}_y^n} \left| \left(\frac{\partial}{\partial x}\right)^a \left(\frac{\partial}{\partial y}\right)^\beta f(x,y) \right| \qquad (0.8.12) $$

for $m = 0, 1, 2, \dots$. We shall denote by $a^{(N)}(\tilde{\lambda}, t, s)$ the function $a^{(N)}(\tilde{\lambda}, t, s, x, y)$
$$ \lim_{\delta(\Delta) \to 0} \mathbf{I}^{(0)}(\Delta|\tilde{\lambda}, t, s, x, y) \qquad (0.8.8) $$

exists and equals the kernel function of the fundamental solution (Green's function) for the Schrödinger equation (0.1.1) or

$$ \frac{\partial}{\partial \tilde{\lambda}} u(t,x) + \frac{1}{2} \sum_{j=1}^{n} \left( \frac{\partial}{\partial \tilde{\lambda}} \partial_{x_j} \right)^2 u(t,x) + V(t,x)u(t,x) = 0, \qquad (0.8.9) $$

$$ u(0,x) = u_0(x). $$

Note that

$$ \mathbf{I}^{(N)}(\Delta|\tilde{\lambda}, t, s, x, y) = \left( \frac{-\tilde{\lambda}}{2\pi(t-s)} \right)^{n/2} a^{(N)}(\tilde{\lambda}, t, s, x, y) \exp[\tilde{\lambda}S(t, s, x, y)] \quad (0.8.10) $$

and

$$ \mathbf{I}^{(N)}(\Delta|\tilde{\lambda}, s, t, x, y) = \left( \frac{-\tilde{\lambda}}{2\pi(s-t)} \right)^{n/2} a^{(N)}(\tilde{\lambda}, s, t, x, y) \exp[\tilde{\lambda}S(s, t, x, y)] \quad (0.8.11) $$

if $|t-s| < \delta(T)$. These amplitude functions $a^{(N)}(\tilde{\lambda}, t, s, x, y)$ and $a^{(N)}(\tilde{\lambda}, s, t, x, y)$ belong to the function space $D(\mathbb{R}_x^n \times \mathbb{R}_y^n)$ of Schwartz [47] with parameters $\Delta, \tilde{\lambda}, t, s$. As usual we shall employ the norm $\|f\|_m$ defined by

$$ \|f\|_m = \sum_{|a+\beta| \le m} \sup_{(x,y) \in \mathbb{R}_x^n \times \mathbb{R}_y^n} \left| \left(\frac{\partial}{\partial x}\right)^a \left(\frac{\partial}{\partial y}\right)^b f(x,y) \right| \quad (0.8.12) $$

for $m = 0, 1, 2, \dots$. We shall denote by $a^{(N)}(\tilde{\lambda}, t, s)$ the function $a^{(N)}(\tilde{\lambda}, t, s, x, y)$
considered as an element of the function space $D(\mathbb{R}_x^n \times \mathbb{R}_y^n)$ with parameters $\Delta, \tilde{\lambda}, t, s$.

Fujiwara [40] prove Feynman assertion (0.8.8) under the assumptions (i)-(iii) concerning the potential $V(t,x)$.

**Theorem.0.8.1.** (i) (Fujiwara [40]) Assume that (i) and (ii) hold. Assume further that $|t-s| < \delta(T)$ Then, for any $N = 0, 1, 2, \dots$ both the limits

$$k(\tilde{\lambda}, t, s) = \lim_{\delta(\Delta) \to 0} a^{(N)}(\Delta|\tilde{\lambda}, t, s) \quad (0.8.13)$$

and

$$k(\tilde{\lambda}, s, t) = \lim_{\delta(\Delta) \to 0} a^{(N)}(\Delta|\tilde{\lambda}, s, t) \quad (0.8.14)$$

exist in the function space $D(\mathbb{R}_x^n \times \mathbb{R}_y^n)$. Moreover, for any integers $m > 0$ and $N > 0$, there exists a positive constant $\gamma = \gamma(m, N, T)$ such that

$$\|k(\tilde{\lambda}, t, s) - a^{(N)}(\Delta|\tilde{\lambda}, t, s)\|_{m} \leq \gamma|\lambda|^{-N}(\delta(\Delta))^{N+1}|t-s| \quad (0.8.15)$$

and

$$\|k(\tilde{\lambda}, s, t) - a^{(N)}(\Delta|\tilde{\lambda}, s, t)\|_{m} \leq \gamma|\lambda|^{-N}(\delta(\Delta))^{N+1}|t-s| \quad (0.8.16)$$

if $|t-s| < \delta(T)$. The constant $\gamma(m, N, T)$ is independent of any particular choice of $t, s, \Delta$ and $\tilde{\lambda}$ if $|\tilde{\lambda}|$ is bounded away from 0. The function $k(\tilde{\lambda}, t, s, x, y)$ of is independent of $N$. For any $t \in [s - \delta(T), s + \delta(T)]$, we define

$$U(\tilde{\lambda}, t, s) \varphi(x) = \left( \frac{-\tilde{\lambda}}{2\pi(t-s)} \right)^{n/2} \int_{\mathbb{R}^n} dy \varphi(y) k(\tilde{\lambda}, t, s, x, y) \exp[\tilde{\lambda}S(t, s, x, y)] \quad (0.8.17)$$
considered as an element of the function space $D(\mathbb{R}_x^n \times \mathbb{R}_y^n)$ with parameters $\Delta, \tilde{\lambda}, t, s$.

Fujiwara [40] prove Feynman assertion (0.8.8) under the assumptions (i)-(iii) concerning the potential $V(t,x)$.

**Theorem.0.8.1.** (i) (Fujiwara [40]) Assume that (i) and (ii) hold. Assume further that $|t-s| < \delta(T)$ Then, for any $N = 0, 1, 2, \dots$ both the limits

$$k(\tilde{\lambda}, t, s) = \lim_{\delta(\Delta) \to 0} a^{(N)}(\Delta|\tilde{\lambda}, t, s) \qquad (0.8.13)$$

and

$$k(\tilde{\lambda}, s, t) = \lim_{\delta(\Delta) \to 0} a^{(N)}(\Delta|\tilde{\lambda}, s, t) \qquad (0.8.14)$$

exist in the function space $D(\mathbb{R}_x^n \times \mathbb{R}_y^n)$. Moreover, for any integers $m > 0$ and $N > 0$, there exists a positive constant $\gamma = \gamma(m, N, T)$ such that

$$\|k(\tilde{\lambda}, t, s) - a^{(N)}(\Delta|\tilde{\lambda}, t, s)\|_{m} \leq \gamma|\lambda|^{-N}(\delta(\Delta))^{N+1}|t-s| \qquad (0.8.15)$$

and

$$\|k(\tilde{\lambda}, s, t) - a^{(N)}(\Delta|\tilde{\lambda}, s, t)\|_{m} \leq \gamma|\lambda|^{-N}(\delta(\Delta))^{N+1}|t-s| \qquad (0.8.16)$$

if $|t-s| < \delta(T)$. The constant $\gamma(m, N, T)$ is independent of any particular choice of $t, s, \Delta$ and $\tilde{\lambda}$ if $|\tilde{\lambda}|$ is bounded away from 0. The function $k(\tilde{\lambda}, t, s, x, y)$ of is independent of $N$. For any $t \in [s - \delta(T), s + \delta(T)]$, we define

$$U(\tilde{\lambda}, t, s)\varphi(x) = \left(\frac{-\tilde{\lambda}}{2\pi(t-s)}\right)^{n/2} \int_{\mathbb{R}^n} dy\varphi(y)k(\tilde{\lambda}, t, s, x, y) \exp[\tilde{\lambda}S(t, s, x, y)] \qquad (0.8.17)$$
Then, $u(t,x) = U(\tilde{\lambda}, t, s)\varphi(x)$ satisfies the Schrödinger equation (0.8.9) and the initial condition $u(s,x) = \varphi(x)$.

(ii) Let $f(x) \in \mathcal{L}_2(\mathbb{R}^n)$ then

$$ U(\tilde{\lambda}, t, s)f(x) = \int_{\mathbb{R}^n} \left\{ \lim_{\delta(\Delta) \to 0} \mathbf{I}^{(N)}(\Delta|\tilde{\lambda}, t, s, x, y) \right\} f(y)dy = \lim_{\delta(\Delta) \to 0} E^{(N)}(\Delta|\tilde{\lambda}, t, s)f(x) \quad (0.8.18) $$

in the sense of convergence in $\mathcal{L}_2(\mathbb{R}^n)$, i.e. one can interchange the order of operations $\lim$ and $\int dy$ if $f(x) \in \mathcal{L}_2(\mathbb{R}^n)$.

**Theorem.0.8.2.** (Fujiwara [40]) If $|t-s| < \delta(T)$, then for any integers $N \ge 0$ and $m \ge 0$, we have

$$ \begin{aligned} \|k(\tilde{\lambda}, t, s) - a^{(N)}(\tilde{\lambda}, t, s)\|_{m} &\le \gamma|\lambda|^{-N}|t-s|^{N+1}, \\ \|k(\tilde{\lambda}, s, t) - a^{(N)}(\tilde{\lambda}, s, t)\|_{m} &\le \gamma|\lambda|^{-N}|t-s|^{N+1} \end{aligned} \quad (0.8.19) $$

with some constant $\gamma = \gamma(m, N, T)$. Here $a^{(N)}(\tilde{\lambda}, t, s)$ is the amplitude function of $E^{(N)}(\Delta|\tilde{\lambda}, t, s)$ of eq. (0.8.4).

In our subsequent consideration, the potential $V(t,x)$ does not satisfy Fujiwara condition (i)-(iii). We remove it by cut-off or another type of the regularization. As a framework we use Colombeau's algebra of generalized functions, i.e. we change potential $V(t,x)$ by Colombeau generalized function $(V_{\varepsilon}(t,x))_{\varepsilon \in (0,1]}$ such that $V_0(t,x) = V(t,x)$ and

$(V_{\varepsilon}i) V_{\varepsilon}(t,x)$ is a real valued function of $(t,x) \in \mathbb{R}_+ \times \mathbb{R}^n$. For any fixed $t \in \mathbb{R}_+$,

$\mathbf{cl}[(V_{\varepsilon}(t,x))_{\varepsilon}] \in G(\mathbb{R}^n)$.

$\forall \varepsilon \in (0,1] [V_{\varepsilon}(t,x)$ is a measurable function of $(t,x) \in \mathbb{R}_+ \times \mathbb{R}^n]$.

$(V_{\varepsilon}ii)$ For any multi-index $\alpha$, with its length $|\alpha| \ge 2$, the non-negative measurable function $\mu_{\alpha,\varepsilon}(t)$ defined by
Then, $u(t,x) = U(\tilde{\lambda}, t, s)\varphi(x)$ satisfies the Schrödinger equation (0.8.9) and the initial condition $u(s,x) = \varphi(x)$.

(ii) Let $f(x) \in \mathcal{L}_2(\mathbb{R}^n)$ then

$$ U(\tilde{\lambda}, t, s)f(x) = \int_{\mathbb{R}^n} \left\{ \lim_{\delta(\Delta) \to 0} \mathbf{I}^{(N)}(\Delta|\tilde{\lambda}, t, s, x, y) \right\} f(y)dy = \lim_{\delta(\Delta) \to 0} E^{(N)}(\Delta|\tilde{\lambda}, t, s)f(x) \quad (0.8.18) $$

in the sense of convergence in $\mathcal{L}_2(\mathbb{R}^n)$, i.e. one can interchange the order of operations $\lim$ and $\int dy$ if $f(x) \in \mathcal{L}_2(\mathbb{R}^n)$.

**Theorem.0.8.2.** (Fujiwara [40]) If $|t-s| < \delta(T)$, then for any integers $N \ge 0$ and $m \ge 0$, we have

$$ \begin{aligned} \|k(\tilde{\lambda}, t, s) - a^{(N)}(\tilde{\lambda}, t, s)\|_{m} &\le \gamma|\lambda|^{-N}|t-s|^{N+1}, \\ \|k(\tilde{\lambda}, s, t) - a^{(N)}(\tilde{\lambda}, s, t)\|_{m} &\le \gamma|\lambda|^{-N}|t-s|^{N+1} \end{aligned} \quad (0.8.19) $$

with some constant $\gamma = \gamma(m, N, T)$. Here $a^{(N)}(\tilde{\lambda}, t, s)$ is the amplitude function of $E^{(N)}(\Delta|\tilde{\lambda}, t, s)$ of eq. (0.8.4).

In our subsequent consideration, the potential $V(t,x)$ does not satisfy Fujiwara condition (i)-(iii). We remove it by cut-off or another type of the regularization. As a framework we use Colombeau's algebra of generalized functions, i.e. we change potential $V(t,x)$ by Colombeau generalized function $(V_{\varepsilon}(t,x))_{\varepsilon \in (0,1]}$ such that $V_0(t,x) = V(t,x)$ and

$(V_{\varepsilon}i) V_{\varepsilon}(t,x)$ is a real valued function of $(t,x) \in \mathbb{R}_+ \times \mathbb{R}^n$. For any fixed $t \in \mathbb{R}_+$,

$\mathbf{cl}[(V_{\varepsilon}(t,x))_{\varepsilon}] \in G(\mathbb{R}^n)$.

$\forall \varepsilon \in (0,1] [V_{\varepsilon}(t,x)$ is a measurable function of $(t,x) \in \mathbb{R}_+ \times \mathbb{R}^n]$.

$(V_{\varepsilon}ii)$ For any multi-index $\alpha$, with its length $|\alpha| \ge 2$, the non-negative measurable function $\mu_{\alpha,\varepsilon}(t)$ defined by
$$ \mu_{\alpha, \varepsilon}(t) = \sup_{x \in \mathbb{R}^n} \left| \left( \frac{\partial}{\partial x} \right)^{\alpha} V_{\varepsilon}(t,x) \right| + \sup_{|x| \le 1} |V_{\varepsilon}(t,x)|, \quad (0.8.20) $$

is essentially bounded on every bounded interval of $\mathbb{R}$ and $\mu_{\alpha,\varepsilon}(t) = O(\varepsilon^{-M(a)})$.

(V_ε iii) We fix a sufficiently large K, and we let $T = \infty$ if $\forall \varepsilon \in (0, 1]$:

$$ \mu_{\varepsilon}^{*} = \sum_{2 \le a \le K} \operatorname{ess. sup}_{t \in \mathbb{R}} \mu_{\alpha, \varepsilon}(t) < \infty \quad (0.8.21) $$

We shall discuss Feynman path integral representation of the Colombeau's solutions for the Colombeau-Schrödinger type equation

$$ \frac{\partial}{\partial \tilde{\lambda} \partial t} (u_{\varepsilon}(t,x))_{\varepsilon} + \frac{1}{2} \sum_{j=1}^{n} \left( \frac{\partial}{\partial \tilde{\lambda} \partial x_j} \right)^2 (u_{\varepsilon}(t,x))_{\varepsilon} + ((V_{\varepsilon}(t,x))_{\varepsilon})(u_{\varepsilon}(t,x))_{\varepsilon} = 0, \quad (0.8.22) $$

$$ (u_{\varepsilon}(0,x))_{\varepsilon} = (u_{0,\varepsilon}(x))_{\varepsilon}. $$

Initial data are strongly singular and regularized with delta sequence. It well known that there exists Colombeau's solutions of the Eq.(0.8.22),which are compatible with classical solutions in a limiting case when $\varepsilon \to 0$ [12],[42-43].

Let $S_{\varepsilon}(t,s,x,y)$ be the classical action along the classical orbit starting from $y$ at time $s$ and reaching $x$ at time $t$. It well known that there exists a positive constant $\delta(T)$ such that $S(t,s,x,y)$ is uniquely defined for any $x$ and $y$ in $\mathbb{R}^n$ if $|t-s| < \delta(T, \varepsilon)$ [40]. Using this, we define the following Colombeau oscillatory integral transformations. For any $N = 0, 1, 2$,
$$ \mu_{\alpha, \varepsilon}(t) = \sup_{x \in \mathbb{R}^n} \left| \left( \frac{\partial}{\partial x} \right)^{\alpha} V_{\varepsilon}(t,x) \right| + \sup_{|x| \le 1} |V_{\varepsilon}(t,x)|, \quad (0.8.20) $$

is essentially bounded on every bounded interval of $\mathbb{R}$ and $\mu_{\alpha, \varepsilon}(t) = O(\varepsilon^{-M(a)})$.

(V_ε iii) We fix a sufficiently large K, and we let $T = \infty$ if $\forall \varepsilon \in (0, 1]$:

$$ \mu_{\varepsilon}^{*} = \sum_{2 \le a \le K} \operatorname{ess. sup}_{t \in \mathbb{R}} \mu_{\alpha, \varepsilon}(t) < \infty \quad (0.8.21) $$

We shall discuss Feynman path integral representation of the Colombeau's solutions for the Colombeau-Schrödinger type equation

$$ \begin{gathered} \frac{\partial}{\partial \lambda} (u_{\varepsilon}(t,x))_{\varepsilon} + \frac{1}{2} \sum_{j=1}^{n} \left( \frac{\partial}{\partial \lambda \partial x_j} \right)^2 (u_{\varepsilon}(t,x))_{\varepsilon} + ((V_{\varepsilon}(t,x))_{\varepsilon}) (u_{\varepsilon}(t,x))_{\varepsilon} = 0, \quad (0.8.22) \\ (u_{\varepsilon}(0,x))_{\varepsilon} = (u_{0,\varepsilon}(x))_{\varepsilon}. \end{gathered} $$

Initial data are strongly singular and regularized with delta sequence. It well known that there exists Colombeau's solutions of the Eq.(0.8.22),which are compatible with classical solutions in a limiting case when $\varepsilon \to 0$ [12],[42-43].

Let $S_{\varepsilon}(t,s,x,y)$ be the classical action along the classical orbit starting from $y$ at time $s$ and reaching $x$ at time $t$. It well known that there exists a positive constant $\delta(T)$ such that $S(t,s,x,y)$ is uniquely defined for any $x$ and $y$ in $\mathbb{R}^n$ if $|t-s| < \delta(T, \varepsilon)$ [40]. Using this, we define the following Colombeau oscillatory integral transformations. For any $N = 0, 1, 2$,
$$ (E_{\varepsilon}^{(N)}(\tilde{\lambda}, t, s)f_{\varepsilon}(x))_{\varepsilon} = $$

$$ \left( \frac{-\tilde{\lambda}}{2\pi(t-s)} \right)^{n/2} \left( \int_{\mathbb{R}^n} d^ny f_{\varepsilon}(y) a_{\varepsilon}^{(N)}(\tilde{\lambda}, t, s, x, y) \exp[\lambda S_{\varepsilon}(t, s, x, y)] \right)_{\varepsilon} = \quad (0.8.23) $$

$$ \left( \frac{-\tilde{\lambda}}{2\pi(t-s)} \right)^{n/2} \int_{\mathbb{R}^n} d^ny (f_{\varepsilon}(y))_{\varepsilon} \left( (a_{\varepsilon}^{(N)}(\tilde{\lambda}, t, s, x, y))_{\varepsilon} \right) \exp[\lambda((S_{\varepsilon}(t, s, x, y))_{\varepsilon})] $$

Here $\tilde{\lambda} = i\hbar^{-1}, \hbar$ being a small positive parameter. In Eq.(0.8.23), the generalized amplitude function is defined by

$$ (a_{\varepsilon}^{(N)}(\tilde{\lambda}, t, s, x, y))_{\varepsilon} = 1, $$

(0.8.24)

$$ (a_{\varepsilon}^{(N)}(\tilde{\lambda}, t, s, x, y))_{\varepsilon} = \sum_{j=1}^{N} \tilde{\lambda}^{1-j} (a_{j,\varepsilon}(t, s, x, y))_{\varepsilon}, $$

where $(a_{j,\varepsilon}(t,s,x,y))_{\varepsilon}$ will be defined by transport equation. Let $[s,t]$ be an arbitrary time interval in $(-T, T)$ and let $\Delta(\varepsilon) = \{s = t_0 < t_1 < \dots < t_L = t\}$ be an arbitrary subdivision of the interval $[s,t]$. We put $\delta(\Delta(\varepsilon)) = \max_{1 \le j \le L_\varepsilon} (|t_j - t_{j-1}|)$. We introduce the iterated generalized integral operators $\left[ (E_{\varepsilon}^{(N)}(\Delta(\varepsilon)|\tilde{\lambda}, t, s))_{\varepsilon} \right]$ and $\left[ (E_{\varepsilon}^{(N)}(\Delta(\varepsilon)|\tilde{\lambda}, s, t))_{\varepsilon} \right]$ associated with the subdivision $\Delta(\varepsilon)$ by the formulae:
$$ (E_{\varepsilon}^{(N)}(\tilde{\lambda}, t, s)f_{\varepsilon}(x))_{\varepsilon} = \\ \left(\frac{-\tilde{\lambda}}{2\pi(t-s)}\right)^{n/2} \left(\int_{\mathbb{R}^n} d^ny f_{\varepsilon}(y) a_{\varepsilon}^{(N)}(\tilde{\lambda}, t, s, x, y) \exp[\lambda S_{\varepsilon}(t, s, x, y)]\right)_{\varepsilon} = \\ \left(\frac{-\tilde{\lambda}}{2\pi(t-s)}\right)^{n/2} \int_{\mathbb{R}^n} d^ny (f_{\varepsilon}(y))_{\varepsilon} \left(\left(a_{\varepsilon}^{(N)}(\tilde{\lambda}, t, s, x, y)\right)_{\varepsilon}\right) \exp[\lambda\left(S_{\varepsilon}(t, s, x, y)\right)_{\varepsilon}] $$

Here $\tilde{\lambda} = i\hbar^{-1}, \hbar$ being a small positive parameter. In Eq.(0.8.23), the generalized amplitude function is defined by

$$ (a_{\varepsilon}^{(N)}(\tilde{\lambda}, t, s, x, y))_{\varepsilon} = 1, \\ (a_{\varepsilon}^{(N)}(\tilde{\lambda}, t, s, x, y))_{\varepsilon} = \sum_{j=1}^{N} \tilde{\lambda}^{1-j} (a_{j,\varepsilon}(t, s, x, y))_{\varepsilon}, $$

where $(a_{j,\varepsilon}(t,s,x,y))_{\varepsilon}$ will be defined by transport equation.Let $[s,t]$ be an arbitrary time interval in $(-T,T)$ and let $\Delta(\varepsilon) = \{s = t_0 < t_1 < \dots < t_L = t\}$ be an arbitrary subdivision of the interval $[s,t]$. We put $\delta(\Delta(\varepsilon)) = \max_{1 \le j \le L_\varepsilon} (|t_j - t_{j-1}|)$. We introduce the iterated generalized integral operators $\left[ (E_{\varepsilon}^{(N)}(\Delta(\varepsilon)|\tilde{\lambda}, t, s))_{\varepsilon}\right]$ and $\left[ (E_{\varepsilon}^{(N)}(\Delta(\varepsilon)|\tilde{\lambda}, s, t))_{\varepsilon}\right]$ associated with the subdivision $\Delta(\varepsilon)$ by the formulae:
$$ \begin{aligned} & \left[ (E_{\varepsilon}^{(N)}(\Delta(\varepsilon)|\tilde{\lambda}, t, s))_{\varepsilon} \right] = \\ & \left[ (E_{\varepsilon}^{(N)}(\tilde{\lambda}, t, t_{L_{\varepsilon}-1}))_{\varepsilon} \right] \left[ (E_{\varepsilon}^{(N)}(\tilde{\lambda}, t_{L_{\varepsilon}-1}, t_{L_{\varepsilon}-2}))_{\varepsilon} \right] \cdots \left[ (E_{\varepsilon}^{(N)}(\tilde{\lambda}, t_{L_{\varepsilon}-1}, t))_{\varepsilon} \right], \end{aligned} \quad (0.8.25) $$

$$ \left[ (E_{\varepsilon}^{(N)}(\Delta(\varepsilon)|\tilde{\lambda}, s, t))_{\varepsilon} \right] = \left[ (E_{\varepsilon}^{(N)}(\tilde{\lambda}, s, t_{1}))_{\varepsilon} \right] \cdots \left[ (E_{\varepsilon}^{(N)}(\tilde{\lambda}, t_{L_{\varepsilon}}, t))_{\varepsilon} \right] $$

for $N = 0, 1, 2, \dots$. Let $(\mathbf{I}_{\varepsilon}^{(N)}(\Delta(\varepsilon)|\tilde{\lambda}, t, s, x, y))_{\varepsilon}$ and $(\mathbf{I}_{\varepsilon}^{(N)}(\Delta(\varepsilon)|\tilde{\lambda}, t, s, x, y))_{\varepsilon}$ be the kernel functions of $(E_{\varepsilon}^{(N)}(A|\tilde{\lambda}, t, s))_{\varepsilon}$ and $(E_{\varepsilon}^{(N)}(A|\tilde{\lambda}, s, t))_{\varepsilon}$, respectively. Then, we have

$$ \begin{aligned} & \left[ (E_{\varepsilon}^{(N)}(\Delta(\varepsilon)|\tilde{\lambda}, t, s))_{\varepsilon} \right] [(f_{\varepsilon}(x))_{\varepsilon}] = \int_{\mathbb{R}^n} d^n y [(f_{\varepsilon}(y))_{\varepsilon}] \left[ (\mathbf{I}_{\varepsilon}^{(N)}(\Delta(\varepsilon)|\tilde{\lambda}, t, s, x, y))_{\varepsilon} \right], \\ & (\mathbf{I}_{\varepsilon}^{(N)}(\Delta(\varepsilon)|\tilde{\lambda}, t, s, x, y))_{\varepsilon} = \\ & \prod_{j=1}^{L_{\varepsilon}} \left( \frac{-\tilde{\lambda}}{2\pi(t_j - t_{j-1})} \right)^{n/2} \int_{\mathbb{R}^n} \cdots \int_{\mathbb{R}^n} \prod_{j=1}^{L_{\varepsilon}} \left[ (a_{\varepsilon}^{(N)}(\tilde{\lambda}, t_j, t_{j-1}, x^j, y^{j-1}))_{\varepsilon} \right] \times \\ & \exp \left\{ \tilde{\lambda} \left[ \sum_{j=1}^{L_{\varepsilon}} (S_{\varepsilon}(t_j, t_{j-1}, x^j, y^{j-1}))_{\varepsilon} \right] \right\} \prod_{j=1}^{L_{\varepsilon}-1} dx^j, \end{aligned} \quad (0.8.26) $$

where $x^0 = y$ and $x^{L_{\varepsilon}} = x$. Similar formulae are valid for $(E_{\varepsilon}^{(N)}(\Delta|\tilde{\lambda}, s, t))_{\varepsilon}$. We stated that $\forall \varepsilon \in (0, 1]$

$$ \lim_{\delta(\Delta(\varepsilon)) \to 0} \mathbf{I}_{\varepsilon}^{(0)}(\Delta(\varepsilon)|\tilde{\lambda}, t, s, x, y) \quad (0.8.27) $$

exists and equals the generalized kernel function of the fundamental solution (Green's function) for the Colombeau-Schrödinger equation (0.8.22).
$$ \begin{aligned} & \left[ (E_{\varepsilon}^{(N)}(\Delta(\varepsilon)|\tilde{\lambda}, t, s))_{\varepsilon} \right] = \\ & \left[ (E_{\varepsilon}^{(N)}(\tilde{\lambda}, t, t_{L_{\varepsilon}-1}))_{\varepsilon} \right] \left[ (E_{\varepsilon}^{(N)}(\tilde{\lambda}, t_{L_{\varepsilon}-1}, t_{L_{\varepsilon}-2}))_{\varepsilon} \right] \cdots \left[ (E_{\varepsilon}^{(N)}(\tilde{\lambda}, t_{L_{\varepsilon}-1}, t))_{\varepsilon} \right], \end{aligned} \quad (0.8.25) $$

$$ \left[ (E_{\varepsilon}^{(N)}(\Delta(\varepsilon)|\tilde{\lambda}, s, t))_{\varepsilon} \right] = \left[ (E_{\varepsilon}^{(N)}(\tilde{\lambda}, s, t_{1}))_{\varepsilon} \right] \cdots \left[ (E_{\varepsilon}^{(N)}(\tilde{\lambda}, t_{L_{\varepsilon}}, t))_{\varepsilon} \right] $$

for $N = 0, 1, 2, \dots$. Let $(\mathbf{I}_{\varepsilon}^{(N)}(\Delta(\varepsilon)|\tilde{\lambda}, t, s, x, y))_{\varepsilon}$ and $(\mathbf{I}_{\varepsilon}^{(N)}(\Delta(\varepsilon)|\tilde{\lambda}, t, s, x, y))_{\varepsilon}$ be the kernel functions of $(E_{\varepsilon}^{(N)}(A|\tilde{\lambda}, t, s))_{\varepsilon}$ and $(E_{\varepsilon}^{(N)}(A|\tilde{\lambda}, s, t))_{\varepsilon}$, respectively. Then, we have

$$ \begin{aligned} & \left[ (E_{\varepsilon}^{(N)}(\Delta(\varepsilon)|\tilde{\lambda}, t, s))_{\varepsilon} \right] [(f_{\varepsilon}(x))_{\varepsilon}] = \int_{\mathbb{R}^n} d^n y [(f_{\varepsilon}(y))_{\varepsilon}] \left[ (\mathbf{I}_{\varepsilon}^{(N)}(\Delta(\varepsilon)|\tilde{\lambda}, t, s, x, y))_{\varepsilon} \right], \\ & (\mathbf{I}_{\varepsilon}^{(N)}(\Delta(\varepsilon)|\tilde{\lambda}, t, s, x, y))_{\varepsilon} = \\ & \prod_{j=1}^{L_{\varepsilon}} \left( \frac{-\tilde{\lambda}}{2\pi(t_j - t_{j-1})} \right)^{n/2} \int_{\mathbb{R}^n} \cdots \int_{\mathbb{R}^n} \prod_{j=1}^{L_{\varepsilon}} \left[ (a_{\varepsilon}^{(N)}(\tilde{\lambda}, t_j, t_{j-1}, x^j, y^{j-1}))_{\varepsilon} \right] \times \\ & \exp \left\{ \tilde{\lambda} \left[ \sum_{j=1}^{L_{\varepsilon}} (S_{\varepsilon}(t_j, t_{j-1}, x^j, y^{j-1}))_{\varepsilon} \right] \right\} \prod_{j=1}^{L_{\varepsilon}-1} dx^j, \end{aligned} \quad (0.8.26) $$

where $x^0 = y$ and $x^{L_{\varepsilon}} = x$. Similar formulae are valid for $(E_{\varepsilon}^{(N)}(\Delta|\tilde{\lambda}, s, t))_{\varepsilon}$. We stated that $\forall \varepsilon \in (0, 1]$

$$ \lim_{\delta(\Delta(\varepsilon)) \to 0} \mathbf{I}_{\varepsilon}^{(0)}(\Delta(\varepsilon)|\tilde{\lambda}, t, s, x, y) \quad (0.8.27) $$

exists and equals the generalized kernel function of the fundamental solution (Green's function) for the Colombeau-Schrödinger equation (0.8.22).
We note that

$$ (\mathbf{I}_{\varepsilon}^{(N)}(\Delta|\tilde{\lambda}, t, s, x, y))_{\varepsilon} = \left(\frac{-\tilde{\lambda}}{2\pi(t-s)}\right)^{n/2} \left(a_{\varepsilon}^{(N)}(\tilde{\lambda}, t, s, x, y) \exp[\tilde{\lambda}S_{\varepsilon}(t, s, x, y)]\right)_{\varepsilon} \quad (0.8.28) $$

and

$$ (\mathbf{I}_{\varepsilon}^{(N)}(\Delta|\tilde{\lambda}, s, t, x, y))_{\varepsilon} = \left(\frac{-\tilde{\lambda}}{2\pi(s-t)}\right)^{n/2} \left(a_{\varepsilon}^{(N)}(\tilde{\lambda}, s, t, x, y) \exp[\tilde{\lambda}S_{\varepsilon}(s, t, x, y)]\right)_{\varepsilon} \quad (0.8.29) $$

if $|t-s| < \delta(T)$. These Colombeau generalized amplitude functions $a^{(N)}(\tilde{\lambda}, t, s, x, y)$

and $\forall \varepsilon \in (0, 1], a_{\varepsilon}^{(N)}(\tilde{\lambda}, s, t, x, y)$ belong to the function space $D(\mathbb{R}_x^n \times \mathbb{R}_y^n)$ of Schwartz [47] with parameters $\varepsilon, \Delta(\varepsilon), \tilde{\lambda}, t, s$. As usual we shall employ the norm $\|f_{\varepsilon}\|_m$ defined by

$$ \|f_{\varepsilon}\|_{m} = \sum_{|a+\beta| \le m} \sup_{(x,y) \in \mathbb{R}_{x}^{n} \times \mathbb{R}_{y}^{n}} \left| \left(\frac{\partial}{\partial x}\right)^{a} \left(\frac{\partial}{\partial y}\right)^{\beta} f_{\varepsilon}(x,y) \right| \quad (0.8.30) $$

for $m = 0, 1, 2, \dots$. We shall denote by $a_{\varepsilon}^{(N)}(\tilde{\lambda}, t, s)$ the function $a_{\varepsilon}^{(N)}(\tilde{\lambda}, t, s, x, y)$ considered as an element of the function space $D(\mathbb{R}_x^n \times \mathbb{R}_y^n)$ with parameters $\varepsilon, \Delta(\varepsilon), \tilde{\lambda}, t, s$.

**Theorem.0.8.3. (i)** Assume that generalized Fujiwara conditions $(V_{\varepsilon}i)$ and $(V_{\varepsilon}ii)$ hold. Assume further that $|t-s| < \delta(T)$ Then, for any $N = 0, 1, 2, \dots$ both the limits

$$ (k_{\varepsilon}(\tilde{\lambda}, t, s))_{\varepsilon} = \left( \lim_{\delta(\Delta(\varepsilon)) \to 0} a_{\varepsilon}^{(N)}(\Delta(\varepsilon)|\tilde{\lambda}, t, s) \right)_{\varepsilon} \quad (0.8.31) $$

and
We note that

$$ (\mathbf{I}_{\varepsilon}^{(N)}(\Delta|\tilde{\lambda}, t, s, x, y))_{\varepsilon} = \left(\frac{-\tilde{\lambda}}{2\pi(t-s)}\right)^{n/2} \left(a_{\varepsilon}^{(N)}(\tilde{\lambda}, t, s, x, y) \exp[\tilde{\lambda}S_{\varepsilon}(t, s, x, y)]\right)_{\varepsilon} \quad (0.8.28) $$

and

$$ (\mathbf{I}_{\varepsilon}^{(N)}(\Delta|\tilde{\lambda}, s, t, x, y))_{\varepsilon} = \left(\frac{-\tilde{\lambda}}{2\pi(s-t)}\right)^{n/2} \left(a_{\varepsilon}^{(N)}(\tilde{\lambda}, s, t, x, y) \exp[\tilde{\lambda}S_{\varepsilon}(s, t, x, y)]\right)_{\varepsilon} \quad (0.8.29) $$

if $|t-s| < \delta(T)$. These Colombeau generalized amplitude functions $a^{(N)}(\tilde{\lambda}, t, s, x, y)$

and $\forall \varepsilon \in (0, 1], a_{\varepsilon}^{(N)}(\tilde{\lambda}, s, t, x, y)$ belong to the function space $D(\mathbb{R}_x^n \times \mathbb{R}_y^n)$ of Schwartz [47] with parameters $\varepsilon, \Delta(\varepsilon), \tilde{\lambda}, t, s$. As usual we shall employ the norm $\|f_{\varepsilon}\|_m$ defined by

$$ \|f_{\varepsilon}\|_{m} = \sum_{|a+\beta| \le m} \sup_{(x,y) \in \mathbb{R}_{x}^{n} \times \mathbb{R}_{y}^{n}} \left| \left(\frac{\partial}{\partial x}\right)^{a} \left(\frac{\partial}{\partial y}\right)^{\beta} f_{\varepsilon}(x,y) \right| \quad (0.8.30) $$

for $m = 0, 1, 2, \dots$. We shall denote by $a_{\varepsilon}^{(N)}(\tilde{\lambda}, t, s)$ the function $a_{\varepsilon}^{(N)}(\tilde{\lambda}, t, s, x, y)$ considered as an element of the function space $D(\mathbb{R}_x^n \times \mathbb{R}_y^n)$ with parameters $\varepsilon, \Delta(\varepsilon), \tilde{\lambda}, t, s$.

**Theorem.0.8.3. (i)** Assume that generalized Fujiwara conditions $(V_{\varepsilon}i)$ and $(V_{\varepsilon}ii)$ hold. Assume further that $|t-s| < \delta(T)$ Then, for any $N = 0, 1, 2, \dots$ both the limits

$$ (k_{\varepsilon}(\tilde{\lambda}, t, s))_{\varepsilon} = \left( \lim_{\delta(\Delta(\varepsilon)) \to 0} a_{\varepsilon}^{(N)}(\Delta(\varepsilon)|\tilde{\lambda}, t, s) \right)_{\varepsilon} \quad (0.8.31) $$

and