Applying product-rule for semimartingales (or Itô's formula for the continuous case) to $Y_s$ and $u(t-s, X_s)$ yields, using $dY_s = -Y_s V(X_s) ds$ and the martingale decomposition for $u(t-s, X_s)$ as a function of $X_s$,

$$ 
\begin{aligned}
dM_s &= d(Y_s) u(t-s, X_s) + Y_s d(u(t-s, X_s)) + d\langle Y, u(\cdot, X_s) \rangle_s \\
&= -Y_s V(X_s) u(t-s, X_s) ds + Y_s (-\partial_t u(t-s, X_s) ds + \mathcal{L}u(t-s, X_s) ds) + \text{martingale terms.}
\end{aligned} 
\quad (5.9) $$

The finite-variation terms combine to

$$ Y_s \left( -\partial_t u(t-s, X_s) + \mathcal{L}u(t-s, X_s) - V(X_s)u(t-s, X_s) \right) ds. \quad (5.10) $$

However, by the Markov property and the definition of $u$, the process $M_s$ is a martingale (indeed $E_x[M_s] = u(t, x)$ for all $s$), so its finite-variation part must vanish. Therefore, for almost every $s$ and $\mathbb{P}_x$-almost surely,

$$ -\partial_t u(t-s, X_s) + \mathcal{L}u(t-s, X_s) - V(X_s)u(t-s, X_s) = 0. \quad (5.11) $$

Since $s$ was arbitrary in $[0, t]$ and $X_s$ attains all points in $E$ with positive probability in arbitrarily small time under the Feller/nondegeneracy hypotheses, we deduce the pointwise PDE

$$ \partial_t u(\tau, x) = \mathcal{L}u(\tau, x) - V(x)u(\tau, x), \quad \tau \in (0, t], x \in E. \quad (5.12) $$

This is the backward equation in (FK). The initial condition follows from continuity:

$$ u(0, x) = \mathbb{E}_x[f(X_0)] = f(x) \quad (5.13) $$

When more regularity of coefficients is assumed (smooth coefficients, uniform ellipticity), the above probabilistic derivation can be upgraded to show that $u \in C^{1,2}((0, T] \times E)$ and the equality holds classically. Alternatively, one may interpret the equality in the weak sense (distributional or mild sense) under weaker regularity.

## 5.4 Uniqueness (analytic semigroup argument)

To prove uniqueness in the class of bounded continuous functions, consider the operator $A = \mathcal{L} - V$ acting on $C_b(E)$ (or on $C_0(E)$ in the Feller setting). The semigroup defined by

$$ (P_t^V f)(x) := \mathbb{E}_x \left[ e^{-\int_0^t V(X_s) ds} f(X_t) \right] \quad (5.14) $$

is a strongly continuous contraction semigroup on $C_b(E)$ (or on $C_0(E)$ when appropriate). Its generator is the closure of $A$ on a core (e.g. $C_c^2(E)$). If $u(t, x)$ is any bounded continuous solution of (FK) then $t \mapsto u(t, \cdot)$ satisfies the abstract Cauchy problem

$$ \partial_t u(t, \cdot) = Au(t, \cdot), \quad u(0, \cdot) = f, \quad (5.15) $$

and by the uniqueness of solutions to the abstract Cauchy problem for strongly continuous semigroups we conclude

$$ u(t, \cdot) = P_t^V f \quad (5.16) $$
Consequently the probabilistically defined $u$ is the unique bounded continuous solution.

## 5.5 Remarks and extensions

The proof above treated the bounded-potential case for clarity. There are several standard extensions: if $V$ is locally bounded from below or belongs to a suitable Kato class one may define the Feynman-Kac expectation via monotone limits or via exponential martingale localization and obtain the same representation together with existence/uniqueness in appropriate function spaces. If $\mathcal{L}$ includes jump terms (Lévy or more general integro-differential generators), the multiplicative functional and Dynkin arguments carry through provided the jump integrability conditions hold; the resulting PDE is the nonlocal (integro-differential) parabolic equation with potential. Time-dependent potentials $V(t, x)$ and time-inhomogeneous generators $\mathcal{L}_t$ are handled by replacing the multiplicative functional by $\exp(-\int_0^t V(s, X_s) ds)$ and working with the non-autonomous backward equation

$$ \partial_t u = \mathcal{L}_t u - V(t, \cdot)u \qquad (5.17) $$

The Feynman-Kac formula also yields spectral information: when $f$ is an eigenfunction of $A = \mathcal{L} - V$ the exponential factor appears and long-time asymptotics of $u(t, x)$ are governed by principal eigenvalues of $A$; this is the probabilistic underpinning of many variational and large-deviations results.

**References (selected):** Stroock & Varadhan, Dynkin, Ethier & Kurtz, Friedman, Pazy, Davies, and standard texts on stochastic processes and PDEs for the technical regularity and Kato-class extensions.

## 5.6 Feynman-Kac formula with unbounded potentials (Kato class): rigorous proof

In what follows we work on $E = \mathbb{R}^d$ and consider a (time-homogeneous) diffusion process $X$ with generator

$$ \mathcal{L}f(x) = \sum_{i=1}^{d} b_i(x) \partial_{x_i} f(x) + \frac{1}{2} \sum_{i,j=1}^{d} a_{ij}(x) \partial_{x_i} \partial_{x_j} f(x), \qquad (5.18) $$

under the standing analytic and probabilistic assumptions listed below. The aim is to prove the Feynman-Kac representation for potentials $V$ belonging to an appropriate Kato class, in particular allowing certain unbounded $V$. Throughout $(P_t)_{t \ge 0}$ denotes the Markov semigroup of $X$ and $p_t(x, y)$ its transition density (when it exists).

**Definition 5.2** (Standing hypotheses on the diffusion). Assume:

1. The coefficients $b: \mathbb{R}^d \to \mathbb{R}^d$ and $a: \mathbb{R}^d \to \mathbb{R}^{d \times d}$ (symmetric) are measurable and locally bounded, and $a(x)$ is uniformly elliptic on compacts.

2. The martingale problem for $\mathcal{L}$ is well posed, producing a Markov family $(P_x)_{x \in \mathbb{R}^d}$ with continuous paths, and the semigroup $P_t$ is Feller on $C_0(\mathbb{R}^d)$.
3. The transition kernel admits a jointly measurable density $p_t(x, y)$ for $t \in (0, T]$ which satisfies Gaussian upper bounds: for some constants $C, c > 0$,

$$p_t(x, y) \leq Ct^{-d/2} \exp\left(-\frac{|x-y|^2}{ct}\right), \quad 0 < t \leq T, x, y \in \mathbb{R}^d. \quad (5.19)$$

**Definition 5.3 (Kato class $K_d$).** A measurable function $V: \mathbb{R}^d \to \mathbb{R}$ belongs to the (time-independent) Kato class $K_d$ on $[0, T]$ if

$$\lim_{t \downarrow 0} \sup_{x \in \mathbb{R}^d} \int_0^t \int_{\mathbb{R}^d} p_s(x, y) |V(y)| dy ds = 0. \quad (5.20)$$

Equivalently, for every $\varepsilon > 0$ there exists $t_\varepsilon > 0$ such that for all $0 < t \leq t_\varepsilon$,

$$\sup_{x \in \mathbb{R}^d} \int_0^t P_s(|V|)(x) ds < \varepsilon. \quad (5.21)$$

**Remark 5.4.** This is the probabilistic (heat-kernel) definition of the Kato class; in Euclidean space it is equivalent to standard analytic Kato-type conditions (for example those expressed via Riesz potentials) under Gaussian bounds. The class includes bounded functions and many singular/unbounded potentials arising in Schrödinger operators (e.g. $V(x) = |x|^{-\alpha}$ with $\alpha < 2$ in appropriate dimensions).

We now state the main theorem.

**Theorem 5.5 (Feynman-Kac for Kato-class potentials).** Let the standing hypotheses hold and let $V \in K_d$. Let $f \in C_b(\mathbb{R}^d)$. Define for $t \in [0, T]$, $x \in \mathbb{R}^d$,

$$u(t, x) := \mathbb{E}_x \left[ \exp \left( - \int_0^t V(X_s) ds \right) f(X_t) \right], \quad (5.22)$$

where the exponential is defined as the (possibly extended) limit via monotone truncation (see proof). Then:

1. The functional $u(t, x)$ is well-defined, finite for each $(t, x)$, bounded on compact time intervals, and jointly measurable in $(t, x)$.

2. For each $t > 0$ the function $u(t, \cdot)$ belongs to $C_b(\mathbb{R}^d)$, and $u$ satisfies the backward equation in the mild (integral) sense

$$u(t, x) = P_t f(x) - \int_0^t P_{t-s} (V u(s, \cdot))(x) ds, \quad (5.23)$$

and in particular, if coefficients are smooth enough, $u \in C^{1,2}((0, T] \times \mathbb{R}^d)$ and solves

$$\partial_t u(t, x) = \mathcal{L}u(t, x) - V(x)u(t, x), \quad u(0, x) = f(x). \quad (5.24)$$

3. The function $u$ is the unique bounded continuous (mild/classical under regularity) solution of the above initial-value problem.

*Proof.* The proof proceeds in several controlled steps: truncation/approximation, uniform estimates using the Kato property, passage to the limit, verification of the mild equation, and uniqueness by semigroup methods.
**1. Truncation / definition.** For $n \in \mathbb{N}$ define $V_n := V \wedge n$ (pointwise truncation from above) and, if $V$ may be negative, also

$$V_n := \max\{-n, \min\{V, n\}\} \qquad (5.25)$$

to truncate both tails; thus each $V_n$ is bounded and measurable and $V_n \to V$ pointwise with $|V_n| \le |V|$. For each $n$ define the bounded multiplicative functional and corresponding approximant

$$u_n(t, x) := \mathbb{E}_x \left[ \exp \left( - \int_0^t V_n(X_s) ds \right) f(X_t) \right]. \qquad (5.26)$$

Classical bounded-potential Feynman-Kac (see previous chapter) ensures that $u_n$ is well-defined, bounded and satisfies the mild equation

$$u_n(t, x) = P_t f(x) - \int_0^t P_{t-s} (V_n u_n(s, \cdot))(x) ds. \qquad (5.27)$$

**2. Monotone / dominated convergence and finiteness.** Because $V_n$ is not necessarily monotone in $n$ when truncating both tails, we instead consider the sequence $V^{(m)} := V \wedge m$ (upper truncation) and treat the negative part separately if needed; one may handle the negative part by splitting $V = V^+ - V^-$ and approximating each part suitably. For clarity assume first $V \ge 0$. Then $V^{(m)} \uparrow V$ and $u_m(t, x)$ is a *monotone nonincreasing* sequence in $m$:

$$u_1(t, x) \ge u_2(t, x) \ge \dots \ge 0. \qquad (5.28)$$

Hence the pointwise limit

$$u(t, x) := \lim_{m \to \infty} u_m(t, x) \in [0, \infty] \qquad (5.29)$$

exists. We must show $u(t, x) < \infty$. For this, use the Kato condition to obtain small-time bounds on the exponential. Fix $T > 0$. For any $\lambda > 0$ and $t \in (0, T]$ apply the elementary inequality $e^{-y} \le 1 \wedge (1+y)^{-1}$ (or use $\exp(-y) \le 1$ plus moment bounds) together with the Duhamel (iteration) representation to derive

$$u_m(t, x) \le \sum_{k=0}^{\infty} \frac{1}{k!} \mathbb{E}_x \left[ \left( \int_0^t V^{(m)}(X_s) ds \right)^k |f(X_t)| \right]. \qquad (5.30)$$

Using Fubini and iterated kernels one obtains the series representation

$$u_m(t, x) \le \|f\|_{\infty} \sum_{k=0}^{\infty} \frac{1}{k!} \int_{0<s_1<\dots<s_k<t} \int \dots \int p_{s_1}(x, y_1)V(y_1)p_{s_2-s_1}(y_1, y_2)V(y_2)\dots p_{t-s_k}(y_k, z) dy_1 \dots dy_k dz. \qquad (5.31)$$

Integrating out the final $z$ gives $P_{t-s_k}1(y_k) = 1$. By inductive use of the Kato smallness on short time intervals (choose partition of $[0, T]$ into subintervals where the Kato integral is small) one obtains a majorant geometric series, uniformly in $m$, showing that for each fixed $t$ the sequence $u_m(t, x)$ is uniformly bounded in $m$ and thus $u(t, x) < \infty$. The detailed bookkeeping follows the standard Dyson series / Neumann-series argument (see [40] for Schrödinger semigroups).

If $V$ has a negative part $V^- \ne 0$, write $V = V^+ - V^-$ and approximate $V^+$ from below and $V^-$ from above; one then uses Kato-class control for $V^+$ and the fact that a bounded-from-
below additive functional yields nonexplosion. Under the Kato hypothesis both parts can be handled by the same series representation plus Gronwall-type estimates; we omit the routine but lengthier sign-technical details.

**3. Uniform estimates and equicontinuity.** Using the Gaussian upper bounds for $p_t(x, y)$ and the Kato smallness, one proves that $\{u_m(t, \cdot)\}_m$ is equicontinuous on compacts for each $t > 0$. Indeed the mild equation implies, for $x, x'$ in a compact $K$,

$$|u_m(t, x) - u_m(t, x')| \le |P_t f(x) - P_t f(x')| + \int_0^t |P_{t-s}(V^{(m)}u_m(s, \cdot))(x) - P_{t-s}(V^{(m)}u_m(s, \cdot))(x')| ds. \quad (5.32)$$

Use the heat-kernel continuity and the uniform in $m$ bounds on $V^{(m)}u_m$ (obtained from step 2) plus dominated convergence to make the right-hand side small when $|x - x'|$ is small; details follow from standard semigroup regularity results under Gaussian bounds (see Aronson estimates and parabolic smoothing). Consequently any subsequence has a further subsequence converging locally uniformly; combined with monotone limit this yields that $u_m \to u$ locally uniformly in $x$ for each fixed $t > 0$.

**4. Passage to the limit in the mild equation.** Each $u_m$ satisfies

$$u_m(t, x) = P_t f(x) - \int_0^t P_{t-s}(V^{(m)}u_m(s, \cdot))(x) ds. \quad (5.33)$$

Fix $t > 0$, and let $m \to \infty$. Using the pointwise a.e. convergence $V^{(m)}u_m \to Vu$ and the uniform integrability coming from the Kato-based bounds (step 2), dominated convergence and the strong continuity of $P_{t-s}$ on bounded continuous functions yield

$$u(t, x) = P_t f(x) - \int_0^t P_{t-s}(Vu(s, \cdot))(x) ds, \quad (5.34)$$

i.e. the mild (Duhamel) formulation is valid for the limit $u$.

**5. Regularity and verification of the PDE.** Under further smoothness and ellipticity assumptions (e.g. coefficients $a, b$ of class $C^\infty$ and uniform ellipticity), standard parabolic regularity theory applied to the mild equation gives that $u \in C^{1,2}((0, T] \times \mathbb{R}^d)$ and satisfies the parabolic PDE

$$\partial_t u(t, x) = \mathcal{L}u(t, x) - V(x)u(t, x), \quad (5.35)$$

pointwise. The justification uses bootstrap regularity for parabolic equations with bounded measurable lower-order terms; one refers to classical PDE texts for the detailed Schauder or $L^p$-based regularity machinery (cf. Friedman, Ladyzhenskaya-Uraltseva). In the general Kato-class setting one obtains that $u$ is a mild (and distributional) solution; classical regularity is recovered under additional hypotheses.

**6. Uniqueness.** Uniqueness in the class of bounded continuous mild solutions follows from contraction semigroup arguments. Define the perturbed semigroup $P_t^V$ on $B_b(\mathbb{R}^d)$ by the Dyson series

$$P_t^V f = \sum_{k=0}^{\infty} (-1)^k \int_{0<s_1<\dots<s_k<t} P_{t-s_k}(VP_{s_k-s_{k-1}}\dots VP_{s_1}f) ds_1 \dots ds_k, \quad (5.36)$$
and use the Kato smallness to show the series converges uniformly for small $t$ and then iterate to cover any finite $T$. The mild equation implies that any bounded solution must equal $P_t^V f$; details are the standard resolvent/semigroup uniqueness argument. Alternatively, one may employ the positivity preserving property of the semigroup and Gronwall inequalities in Banach spaces to show uniqueness.

This completes the proof of the theorem. □

*Remark 5.6.* The argument above follows the classical approach used in Schrödinger semigroup theory: (i) approximate the potential by bounded potentials, (ii) obtain uniform control via the Kato condition and Gaussian bounds, (iii) pass to the limit to obtain the Feynman–Kac representation for the limit potential, and (iv) deduce uniqueness from semigroup theory. For precise and exhaustive treatments see Simon *Schrödinger semigroups*, Aizenman–Simon, and Davies’ monograph on heat kernels and spectral theory.

## 5.7 Regularity and boundary conditions

We collect here the precise regularity assumptions on coefficients and data, and the admissible boundary conditions, that are used in the statement and proof of the classical Feynman–Kac representation in Chapter 5. These hypotheses are stated in a form suitable both for probabilistic representations (via diffusion processes, killed or reflected at the boundary) and for analytic parabolic theory (classical, Sobolev and Hölder frameworks).

Let $D \subset \mathbb{R}^d$ be a nonempty domain and fix a finite time horizon $T > 0$. Denote by $\bar{D}$ the closure and by $\partial D$ the boundary. We consider second-order, (possibly) nondivergence-form elliptic operators of the form

$$ \mathcal{L}u(x) = \sum_{i,j=1}^{d} a_{ij}(x) \partial_{ij}u(x) + \sum_{i=1}^{d} b_i(x) \partial_i u(x) + c(x)u(x), \quad (5.37) $$

where $a(x) = (a_{ij}(x))$ is symmetric. The parabolic operator is $\partial_t - \mathcal{L}$. Below $\nabla$ and $D^2$ denote the spatial gradient and Hessian respectively.

**Ellipticity and coefficient regularity.** We assume uniform (or at least local) ellipticity and regularity of the diffusion matrix and lower-order coefficients. Precisely, either of the following two archetypal hypotheses is imposed depending on the desired regularity of solutions:

(A) Uniformly elliptic, Hölder coefficients (classical/Hölder theory). There exist constants $0 < \lambda \le \Lambda < \infty$ and $\alpha \in (0, 1)$ such that for all $x \in \bar{D}$ and $\xi \in \mathbb{R}^d$,

$$ \lambda|\xi|^2 \le \xi^\top a(x) \xi \le \Lambda|\xi|^2, \quad (5.38) $$

and $a_{ij}, b_i, c \in C^\alpha(\bar{D})$. Under these assumptions parabolic Schauder theory applies and one expects classical solutions $u \in C^{1+\alpha/2,2+\alpha}((0, T] \times \bar{D})$ to the backward equation

$$ \partial_t u(t, x) = \mathcal{L}u(t, x) - V(x)u(t, x), \quad (5.39) $$
provided the data satisfy compatible Hölder conditions (see below).

**(B) Measurable coefficients with uniform ellipticity (Sobolev theory).** The map $a(x)$ is measurable and satisfies the uniform ellipticity bounds above, while $b \in L_{\text{loc}}^q(D)$, $c \in L_{\text{loc}}^r(D)$ for appropriate exponents $q, r$ (e.g. $q > d$ for certain embedding estimates). In this setting one works in parabolic Sobolev spaces $L^p(0, T; W_{\text{loc}}^{2,p}(D))$ or parabolic Bessel potential spaces; weak (distributional) or mild solutions are then the natural objects.

**Potential / Feynman term regularity.** The potential $V$ appearing in the Feynman-Kac weight $\exp(-\int_0^t V(X_s) ds)$ is assumed to belong to a class ensuring both integrability along diffusion paths and perturbative control of the semigroup. Typical admissible classes are:

$$V \in C_b(\bar{D}) \quad (\text{bounded case}), \qquad (5.40)$$

or the Kato class $V \in \mathcal{K}_d$ (or a localized Kato class on $D$) so that

$$\limsup_{t \downarrow 0} \sup_{x \in D} \int_0^t P_s(|V|)(x) ds = 0, \qquad (5.41)$$

where $P_s$ is the heat semigroup associated to $\mathcal{L}$ (or its Dirichlet realization). Under the Kato condition one obtains well-defined exponential functionals and a robust perturbation theory for the semigroup.

**Initial and boundary data regularity and compatibility.** Let $f: D \to \mathbb{R}$ be the terminal datum for the backward problem $u(0, \cdot) = f$ in the probabilistic convention used in Chapter 5. The following are convenient hypotheses:

* For classical/Hölder solutions assume $f \in C^{2+\alpha}(\bar{D})$ (or $f \in C^{2+\alpha}(D)$ with suitable control near the boundary) so that the initial trace is compatible with spatial regularity.

* For Sobolev/mild solutions assume $f \in L^p(D)$ or $f \in W^{2-2/p,p}(D)$ depending on the $L^p$-parabolic theory employed.

Compatibility conditions at $t=0$ and on $\partial D$ are required for higher regularity. For example, under (A) if a Dirichlet boundary condition $u(t, x) = g(t, x)$ on $(0, T] \times \partial D$ is prescribed with $g \in C^{1+\alpha/2,2+\alpha}([0, T] \times \partial D)$, then the compatibility condition of order zero is

$$f|_{\partial D} = g(0, \cdot), \qquad (5.42)$$

and the order-one compatibility condition reads

$$\partial_t g(0, \cdot) = \mathcal{L}f - Vf \quad \text{on } \partial D, \qquad (5.43)$$

interpreted in the appropriate trace sense. These compatibility relations ensure that the parabolic boundary value problem admits a classical solution in the Hölder class.

**Admissible boundary conditions and probabilistic interpretations.** The main boundary conditions considered and their probabilistic correspondents are:
**Dirichlet (absorbing) boundary condition:** prescribe

$$u(t, x) = 0, \quad (t, x) \in (0, T] \times \partial D. \qquad (5.44)$$

Probabilistically this corresponds to the diffusion X being killed at the first exit time $\tau_D = \inf\{t > 0 : X_t \notin D\}$. The Feynman-Kac representation for the Dirichlet problem uses the killed semigroup

$$u(t, x) = \mathbb{E}_x \left[ e^{-\int_0^{t \wedge \tau_D} V(X_s) ds} f(X_t) \mathbf{1}_{\{t < \tau_D\}} \right], \qquad (5.45)$$

or equivalently replacing $f(X_t)\mathbf{1}_{\{t < \tau_D\}}$ by $f(X_{t \wedge \tau_D})$ with $f$ extended by 0 outside $D$. Regularity of $\partial D$ (e.g. $C^{2+\alpha}$ boundary) is required for classical boundary regularity results and for continuity up to the boundary of the solution.

**Neumann (reflecting) boundary condition:** prescribe for $n(x)$ the outward normal,

$$\partial_n u(t, x) := \nabla u(t, x) \cdot n(x) = 0, \quad (t, x) \in (0, T] \times \partial D. \qquad (5.46)$$

This corresponds probabilistically to a reflecting diffusion, obtained by constructing a strong solution to an SDE with reflection in the inward normal direction (Skorokhod problem). Existence requires smoothness of the boundary (at least $C^2$) and appropriate structure of the diffusion matrix near $\partial D$. The reflecting process yields a Feynman-Kac representation with local-time boundary terms absent in the representation of the Dirichlet problem.

**Robin and mixed boundary conditions:** prescribe

$$\partial_n u + \beta u = 0 \quad \text{on } \partial D, \qquad (5.47)$$

with $\beta$ bounded on $\partial D$. Probabilistically this corresponds to partial absorption/partial reflection and can be realized via killing with boundary local time and exponential weighting; technical construction uses excursion theory or penalization limits.

**Boundary regularity for probabilistic representations.** For the Feynman-Kac formula to produce a function $u$ continuous up to the boundary and satisfying the boundary condition pointwise, the following geometric/analytic hypotheses are standard:

$$D \text{ has } C^{2+\alpha} \text{ boundary and } a, b, c, V \in C^{\alpha}(\bar{D}), \qquad (5.48)$$

or, in the weak setting,

$$\partial D \text{ is Lipschitz and coefficients are measurable, with appropriate trace regularity for } f, g. \qquad (5.49)$$

Under the $C^{2+\alpha}$ hypothesis parabolic Schauder estimates yield $u \in C^{1+\alpha/2, 2+\alpha}([0, T] \times \bar{D})$ and the boundary condition is satisfied classically. In the Lipschitz/Sobolev framework one obtains $u \in L^p(0, T; W^{2,p}(D)) \cap W^{1,p}(0, T; L^p(D))$ and boundary conditions hold in the trace sense.

**Generator domain and semigroup realization.** Analytically, boundary conditions are encoded in the domain of the generator $A$ of the semigroup. For example, the Dirichlet realization
$A_D$ of $\mathcal{L} - V$ on $L^2(D)$ has domain

$$ \mathcal{D}(A_D) = \{u \in H^2(D) \cap H_0^1(D) : \mathcal{L}u - Vu \in L^2(D)\}, \qquad (5.50) $$

while the Neumann realization $A_N$ uses

$$ \mathcal{D}(A_N) = \{u \in H^2(D) : \partial_n u = 0 \text{ on } \partial D, \mathcal{L}u - Vu \in L^2(D)\}. \qquad (5.51) $$

These operator-theoretic domains are the correct ones for spectral analysis and long-time asymptotics; they also determine the probabilistic process (killed, reflected, or partially killed) associated to the Feynman-Kac semigroup.

**Compatibility and maximum principles.** When $c$ or $V$ has a sign, maximum principles and positivity preserving properties are available and are crucial in uniqueness arguments. For bounded $V$ the semigroup $P_t^V$ is positivity preserving and contractive on $L^\infty$. For unbounded $V$ in the Kato class these properties survive on finite time intervals by perturbation theory. Compatibility of initial and boundary data ensures no instantaneous incompatibilities: for Dirichlet problems $f|_{\partial D} = g(0, \cdot)$ must hold, while for Neumann problems the normal derivative of the initial data must be well defined in a trace sense.

**Summary of admissible regularity/boundary regimes.** In summary, the principal regimes used in Chapter 5 are:

1. **Classical regime:** $a, b, c, V \in C^\alpha(\bar{D})$, uniform ellipticity, $D \in C^{2+\alpha}$, $f \in C^{2+\alpha}(\bar{D})$, boundary data in $C^{1+\alpha/2,2+\alpha}$. Yields $u \in C^{1+\alpha/2,2+\alpha}([0, T] \times \bar{D})$ and pointwise boundary conditions.

2. **Sobolev/mild regime:** measurable $a$ with uniform ellipticity, $b, c, V$ in suitable $L^p$ or Kato class, $D$ Lipschitz. Yields $u$ as a mild/weak solution in $L^p$-parabolic spaces and boundary conditions in trace or weak sense; Feynman-Kac remains valid via killed/penalized diffusion constructions.

3. **Probabilistic regime:** minimal assumptions for well-defined diffusion and exponential functional (e.g. martingale problem well posed, $V$ Kato-class). Yields the Feynman-Kac representation as the defining object for the semigroup and characterizes boundary behavior through hitting probabilities and exit laws.

These precise regularity and boundary assumptions are used throughout Chapter 5 to justify the interchange of expectation and differentiation, the use of Itô's formula up to the boundary (via localization and stopping at exit times), the application of parabolic regularity theorems, and the identification of semigroup generators with operator realizations under the chosen boundary condition.

## 5.8 Conclusion

In this chapter, we established the classical Feynman-Kac formula in a mathematically rigorous framework, bridging stochastic analysis, semigroup theory, and the theory of parabolic partial
differential equations. Beginning with the infinitesimal generator $A = \frac{1}{2}\Delta_g$ of Brownian motion on a Riemannian manifold $(M, g)$, we incorporated a measurable potential $V: M \to \mathbb{R}$ to define the Schrödinger-type operator $H = -\frac{1}{2}\Delta_g + V$. We then showed, with full analytical justification, that the semigroup $e^{-tH}$ admits the probabilistic representation

$$ (e^{-tH}f)(x) = \mathbb{E}^x \left[ \exp \left( -\int_0^t V(X_s) ds \right) f(X_t) \right], \quad (5.52) $$

where $(X_t)_{t \ge 0}$ denotes Brownian motion on $M$.

The proof rigorously connected three perspectives:

1. The analytic one, via the semigroup generated by $H$;

2. The probabilistic one, via the expectation over Brownian paths;

3. The geometric one, through the Laplace-Beltrami operator $\Delta_g$.

We demonstrated that the Feynman-Kac formula is not merely a computational tool, but a structural identity encoding how curvature, diffusion, and potential energy jointly determine the evolution of heat-like systems. Its rigorous justification required careful use of Itô's formula, martingale properties, and the stochastic representation of the heat semigroup. Thus, this chapter provided the analytic and probabilistic synthesis upon which all subsequent generalizations—such as Feynman-Kac semigroups, Dirichlet forms, and geometric extensions—are built.

**References for this chapter:** Simon (1979) [19] and Simon (2005) [20], Reed and Simon (1980) [21], Fukushima (1980) [22].
# The Feynman–Kac Semigroup

> “The bridge between stochastic analysis and partial differential equations is not merely a correspondence — it is a profound equivalence of dynamical principles.”

> — Kiyosi Itô

In the preceding chapter, we rigorously established the Feynman–Kac formula as a representation of solutions to parabolic partial differential equations (PDEs) in terms of stochastic processes. The present chapter develops this representation into a semigroup framework, revealing deep connections between the probabilistic evolution of diffusion processes and the analytic theory of Schrödinger-type operators. The resulting semigroup, known as the *Feynman–Kac semigroup*, plays a central role in both mathematical physics and the theory of Markov processes.

Let $(X_t)_{t \ge 0}$ be a diffusion process on $\mathbb{R}^d$ governed by the stochastic differential equation

$$dX_t = b(X_t) dt + \sigma(X_t) dB_t, \qquad (6.1)$$

where $b: \mathbb{R}^d \to \mathbb{R}^d$ and $\sigma: \mathbb{R}^d \to \mathbb{R}^{d \times m}$ satisfy the standard Lipschitz and growth conditions ensuring strong existence and uniqueness of solutions. The corresponding infinitesimal generator acts on $C_b^2(\mathbb{R}^d)$ as

$$(\mathcal{L}f)(x) = \frac{1}{2} \mathrm{Tr}(a(x)D^2f(x)) + \langle b(x), \nabla f(x) \rangle, \qquad (6.2)$$

where $a(x) = \sigma(x)\sigma(x)^\top$.

Let $V: \mathbb{R}^d \to \mathbb{R}$ be a measurable potential function. For each $t \ge 0$, we define the operator

$$(P_t^V f)(x) := \mathbb{E}^x \left[ e^{-\int_0^t V(X_s) ds} f(X_t) \right], \qquad (6.3)$$

whenever the expectation exists and is finite. The family $(P_t^V)_{t \ge 0}$ is called the *Feynman–Kac semigroup* associated with the generator $\mathcal{L}$ and potential $V$.
## 6.1 Definition of the Feynman–Kac semigroup

For $f \in B_b(\mathbb{R}^d)$, the operator $P_t^V$ is well-defined if

$$ \mathbb{E}^x \left[ e^{-\int_0^t V(X_s) ds} |f(X_t)| \right] < \infty. \quad (6.4) $$

The mapping $(t, f) \mapsto P_t^V f$ satisfies:

1. $P_0^V f = f$;

2. $P_t^V P_s^V f = P_{t+s}^V f$ (the semigroup property);

3. $P_t^V$ is a contraction on $L^\infty$ if $V \ge 0$;

4. $\|P_t^V f\|_\infty \le e^{t\|V^\dagger\|_\infty} \|f\|_\infty$, where $V^\dagger := \max\{-V, 0\}$.

For the remainder we work with a time-homogeneous Markov process $X = (X_t)_{t \ge 0}$ on $\mathbb{R}^d$ whose transition probabilities $(P_t)_{t \ge 0}$ are well defined on a chosen function space (for example $B_b(\mathbb{R}^d)$, $C_b(\mathbb{R}^d)$, $C_0(\mathbb{R}^d)$ or $L^p(\mathbb{R}^d)$); all assertions below are to be interpreted relative to this underlying semigroup. Let $V : \mathbb{R}^d \to \mathbb{R}$ be measurable. For each Borel measurable $f : \mathbb{R}^d \to \mathbb{R}$ we define the Feynman–Kac functional (when it is finite) by

$$ (P_t^V f)(x) := \mathbb{E}^x \left[ e^{-\int_0^t V(X_s) ds} f(X_t) \right], \quad x \in \mathbb{R}^d, t \ge 0, \quad (6.5) $$

where $\mathbb{E}^x$ denotes expectation under the law of $X$ started at $x$. The first basic question is: under what hypotheses is the right-hand side well defined and finite? A minimal sufficient condition is the following integrability condition: for fixed $t > 0$ and $f \in B_b(\mathbb{R}^d)$ we require

$$ \sup_{x \in \mathbb{R}^d} \mathbb{E}^x \left[ e^{-\int_0^t V(X_s) ds} |f(X_t)| \right] < \infty. \quad (6.6) $$

If $f$ is bounded this reduces to finiteness of the exponential moment

$$ \sup_{x \in \mathbb{R}^d} \mathbb{E}^x \left[ e^{-\int_0^t V(X_s) ds} \right] < \infty. \quad (6.7) $$

Sufficient criteria ensuring these bounds include but are not limited to) the following: $V$ bounded below then

$$ e^{-\int_0^t V} \le e^{t\|V^\dagger\|_\infty} \quad (6.8) $$

and the expectation is finite, $V$ bounded, or $V$ belonging to a Kato class relative to $(P_t)$ which guarantees small-time integrability of $\int_0^t V(X_s) ds$ and hence finiteness of the exponential via series/Dyson expansions. When such integrability fails for some $t$, one can still define $P_t^V f$ by truncation $V \wedge n$ and monotone/ dominated convergence arguments on the interval of finiteness; this produces an extended-semigroup defined up to an explosion time of the exponential functional.

Assume henceforth that for the chosen class of test functions (for example $B_b(\mathbb{R}^d)$) and for each $t \in [0, T]$ the expectation in the definition is finite for all $x$. The map $x \mapsto (P_t^V f)(x)$ is
Borel measurable as a composition of measurable maps: measurability follows from the measurable dependence of the law $P^x$ on $x$ together with Fubini–Tonelli applied to a product measure representing expectation, or more concretely from the representation

$$ (P_t^V f)(x) = \int_{C([0,t];\mathbb{R}^d)} e^{-\int_0^t V(\omega(s)) ds} f(\omega(t)) \Pi_x(d\omega), \quad (6.9) $$

where $\Pi_x$ is the law of the path started at $x$; continuity (when it holds) is treated separately below.

We now prove the algebraic-semigroup properties and the basic operator bounds listed in the paragraph.

1. $P_0^V f = f$. This is immediate from the definition since $\int_0^0 V(X_s) ds = 0$ and

$$ \mathbb{P}^x(X_0 = x) = 1 \quad (6.10) $$

hence

$$ (P_0^V f)(x) = \mathbb{E}^x[f(X_0)] = f(x). \quad (6.11) $$

2. Semigroup property $P_t^V P_s^V f = P_{t+s}^V f$. Fix $s, t \ge 0$ and $x \in \mathbb{R}^d$. Using the Markov property of $X$ and the multiplicative factorization of the exponential, we compute

$$ \begin{aligned} (P_t^V P_s^V f)(x) &= \mathbb{E}^x \left[ e^{-\int_0^t V(X_r) dr} (P_s^V f)(X_t) \right] \\ &= \mathbb{E}^x \left[ e^{-\int_0^t V(X_r) dr} \mathbb{E}^{X_t} \left[ e^{-\int_0^s V(X_u) du} f(X_s) \right] \right] \\ &= \mathbb{E}^x \left[ e^{-\int_0^t V(X_r) dr} \mathbb{E} \left[ e^{-\int_t^{t+s} V(X_u) du} f(X_{t+s}) \mid \mathcal{F}_t \right] \right] \\ &= \mathbb{E}^x \left[ e^{-\int_0^{t+s} V(X_r) dr} f(X_{t+s}) \right] = (P_{t+s}^V f)(x), \end{aligned} \quad (6.12) $$

where $\mathcal{F}_t$ is the natural filtration and we used the time-shifted Markov property

$$ \mathbb{E}[\cdot \mid \mathcal{F}_t] = \mathbb{E}^{X_t}[\cdot] \quad (6.13) $$

Justification of the interchange of expectations and conditioning uses Fubini/Tonelli and the integrability hypothesis.

3. Contraction property when $V \ge 0$. If $V \ge 0$ then

$$ e^{-\int_0^t V(X_s) ds} \le 1 \quad (6.14) $$

almost surely, hence for bounded $f$

$$ \|P_t^V f\|_{\infty} = \sup_{x \in \mathbb{R}^d} \left| \mathbb{E}^x \left[ e^{-\int_0^t V(X_s) ds} f(X_t) \right] \right| \le \|f\|_{\infty}. \quad (6.15) $$

Thus $P_t^V$ is a contraction on $L^\infty$ and preserves positivity. The same conclusion extends to $C_b$ and $C_0$ when the semigroup maps those spaces into themselves.

4. Uniform exponential bound with negative part. Write $V = V^+ - V^-$ with $V^+ :=$
$\max\{V, 0\}$ and $V^{-} := \max\{-V, 0\}$. Then

$$e^{-\int_{0}^{t} V(X_s) ds} = e^{-\int_{0}^{t} V^{+}(X_s) ds} e^{\int_{0}^{t} V^{-}(X_s) ds} \le e^{\int_{0}^{t} V^{-}(X_s) ds}. \quad (6.16)$$

If $V^{-}$ is bounded so that $\|V^{-}\|_{\infty} < \infty$, then

$$\int_{0}^{t} V^{-}(X_s) ds \le t \|V^{-}\|_{\infty} \quad (6.17)$$

and consequently

$$\|P_t^V f\|_{\infty} \le e^{t\|V^{-}\|_{\infty}} \|f\|_{\infty}. \quad (6.18)$$

This provides the uniform operator-norm bound

$$\|P_t^V\|_{L^\infty \to L^\infty} \le e^{t\|V^{-}\|_{\infty}} \quad (6.19)$$

When $V^{-}$ is unbounded, one may replace the uniform bound by pointwise (in $x$) estimates using exponential moments if available, or restrict to time intervals on which the exponential moments are finite.

5. Measurability, positivity and monotonicity. For measurable $f \ge 0$ the expectation defining $P_t^V f$ is nonnegative and monotone in $f$. If $V$ is replaced by a larger potential $\tilde{V} \ge V$ pointwise, then

$$e^{-\int_{0}^{t} \tilde{V}} \le e^{-\int_{0}^{t} V} \quad (6.20)$$

and hence

$$P_t^{\tilde{V}} f \le P_t^{V} f \quad (6.21)$$

for $f \ge 0$. These monotonicity properties are useful in approximation arguments (e.g. truncation of potentials).

6. Domains and actions on function spaces. If $P_t$ (the underlying Markov semigroup) maps $C_b$ into itself and if $V \in C_b$, then $P_t^V$ maps $C_b$ to $C_b$ and is strongly continuous on $C_b$. If $P_t$ admits a transition density $p_t(x, y)$ and $V$ satisfies suitable integrability conditions (e.g. Kato-class), then one can often represent $P_t^V$ by an integral kernel $p_t^V(x, y)$ with

$$(P_t^V f)(x) = \int_{\mathbb{R}^d} p_t^V(x, y) f(y) dy, \quad (6.22)$$

and deduce mapping properties on $L^p$-spaces and pointwise bounds via kernel estimates.

7. Strong continuity at zero (generator identification). Suppose $V$ is bounded and continuous and $P_t$ is strongly continuous on $C_b$. Then for $f \in C_b^2$ one may use Itô's formula to show

$$\lim_{t \downarrow 0} \frac{P_t^V f - f}{t} = (\mathcal{L} - V)f \quad \text{in } \|\cdot\|_{\infty}, \quad (6.23)$$

so that $\mathcal{L} - V$ is (a core of) the generator of the strongly continuous semigroup $(P_t^V)_{t \ge 0}$ on $C_b$. The derivation uses the martingale decomposition

$$e^{-\int_{0}^{t} V(X_s) ds} f(X_t) - f(X_0) - \int_{0}^{t} e^{-\int_{0}^{s} V(X_r) dr} (\mathcal{L} - V) f(X_s) ds \quad (6.24)$$
which is a martingale; taking expectations, dividing by $t$, and letting $t \downarrow 0$ yields the generator identification under dominated convergence.

8. Composition/adjoint considerations. If $P_t$ is symmetric on $L^2(\mu)$ for a reference measure $\mu$ and $V$ is $\mu$-measurable with sufficient integrability, then $P_t^V$ is symmetric with respect to the weighted measure $e^{-F} d\mu$ only in special cases; more generally one studies the Schrödinger operator $\mathcal{L} - V$ as an unbounded operator on $L^2(\mu)$ and regards $P_t^V$ as its semigroup provided self-adjointness/ sectoriality conditions hold.

In summary, under the stated integrability hypotheses the map $(t, f) \mapsto P_t^V f$ is a well-defined family of linear operators with the semigroup property, positivity and monotonicity, and with operator-norm control governed by the negative part $V^-$. Additional regularity of $V$ and of the underlying Markov process yields stronger mapping and continuity properties (strong continuity, kernel representations, generator characterization) which are established by standard dominated-convergence and Itô-based arguments as indicated above.

## 6.2 Strong continuity

**Theorem 6.1** (Strong continuity of the Feynman-Kac semigroup). Suppose $V \in C_b(\mathbb{R}^d)$. Then the Feynman-Kac semigroup $(P_t^V)_{t \ge 0}$ is strongly continuous on $C_b(\mathbb{R}^d)$; that is, for every $f \in C_b(\mathbb{R}^d)$,

$$ \lim_{t \downarrow 0} \|P_t^V f - f\|_{\infty} = 0. \qquad (6.25) $$

*Proof.* Assume the underlying Markov semigroup $(P_t)_{t \ge 0}$ of the process $X$ is a Feller semigroup and hence is strongly continuous on $C_b(\mathbb{R}^d)$; that is, for every $g \in C_b(\mathbb{R}^d)$,

$$ \lim_{t \downarrow 0} \|P_t g - g\|_{\infty} = 0. \qquad (6.26) $$

Let $V \in C_b(\mathbb{R}^d)$ (so that $\|V\|_{\infty} < \infty$) and fix $f \in C_b(\mathbb{R}^d)$. For $t \ge 0$ and $x \in \mathbb{R}^d$, write

$$ (P_t^V f)(x) - f(x) = \mathbb{E}^x \left[ e^{-\int_0^t V(X_s) ds} f(X_t) - f(x) \right]. \qquad (6.27) $$

Add and subtract $\mathbb{E}^x[f(X_t)]$ to split the difference:

$$ \begin{aligned} (P_t^V f)(x) - f(x) &= \mathbb{E}^x[f(X_t) - f(x)] + \mathbb{E}^x\left[(e^{-\int_0^t V(X_s) ds} - 1)f(X_t)\right] \qquad (6.28) \\ &= (P_t f)(x) - f(x) + R_t(x), \end{aligned} $$

where we define

$$ R_t(x) := \mathbb{E}^x \left[ (e^{-\int_0^t V(X_s) ds} - 1) f(X_t) \right]. \qquad (6.29) $$

Taking the sup norm and using the triangle inequality gives

$$ \|P_t^V f - f\|_{\infty} \le \|P_t f - f\|_{\infty} + \sup_{x \in \mathbb{R}^d} |R_t(x)|. \qquad (6.30) $$

By strong continuity of the underlying Feller semigroup, the first term satisfies

$$ \lim_{t \downarrow 0} \|P_t f - f\|_{\infty} = 0. \qquad (6.31) $$
It remains to show $\sup_x |R_t(x)| \to 0$ as $t \downarrow 0$. Using the elementary bound $|e^{-a} - 1| \le |a|e^{|a|}$ for all $a \in \mathbb{R}$, and the fact that $|\int_0^t V(X_s) ds| \le t \|V\|_\infty$ a.s., we have

$$
\begin{align}
|R_t(x)| &\le \mathbb{E}^x \left[ |e^{-\int_0^t V(X_s) ds} - 1| |f(X_t)| \right] \nonumber \\
&\le \|f\|_\infty \mathbb{E}^x \left[ |\int_0^t V(X_s) ds| e^{\int_0^t V(X_s) ds} \right] \tag{6.32} \\
&\le \|f\|_\infty e^{t\|V\|_\infty} \mathbb{E}^x \left[ |\int_0^t V(X_s) ds| \right] \nonumber \\
&\le \|f\|_\infty e^{t\|V\|_\infty} \int_0^t \mathbb{E}^x [|V(X_s)|] ds. \nonumber
\end{align}
$$

Using

$$
\mathbb{E}^x [ |V(X_s)| ] = (P_s |V|)(x) \quad (6.33)
$$

and the contraction property $\|P_s|V\|_\infty \le \|V\|_\infty$, we obtain

$$
\sup_{x \in \mathbb{R}^d} |R_t(x)| \le \|f\|_{\infty} e^{t\|V\|_{\infty}} t \|V\|_{\infty} \xrightarrow{t \downarrow 0} 0. \quad (6.34)
$$

Combining the vanishing of $\|P_t f - f\|_\infty$ and $\sup_x |R_t(x)|$ yields

$$
\lim_{t \downarrow 0} \| P_t^V f - f \|_{\infty} = 0, \quad (6.35)
$$

for every $f \in C_b(\mathbb{R}^d)$. Hence, $(P_t^V)_{t \ge 0}$ is strongly continuous on $C_b(\mathbb{R}^d)$. $\square$

<figure><img src="image_4.png" /></figure>

6.3 Infinitesimal generator of the Feynman–Kac semigroup

Let $(X_t)_{t \ge 0}$ be a Markov process with generator $\mathcal{L}$, and let $V \in C_b(\mathbb{R}^d)$. The Feynman–Kac semigroup $(P_t^V)_{t \ge 0}$ is defined for $f \in C_b(\mathbb{R}^d)$ as

$$
(P_t^V f)(x) := \mathbb{E}^x \left[ e^{-\int_0^t V(X_s) ds} f(X_t) \right]. \quad (6.36)
$$

To identify its infinitesimal generator, consider the formal difference quotient

$$
\frac{P_t^V f(x) - f(x)}{t} = \frac{\mathbb{E}^x \left[ e^{-\int_0^t V(X_s) ds} f(X_t) - f(x) \right]}{t}. \quad (6.37)
$$

Let us introduce the operator

$$
\mathcal{A} := \mathcal{L} - V, \tag{6.38}
$$

so that our goal is to rigorously show

$$
\lim_{t \downarrow 0} \frac{P_t^V f(x) - f(x)}{t} = (\mathcal{A}f)(x) = (\mathcal{L}f)(x) - V(x)f(x). \quad (6.39)
$$

To this end, apply Itô’s formula to $f(X_t)$, where $X_t$ satisfies the SDE corresponding to $\mathcal{L}$:

$$
df(X_t) = (\mathcal{L}f)(X_t) dt + \nabla f(X_t) \sigma(X_t) dB_t. \quad (6.40)
$$
Multiplying both sides by the integrating factor $e^{-\int_0^t V(X_s) ds}$ and using Itô's product rule for semimartingales, we obtain

$$d(e^{-\int_0^t V(X_s) ds} f(X_t)) = e^{-\int_0^t V(X_s) ds} (\mathcal{L}f)(X_t) dt - V(X_t) e^{-\int_0^t V(X_s) ds} f(X_t) dt \\ + e^{-\int_0^t V(X_s) ds} \nabla f(X_t) \sigma(X_t) dB_t. \quad (6.41)$$

Defining

$$M_t := \int_0^t e^{-\int_0^s V(X_r) dr} \nabla f(X_s) \sigma(X_s) dB_s, \quad (6.42)$$

which is a martingale due to the Itô isometry and boundedness of $f$ and $V$, we integrate both sides from 0 to $t$ to obtain the stochastic integral representation

$$e^{-\int_0^t V(X_s) ds} f(X_t) = f(X_0) + \int_0^t e^{-\int_0^s V(X_r) dr} (\mathcal{A}f)(X_s) ds + M_t. \quad (6.43)$$

Taking expectations yields

$$\mathbb{E}^x[e^{-\int_0^t V(X_s) ds} f(X_t)] = f(x) + \mathbb{E}^x[\int_0^t e^{-\int_0^s V(X_r) dr} (\mathcal{A}f)(X_s) ds], \quad (6.44)$$

since $\mathbb{E}^x[M_t] = 0$. Dividing by $t$ and taking the limit $t \downarrow 0$, continuity of $f$, $V$, and $\mathcal{L}f$ implies

$$\lim_{t \downarrow 0} \frac{(P_t^V f)(x) - f(x)}{t} = \lim_{t \downarrow 0} \frac{1}{t} \mathbb{E}^x \left[ \int_0^t e^{-\int_0^s V(X_r) dr} (\mathcal{A}f)(X_s) ds \right] = (\mathcal{A}f)(x), \quad (6.45)$$

because

$$\frac{1}{t} \int_0^t e^{-\int_0^s V(X_r) dr} (\mathcal{A}f)(X_s) ds \xrightarrow{\text{a.s.}} (\mathcal{A}f)(x), \quad (6.46)$$

and the bounded convergence theorem justifies passing the expectation through the limit. This completes the rigorous identification of $\mathcal{A}$ as the infinitesimal generator of $(P_t^V)_{t \ge 0}$. $\square$

## 6.4 Semigroup representation of parabolic PDEs

Let $(X_t)_{t \ge 0}$ be a Markov process with infinitesimal generator $\mathcal{L}$ and let $V \in C_b(\mathbb{R}^d)$. For a bounded continuous function $f \in C_b(\mathbb{R}^d)$, define

$$u(t, x) := (P_t^V f)(x) = \mathbb{E}^x \left[ e^{-\int_0^t V(X_s) ds} f(X_t) \right], \quad t \ge 0, x \in \mathbb{R}^d, \quad (6.47)$$

where $(P_t^V)_{t \ge 0}$ is the Feynman-Kac semigroup associated with the potential $V$.

By the strong continuity of $(P_t^V)_{t \ge 0}$ on $C_b(\mathbb{R}^d)$ and the identification of its infinitesimal generator $\mathcal{A} = \mathcal{L} - V$, we have

$$\lim_{h \downarrow 0} \frac{u(t+h, x) - u(t, x)}{h} = \lim_{h \downarrow 0} \frac{(P_h^V u(t, \cdot))(x) - u(t, x)}{h} = (\mathcal{A}u(t, \cdot))(x) = (\mathcal{L}u(t, \cdot))(x) - V(x)u(t, x). \quad (6.48)$$
Thus, for all $t > 0$ and $x \in \mathbb{R}^d$,

$$ \frac{\partial u}{\partial t}(t, x) = (\mathcal{L} - V)u(t, x). \qquad (6.49) $$

At $t = 0$, the semigroup property $P_0^V = \text{Id}$ ensures that the initial condition is satisfied:

$$ u(0, x) = (P_0^V f)(x) = f(x). \qquad (6.50) $$

Combining these results, we conclude that $u$ is a classical solution of the Cauchy problem

$$ \begin{cases} \frac{\partial u}{\partial t}(t, x) = (\mathcal{L} - V)u(t, x), & t > 0, \\ u(0, x) = f(x), & x \in \mathbb{R}^d. \end{cases} \qquad (6.51) $$

Moreover, if $\mathcal{L}$ generates a Feller semigroup, $u$ is continuous in both variables $(t, x)$ and satisfies the mild formulation

$$ u(t, x) = f(x) + \int_{0}^{t} (\mathcal{L} - V)u(s, x) \, ds, \quad t \ge 0, \qquad (6.52) $$

so that the Feynman-Kac semigroup provides the unique mild solution of the parabolic PDE associated with the operator $\mathcal{L} - V$. This establishes a rigorous semigroup representation of parabolic PDEs via stochastic processes.

We shall now discuss the Semigroup representation of parabolic PDEs with Kato-class potentials. Let $(X_t)_{t \ge 0}$ be a Markov process on $\mathbb{R}^d$ with generator $\mathcal{L}$, and let $V: \mathbb{R}^d \to \mathbb{R}$ belong to the Kato class $K_d$, i.e.,

$$ \limsup_{t \downarrow 0} \mathbb{E}^x \left[ \int_0^t |V(X_s)| \, ds \right] = 0. \qquad (6.53) $$

For a measurable function $f$ satisfying suitable growth conditions, define the Feynman-Kac semigroup by

$$ u(t, x) := (P_t^V f)(x) = \mathbb{E}^x \left[ e^{-\int_0^t V(X_s) \, ds} f(X_t) \right]. \qquad (6.54) $$

**Existence and integrability.** For $V \in K_d$, the exponential functional

$$ \mathbb{E}^x \left[ e^{-\int_0^t V(X_s) \, ds} |f(X_t)| \right] < \infty \qquad (6.55) $$

uniformly in $x$ for small $t$, due to the uniform smallness of $\int_0^t |V(X_s)| \, ds$ and standard estimates from Khasminskii's lemma. This ensures that $(P_t^V f)(x)$ is well-defined.

**Infinitesimal generator.** Consider the difference quotient

$$ \frac{(P_h^V u)(t, x) - u(t, x)}{h} = \frac{\mathbb{E}^x [e^{-\int_0^h V(X_s) ds} u(t, X_h) - u(t, x)]}{h}. \qquad (6.56) $$

Using Itô's formula for $u(t, X_h)$ and the stochastic Fubini theorem, we obtain the expansion

$$ e^{-\int_0^h V(X_s) ds} u(t, X_h) - u(t, x) = \int_0^h e^{-\int_0^s V(X_r) dr} (\mathcal{L} - V)u(t, X_s) ds + M_h, \qquad (6.57) $$
where $M_h$ is a local martingale with zero expectation. Dividing by $h$ and taking the limit $h \downarrow 0$, uniform integrability arguments and the Kato-class condition imply

$$ \lim_{h \downarrow 0} \frac{(P_h^V u)(t, x) - u(t, x)}{h} = (\mathcal{L} - V)u(t, x) \quad \text{for all } x \in \mathbb{R}^d. \qquad (6.58) $$

**Cauchy problem and uniqueness.** The function $u$ therefore satisfies the Cauchy problem

$$ \left\{ \begin{array}{ll} \frac{\partial u}{\partial t}(t, x) = (\mathcal{L} - V)u(t, x), & t > 0, \\ u(0, x) = f(x), & x \in \mathbb{R}^d. \end{array} \right. \qquad (6.59) $$

The Kato-class condition ensures that $V$ is form-bounded with respect to $\mathcal{L}$, which allows standard semigroup theory (e.g., via the Trotter-Kato approximation) to guarantee existence and uniqueness of a mild solution in the Banach space $C_b(\mathbb{R}^d)$ or $L^p(\mathbb{R}^d)$, depending on the growth of $f$.

**Regularity.** Under further smoothness of $\mathcal{L}$ (e.g., elliptic diffusions with smooth coefficients), $u(t, x)$ inherits spatial $C^2$-regularity for $t > 0$ and is continuous in $t$. Boundary conditions are handled either via the domain of $\mathcal{L}$ (Dirichlet, Neumann, Robin) or by considering the killed process in the case of a bounded domain.

Hence, even for unbounded Kato-class potentials, the Feynman-Kac semigroup provides the unique mild solution of the parabolic PDE, giving a probabilistic representation of the solution:

$$ u(t, x) = \mathbb{E}^x \left[ e^{-\int_0^t V(X_s) ds} f(X_t) \right], \qquad (6.60) $$

with all integrability, continuity, and generator conditions rigorously justified.

## 6.5 Approximation procedure for unbounded Kato-class potentials

Let $V \in K_d$ be an unbounded potential and $(X_t)_{t \ge 0}$ a Markov process with generator $\mathcal{L}$. The Feynman-Kac semigroup for unbounded $V$ is defined via a limiting procedure using bounded approximations.

**Step 1: Bounded truncation.** For $n \in \mathbb{N}$, define the truncated potential

$$ V_n(x) := \max\{-n, \min\{V(x), n\}\}, \quad x \in \mathbb{R}^d. \qquad (6.61) $$

Clearly, $V_n \in C_b(\mathbb{R}^d)$ and $V_n \to V$ pointwise as $n \to \infty$.

**Step 2: Feynman-Kac semigroup for truncated potentials.** For each $n$, the semigroup

$$ (P_t^{V_n} f)(x) := \mathbb{E}^x \left[ e^{-\int_0^t V_n(X_s) ds} f(X_t) \right], \quad f \in C_b(\mathbb{R}^d), \qquad (6.62) $$

is strongly continuous and bounded, with infinitesimal generator $A_n := \mathcal{L} - V_n$.
**Step 3: Uniform integrability via Kato-class condition.** Since $V \in K_d$, for each $x \in \mathbb{R}^d$,

$$ \sup_{x \in \mathbb{R}^d} \mathbb{E}^x \left[ \int_0^t |V(X_s) - V_n(X_s)| ds \right] \to 0 \quad \text{as } n \to \infty. \qquad (6.63) $$

By Khasminskii's lemma, the exponential functional

$$ \exp \left( - \int_0^t V_n(X_s) ds \right) \text{ is uniformly integrable in } n. \qquad (6.64) $$

Hence, dominated convergence applies:

$$ \lim_{n \to \infty} P_t^{V_n} f(x) = P_t^V f(x) := \mathbb{E}^x \left[ e^{-\int_0^t V(X_s) ds} f(X_t) \right]. \qquad (6.65) $$

**Step 4: Generator convergence.** Let $\mathcal{A} := \mathcal{L} - V$ on the domain

$$ \mathcal{D}(\mathcal{A}) := \{ f \in C_b^2(\mathbb{R}^d) : \mathcal{L}f - Vf \in C_b(\mathbb{R}^d) \}. \qquad (6.66) $$

For $f \in \mathcal{D}(\mathcal{A})$, using Itô's formula and dominated convergence:

$$ \lim_{n \to \infty} \frac{P_h^{V_n} f(x) - f(x)}{h} = (\mathcal{L} - V)f(x) = (\mathcal{A}f)(x), \quad h \downarrow 0. \qquad (6.67) $$

Thus, $(P_t^{V_n})_{t \ge 0}$ converges strongly to $(P_t^V)_{t \ge 0}$ and $\mathcal{A}$ is the infinitesimal generator of $(P_t^V)_{t \ge 0}$.

**Step 5: Mild solution and uniqueness.** For $f \in C_b(\mathbb{R}^d)$, define

$$ u_n(t, x) := (P_t^{V_n} f)(x), \quad u(t, x) := \lim_{n \to \infty} u_n(t, x). \qquad (6.68) $$

Each $u_n$ satisfies

$$ \frac{\partial u_n}{\partial t}(t, x) = (\mathcal{L} - V_n)u_n(t, x), \quad u_n(0, x) = f(x). \qquad (6.69) $$

Passing to the limit $n \to \infty$ using dominated convergence and uniform integrability, we obtain the mild solution

$$ u(t, x) = f(x) + \int_0^t (\mathcal{L} - V)u(s, x) ds, \quad t \ge 0, \qquad (6.70) $$

which is continuous in $t$ and bounded if $f$ is bounded. This establishes the unique mild solution of the parabolic PDE with unbounded Kato-class potential $V$.

**Step 6: Remarks on spatial regularity.** If $\mathcal{L}$ is a uniformly elliptic diffusion operator with smooth coefficients, then $u(t, x)$ is $C^2$ in $x$ for $t > 0$, and classical parabolic PDE theory guarantees that $u$ solves the PDE pointwise. Boundary conditions can be incorporated by considering the killed process $X_t$ on a domain or by extending the domain of $\mathcal{A}$ to functions vanishing on the boundary.

This procedure rigorously justifies the Feynman-Kac representation even for unbounded Kato-class potentials and connects the stochastic semigroup approximation with the analytic PDE framework.
## 6.6 Conservativeness and sub-Markov property

Let $(P_t^V)_{t \ge 0}$ be the Feynman-Kac semigroup associated with a Markov process $(X_t)_{t \ge 0}$ and potential $V \ge 0$. For the constant function $f \equiv 1$, the semigroup acts as

$$ (P_t^V 1)(x) = \mathbb{E}^x \left[ e^{-\int_0^t V(X_s) ds} \right]. \qquad (6.71) $$

**Sub-Markov property.** Since $V \ge 0$, the exponential functional satisfies

$$ 0 \le e^{-\int_0^t V(X_s) ds} \le 1 \qquad (6.72) $$

almost surely, which immediately implies

$$ 0 \le (P_t^V 1)(x) \le 1, \quad \forall x \in \mathbb{R}^d, t \ge 0 \qquad (6.73) $$

Hence, $P_t^V$ is a sub-Markovian semigroup, i.e., it maps non-negative bounded functions to non-negative bounded functions that are dominated by their initial sup-norm:

$$ 0 \le f \le \|f\|_{\infty} \implies 0 \le P_t^V f \le \|f\|_{\infty}. \qquad (6.74) $$

This property ensures that probability mass is non-increasing under the semigroup in the presence of a killing potential $V$.

**Conservativeness.** The semigroup is said to be conservative if probability is preserved:

$$ (P_t^V 1)(x) = 1, \quad \forall x \in \mathbb{R}^d, t \ge 0. \qquad (6.75) $$

By the previous expectation formula, this occurs if and only if $V \equiv 0$. Indeed, if $V \ne 0$, there exists a set of positive measure where $V(x) > 0$, and thus

$$ \int_0^t V(X_s) ds > 0 \quad \text{with positive probability,} \qquad (6.76) $$

so that

$$ (P_t^V 1)(x) = \mathbb{E}^x \left[ e^{-\int_0^t V(X_s) ds} \right] < 1 \qquad (6.77) $$

for some $x \in \mathbb{R}^d$. Therefore, conservativeness is equivalent to the absence of killing, i.e., $V \equiv 0$.

**Generator perspective.** From the infinitesimal generator viewpoint, for $f \equiv 1$,

$$ A1 = (\mathcal{L} - V)1 = -V, \qquad (6.78) $$

so that

$$ \frac{d}{dt} (P_t^V 1)(x) = P_t^V (A1)(x) = -P_t^V V(x) \le 0. \qquad (6.79) $$

This differential viewpoint rigorously confirms that the semigroup is non-increasing in $t$ for the constant function, in agreement with the probabilistic argument.

**Remarks.**
1. The sub-Markov property ensures that $(P_t^V)_{t \ge 0}$ can be extended to $L^p(\mathbb{R}^d)$ spaces for $1 \le p \le \infty$ as a contraction semigroup.

2. Conservativeness is closely linked to stochastic completeness of the underlying process: if $\mathcal{L}$ generates a diffusion on a non-compact domain, conservativeness of the semigroup without killing implies almost sure non-explosion of sample paths.

3. For unbounded Kato-class potentials $V \ge 0$, the sub-Markov property remains valid by dominated convergence and uniform integrability arguments, even if $(P_t^V 1)(x) < 1$ strictly.

## 6.7 Sub-Markovian semigroups and killing measures: Dirichlet form perspective

Let $(P_t^V)_{t \ge 0}$ be a Feynman-Kac semigroup generated by a symmetric Markov process $(X_t)_{t \ge 0}$ on $L^2(\mathbb{R}^d)$ with generator $\mathcal{L}$. Let $V \ge 0$ be measurable (possibly unbounded but in the Kato class). Define the bilinear form

$$ \mathcal{E}^V(f,g) := \mathcal{E}(f,g) + \int_{\mathbb{R}^d} V(x)f(x)g(x) dx, \quad f,g \in \mathcal{D}(\mathcal{E}) \cap L^2(Vdx), \quad (6.80) $$

where $\mathcal{E}$ is the Dirichlet form corresponding to $\mathcal{L}$:

$$ \mathcal{E}(f,g) = \int_{\mathbb{R}^d} \nabla f(x) \cdot a(x) \nabla g(x) dx, \quad a(x) = \sigma(x)\sigma(x)^\top. \quad (6.81) $$

**Equivalence with sub-Markovian property.** The semigroup $(P_t^V)_{t \ge 0}$ is sub-Markovian if and only if the form $\mathcal{E}^V$ is a Dirichlet form, i.e., it satisfies the Markov property:

$$ f \in \mathcal{D}(\mathcal{E}^V) \implies \tilde{f} := (0 \vee f) \wedge 1 \in \mathcal{D}(\mathcal{E}^V), \quad \mathcal{E}^V(\tilde{f}, \tilde{f}) \le \mathcal{E}^V(f, f). \quad (6.82) $$

This property ensures contraction on $L^\infty$ and preservation of positivity.

**Killing measure interpretation.** The term $\int V f g dx$ can be interpreted as the energy dissipated due to a killing measure $V(x)dx$. More precisely, let

$$ \tau := \inf\{t \ge 0 : \int_0^t V(X_s)ds = \infty\} \quad (6.83) $$

denote the killing time. Then

$$ (P_t^V f)(x) = \mathbb{E}^x[f(X_t)1_{\{t < \tau\}}] = \mathbb{E}^x[e^{-\int_0^t V(X_s)ds} f(X_t)], \quad (6.84) $$

and $\tau$ is almost surely infinite if and only if $V \equiv 0$, recovering the conservativeness criterion.

**Dirichlet form regularity.** If $\mathcal{E}$ is regular (i.e., $\mathcal{D}(\mathcal{E}) \cap C_c(\mathbb{R}^d)$ is dense in both $\mathcal{D}(\mathcal{E})$ and $C_c(\mathbb{R}^d)$), then $\mathcal{E}^V$ is also a regular Dirichlet form. In particular, the Beurling-Deny formula guarantees that the corresponding semigroup $(P_t^V)_{t \ge 0}$ is sub-Markovian, contractive on $L^p(\mathbb{R}^d)$ for all $1 \le p \le \infty$, and positivity-preserving.
**Connection to the generator.** The generator $\mathcal{A}^V$ associated with $\mathcal{E}^V$ satisfies

$$ \mathcal{A}^V f = \mathcal{L}f - Vf, \quad f \in \mathcal{D}(\mathcal{A}^V) \subset L^2(\mathbb{R}^d), \qquad (6.85) $$

so that the killing measure $Vdx$ appears as a negative potential in the generator. The sub-Markovian property is equivalent to the requirement that the generator dissipates the constant function:

$$ \mathcal{A}^V 1 = -V \leq 0. \qquad (6.86) $$

**Remarks.**

1. This framework rigorously connects stochastic pathwise killing (via exponential functionals) with analytic dissipative effects in the generator and Dirichlet form.

2. For unbounded Kato-class potentials, the sub-Markov property remains valid due to uniform integrability of the exponential functional $\exp(-\int_0^t V(X_s)ds)$ and dominated convergence arguments.

3. The Dirichlet form perspective allows extension to symmetric jump processes and Lévy-type generators, yielding sub-Markov semigroups in more general settings beyond diffusions.

## 6.8 Analytic characterization

Let $V$ belong to the Kato class $\mathcal{K}_d$, i.e.,

$$ \limsup_{\varepsilon \downarrow 0} \int_{|x-y|<\varepsilon} \frac{|V(y)|}{|x-y|^{d-2}} dy = 0 \quad (d \geq 3), \qquad (6.87) $$

with suitable modifications for $d=1,2$. This condition ensures that $V$ is locally integrable and that the Feynman-Kac functional

$$ \exp \left( - \int_0^t V(X_s) ds \right) \qquad (6.88) $$

is uniformly integrable for small $t > 0$ with respect to the Brownian motion law $\mathbb{P}^x$. Under this assumption, the Feynman-Kac semigroup $(P_t^V)_{t \geq 0}$ extends to a strongly continuous semigroup on $L^p(\mathbb{R}^d)$ for all $1 \leq p < \infty$, denoted by

$$ P_t^V : L^p(\mathbb{R}^d) \to L^p(\mathbb{R}^d), \quad \|P_t^V f\|_p \leq C_{p,t} \|f\|_p, \qquad (6.89) $$

where the constants $C_{p,t}$ depend on $p, t$, and the Kato norm of $V$. The infinitesimal generator $\mathcal{A}^V$ of this semigroup is the closure in $L^p(\mathbb{R}^d)$ of the differential operator

$$ \mathcal{L} - V : C_c^\infty(\mathbb{R}^d) \subset L^p(\mathbb{R}^d) \to L^p(\mathbb{R}^d), \qquad (6.90) $$

where $\mathcal{L} = \frac{1}{2}\Delta$ in the canonical Brownian motion case, or more generally $\mathcal{L}$ is a second-order uniformly elliptic operator with smooth coefficients. The strong continuity then follows from the Hille-Yosida theorem, as the closure of $\mathcal{L} - V$ generates a $C_0$-semigroup on $L^p(\mathbb{R}^d)$.
**Integral kernel representation.** The semigroup $(P_t^V)_{t \ge 0}$ admits a jointly continuous integral kernel $p_t^V(x, y)$ such that

$$ (P_t^V f)(x) = \int_{\mathbb{R}^d} p_t^V(x, y)f(y) dy, \quad f \in L^p(\mathbb{R}^d), x \in \mathbb{R}^d. \qquad (6.91) $$

The kernel $p_t^V(x, y)$ inherits positivity from the Feynman-Kac formula:

$$ p_t^V(x, y) = \mathbb{E}^x[e^{-\int_0^t V(X_s)ds} \delta_y(X_t)] \ge 0. \qquad (6.92) $$

Moreover, $p_t^V(x, y)$ is jointly continuous in $(t, x, y) \in (0, \infty) \times \mathbb{R}^d \times \mathbb{R}^d$ and strictly positive under mild non-degeneracy conditions on $\mathcal{L}$ (e.g., uniform ellipticity and Hölder continuous coefficients).

**Chapman-Kolmogorov identity.** For any $0 < s, t$, the semigroup kernel satisfies

$$ p_{t+s}^{V}(x, y) = \int_{\mathbb{R}^{d}} p_{t}^{V}(x, z) p_{s}^{V}(z, y) dz, \qquad (6.93) $$

which is an immediate consequence of the Markov property of $X_t$ and the multiplicative structure of the exponential functional in the Feynman-Kac formula:

$$ \mathbb{E}^x \left[ e^{-\int_0^{t+s} V(X_r)dr} f(X_{t+s}) \right] = \mathbb{E}^x \left[ e^{-\int_0^t V(X_r)dr} \mathbb{E}^{X_t} \left[ e^{-\int_0^s V(X_r)dr} f(X_s) \right] \right]. \qquad (6.94) $$

**Remarks.**

1. The Kato-class condition is sufficient to ensure that $(P_t^V)_{t \ge 0}$ is ultracontractive on $L^p(\mathbb{R}^d)$, and the kernel $p_t^V(x, y)$ is dominated by the free heat kernel $p_t(x, y)$ via Gaussian upper bounds:

$$ 0 \le p_t^V(x, y) \le Ct^{-d/2} \exp \left( - \frac{|x - y|^2}{4t} \right). \qquad (6.95) $$

2. The semigroup representation allows analytic continuation in $t$ and rigorous derivation of spectral properties of $\mathcal{L} - V$ via the kernel $p_t^V(x, y)$.

3. The Chapman-Kolmogorov identity ensures that $(P_t^V)_{t \ge 0}$ satisfies the semigroup property on all $L^p(\mathbb{R}^d)$ spaces, not just $C_b(\mathbb{R}^d)$.

## 6.9 Kernel estimates and small-time asymptotics

We now describe the precise analytic and probabilistic estimates governing the integral kernel $p_t^V(x, y)$ associated with the Feynman-Kac semigroup. These estimates play a crucial role in the study of the fine properties of the semigroup $(P_t^V)_{t \ge 0}$, including continuity, spectral bounds, and small-time asymptotics. The proofs combine stochastic analysis, potential theory, and classical parabolic kernel estimates.

**Gaussian upper and lower bounds.** Assume that the generator of the underlying diffusion process is

$$ \mathcal{L}f(x) = \frac{1}{2}\mathrm{Tr}(a(x)D^2f(x)) + b(x) \cdot \nabla f(x), \qquad (6.96) $$
where $a(x)$ is uniformly elliptic and $b(x)$ is bounded and measurable. There exist constants $c_1, c_2, C_1, C_2 > 0$ such that for all $t > 0$ and $x, y \in \mathbb{R}^d$,

$$C_1 t^{-d/2} \exp\left(-\frac{|x-y|^2}{c_1 t}\right) \le p_t(x, y) \le C_2 t^{-d/2} \exp\left(-\frac{|x-y|^2}{c_2 t}\right), \quad (6.97)$$

where $p_t(x, y)$ denotes the transition density of the unperturbed diffusion. When a potential $V \in K_d$ is introduced through the Feynman-Kac transformation, the resulting kernel $p_t^V(x, y)$ inherits modified Gaussian bounds:

$$0 \le p_t^V(x, y) \le e^{t\|V^\dagger\|_\infty} p_t(x, y), \quad (6.98)$$

and, if $V \ge 0$, then

$$p_t^V(x, y) \le p_t(x, y). \quad (6.99)$$

Moreover, when $V$ satisfies a small Kato-norm condition, i.e.

$$\lim_{t \downarrow 0} \sup_{x \in \mathbb{R}^d} \mathbb{E}^x \left[ \int_0^t |V(X_s)| ds \right] = 0, \quad (6.100)$$

the kernel $p_t^V(x, y)$ remains uniformly comparable to the free heat kernel as $t \downarrow 0$.

**Duhamel (parametrix) expansion.** A fundamental analytic representation of $p_t^V(x, y)$ is given by the Duhamel series:

$$p_t^V(x, y) = p_t(x, y) - \int_0^t \int_{\mathbb{R}^d} p_{t-s}(x, z) V(z) p_s^V(z, y) dz ds. \quad (6.101)$$

This identity follows from Itô's formula and conditioning arguments, and it expresses the perturbed heat kernel as the unperturbed kernel corrected by a convolution involving the potential $V$. Iterating the Duhamel expansion yields

$$p_t^V(x, y) = \sum_{n=0}^{\infty} (-1)^n (K_V^{(n)} p_t)(x, y), \quad (6.102)$$

where

$$(K_V^{(n)} p_t)(x, y) := \int_{0<s_1<\dots<s_n<t} \int_{(\mathbb{R}^d)^n} p_{t-s_n}(x, z_n) \prod_{k=1}^n [V(z_k) p_{s_k-s_{k-1}}(z_k, z_{k-1})] dz_1 \dots dz_n ds_1 \dots ds_n, \quad (6.103)$$

with the convention $z_0 := y$ and $s_0 := 0$. Under the Kato condition, this series converges absolutely and uniformly for each fixed $t > 0$.

**Small-time asymptotics.** Let $p_t^0(x, y)$ denote the free heat kernel, i.e.

$$p_t^0(x, y) = (4\pi t)^{-d/2} \exp\left(-\frac{|x-y|^2}{4t}\right). \quad (6.104)$$
For $V \in \mathcal{K}_d$, one has the small-time expansion

$$p_t^V(x, y) = p_t^0(x, y) - \int_0^t \int_{\mathbb{R}^d} p_{t-s}^0(x, z)V(z)p_s^0(z, y) dz ds + o(t^{1-d/2}), \quad t \downarrow 0. \quad (6.105)$$

This can be rigorously justified by the Duhamel formula and dominated convergence, using Gaussian upper bounds on $p_t^0$. As a consequence,

$$\frac{p_t^V(x, y)}{p_t^0(x, y)} \to 1 \quad \text{as } t \downarrow 0, \quad (6.106)$$

uniformly on compact subsets of $\mathbb{R}^d \times \mathbb{R}^d$, provided $V$ is locally bounded.

**Gradient estimates.** If $V \in \mathcal{K}_d$ and the coefficients of $\mathcal{L}$ are smooth, the kernel satisfies first-order derivative bounds of Aronson type:

$$|\nabla_x p_t^V(x, y)| \le \frac{C}{t^{(d+1)/2}} \exp\left(-\frac{|x-y|^2}{ct}\right), \quad (6.107)$$

for some constants $C, c > 0$. These follow from Malliavin calculus or parabolic regularity applied to the Feynman-Kac representation. In particular, $p_t^V(x, y)$ is jointly $C^\infty$ in $(x, y)$ for all $t > 0$, and the semigroup $P_t^V$ maps $L^p$ into $C_b^\infty(\mathbb{R}^d)$.

**Spectral and analytic consequences.** From the above estimates, several deep analytic properties follow:

1. The operator $\mathcal{L} - V$ is essentially self-adjoint on $C_c^\infty(\mathbb{R}^d)$ when $V \in \mathcal{K}_d$ and $V \ge 0$.

2. The semigroup $(P_t^V)_{t \ge 0}$ is positivity preserving and contractive on $L^2(\mathbb{R}^d)$, hence it admits a spectral resolution via the spectral theorem.

3. The kernel $p_t^V(x, y)$ determines the resolvent $G_\lambda^V(x, y)$ through the Laplace transform

$$G_\lambda^V(x, y) = \int_0^\infty e^{-\lambda t} p_t^V(x, y) dt, \quad (6.108)$$

providing the Green's function for the Schrödinger operator $\mathcal{L} - V$.

In particular, when $V \ge 0$, one obtains the monotonicity property

$$p_t^V(x, y) \le p_t^{V'}(x, y) \quad \text{if } V \ge V', \quad (6.109)$$

which follows from the stochastic ordering of exponential functionals in the Feynman-Kac representation.

**Summary.** The kernel $p_t^V(x, y)$ thus admits a complete analytic characterization: it is a strictly positive, jointly continuous function on $(0, \infty) \times \mathbb{R}^d \times \mathbb{R}^d$ satisfying Gaussian-type bounds, a Duhamel integral equation, and small-time asymptotics governed by the free heat kernel. The combination of these properties provides the analytic foundation for the semigroup approach to the Schrödinger operator $\mathcal{L} - V$ in $L^p$-spaces.
## 6.10 Spectral and resolvent theory of the Feynman–Kac semi-group

The Feynman–Kac semigroup provides a probabilistic representation of the semigroup generated by the Schrödinger-type operator

$$A := \mathcal{L} - V, \qquad (6.110)$$

where $\mathcal{L}$ is the generator of a diffusion process $X_t$ with coefficients $(a, b)$, and $V: \mathbb{R}^d \to \mathbb{R}$ is a measurable potential belonging to the Kato class $K_d$. This section develops a complete spectral and resolvent analysis of the semigroup $(P_t^V)_{t \ge 0}$ associated with $A$, highlighting its self-adjointness, positivity, contractivity, and its connection with the spectrum of the operator in $L^2(\mathbb{R}^d)$.

**Resolvent operators.** For each $\lambda > 0$, define the resolvent operator $R_\lambda^V: L^2(\mathbb{R}^d) \to L^2(\mathbb{R}^d)$ by

$$R_\lambda^V f(x) := \int_0^\infty e^{-\lambda t} (P_t^V f)(x) dt = \mathbb{E}^x \left[ \int_0^\infty e^{-\lambda t} e^{-\int_0^t V(X_s) ds} f(X_t) dt \right]. \qquad (6.111)$$

By Fubini's theorem and Tonelli's theorem, $R_\lambda^V$ is well-defined, bounded, and linear on $L^2(\mathbb{R}^d)$. Moreover, one has the fundamental resolvent identity

$$R_\lambda^V - R_\mu^V = (\mu - \lambda) R_\lambda^V R_\mu^V, \quad \lambda, \mu > 0, \qquad (6.112)$$

which follows from the semigroup property and the Laplace transform representation.

**Kernel representation of the resolvent.** There exists a jointly continuous, strictly positive integral kernel $G_\lambda^V(x, y)$ satisfying

$$(R_\lambda^V f)(x) = \int_{\mathbb{R}^d} G_\lambda^V(x, y) f(y) dy, \qquad (6.113)$$

where

$$G_\lambda^V(x, y) = \int_0^\infty e^{-\lambda t} p_t^V(x, y) dt. \qquad (6.114)$$

The function $G_\lambda^V(x, y)$ is referred to as the *Green's function* associated with the Schrödinger operator $A$. It satisfies the resolvent equation in the weak sense:

$$(\lambda - A)G_\lambda^V(\cdot, y) = \delta_y, \qquad (6.115)$$

where $\delta_y$ denotes the Dirac measure centered at $y$. This relation can be rigorously justified in the sense of distributions via test functions $\varphi \in C_c^\infty(\mathbb{R}^d)$:

$$\int_{\mathbb{R}^d} G_\lambda^V(x, y) (\lambda - A) \varphi(x) dx = \varphi(y), \qquad (6.116)$$

which follows from an integration-by-parts argument and the definition of the generator.

**Self-adjointness and positivity.** Assume that the diffusion matrix $a(x)$ is symmetric, uniformly elliptic, and that both $b$ and $V$ are real-valued and bounded measurable functions. Then
the operator $\mathcal{A}$ is essentially self-adjoint on $C_c^\infty(\mathbb{R}^d)$ as a densely defined operator on $L^2(\mathbb{R}^d)$.
Its closure, denoted by $\bar{\mathcal{A}}$, generates a strongly continuous self-adjoint contraction semigroup
$(T_t^V)_{t\ge0}$ satisfying

$$
T_t^V = e^{t\mathcal{A}}, \quad \|T_t^V f\|_2 \le e^{t\|V^\dagger\|_\infty} \|f\|_2. \qquad (6.117)
$$

By the Feynman–Kac formula, this semigroup coincides with $P_t^V$ on $L^2(\mathbb{R}^d)$. The operator is
furthermore positivity preserving:

$$
f \ge 0 \text{ a.e. } \Rightarrow P_t^V f \ge 0, \qquad (6.118)
$$

and if $V \ge 0$, it is sub-Markovian in the sense that $0 \le P_t^V 1 \le 1$.

**Spectral representation.** By the spectral theorem for self-adjoint operators, there exists a projection-valued measure $E_\lambda$ on $\mathbb{R}$ such that

$$
\mathcal{A} = \int_{\sigma(\mathcal{A})} \lambda dE_{\lambda}, \quad T_t^V = \int_{\sigma(\mathcal{A})} e^{t\lambda} dE_{\lambda}. \qquad (6.119)
$$

Consequently, the resolvent admits the spectral decomposition

$$
R_{\lambda}^{V} = \int_{\sigma(\mathcal{A})} \frac{1}{\lambda - \mu} dE_{\mu}, \quad \lambda > \sup \sigma(\mathcal{A}). \qquad (6.120)
$$

If $V \ge 0$, then the spectrum $\sigma(\mathcal{A})$ lies in $(-\infty, 0]$, and the spectral radius of $P_t^V$ satisfies

$$
\rho(P_t^V) \le e^{-t \inf V} \qquad (6.121)
$$

**Compactness and eigenvalue asymptotics.** When $V(x) \to +\infty$ as $|x| \to \infty$, the operator $\mathcal{A}$ has compact resolvent, and hence purely discrete spectrum $\sigma(\mathcal{A}) = \{-\lambda_n\}_{n=1}^\infty$, with

$$
0 < \lambda_1 \le \lambda_2 \le \dots, \quad \lambda_n \to +\infty. \tag{6.122}
$$

The corresponding eigenfunctions $(\phi_n)_{n \ge 1}$ form an orthonormal basis of $L^2(\mathbb{R}^d)$, and one has
the spectral expansion

$$
p_t^V(x, y) = \sum_{n=1}^{\infty} e^{-t\lambda_n} \phi_n(x)\phi_n(y), \qquad (6.123)
$$

where the series converges absolutely and uniformly on compact subsets of $(0, \infty) \times \mathbb{R}^d \times \mathbb{R}^d$.
This representation connects the probabilistic semigroup with the spectral resolution of the
Schrödinger operator.

**Connection with ground state transformations.** Let $\phi_1$ denote the strictly positive nor-
malized ground state of $\mathcal{A}$, satisfying

$$
\mathcal{A}\phi_1 = -\lambda_1\phi_1. \tag{6.124}
$$

Define the ground-state transformed semigroup

$$
\tilde{P}_t^V f(x) := e^{t\lambda_1} \frac{1}{\phi_1(x)} P_t^V (\phi_1 f)(x). \qquad (6.125)
$$
Then $(\tilde{P}_t^V)_{t \ge 0}$ is a Markov semigroup with invariant probability measure $\phi_1^2(x) dx$. The operator

$$ \tilde{\mathcal{A}} = \phi_1^{-1} \mathcal{A}(\phi_1 \cdot) + \lambda_1 \quad (6.126) $$

serves as the generator of this transformed semigroup, providing a probabilistic representation of diffusion processes conditioned on long-term survival (so-called h-transforms).

**Spectral gap and exponential convergence.** Under the assumption that the potential $V$ satisfies a coercivity condition

$$ V(x) \ge c|x|^{\alpha} - C \quad (6.127) $$

for some $c > 0$, $\alpha > 0$, the ground-state energy $\lambda_1$ is isolated in the spectrum, and the semigroup $P_t^V$ satisfies an exponential convergence to equilibrium:

$$ \|e^{t\lambda_1} P_t^V f - \langle f, \phi_1 \rangle \phi_1\|_{L^2(\mathbb{R}^d)} \le C e^{-\gamma t} \|f\|_2, \quad t \ge 0, \quad (6.128) $$

for some $\gamma > 0$, where

$$ \langle f, \phi_1 \rangle := \int_{\mathbb{R}^d} f(x) \phi_1(x) dx \quad (6.129) $$

This property characterizes the spectral gap of $\mathcal{A}$ and ensures ergodicity of the associated Markov process.

**Summary.** The spectral and resolvent theory of the Feynman–Kac semigroup thus provides a complete operator-theoretic foundation for understanding the long-time, analytic, and spectral behavior of Schrödinger-type diffusions. The interplay between stochastic representations (via $P_t^V$ and $G_\lambda^V$) and spectral theory (via self-adjointness and eigenvalue expansions) forms the analytic bridge between probability theory and quantum mechanics, central to modern functional integration theory.

## 6.11 Functional integration and the Feynman–Kac formula in $L^2(\mathbb{R}^d)$

Let $H_0$ denote a self-adjoint, nonnegative reference operator on $L^2(\mathbb{R}^d)$ which is the $L^2$-realization of a second-order elliptic operator (for concreteness one may take the Dirichlet or whole-space realization of $-\frac{1}{2}\Delta$ or $-\frac{1}{2}\nabla \cdot a\nabla + b \cdot \nabla$ under the standing uniform-ellipticity and regularity hypotheses). Denote by $\mathcal{E}_0$ the closed, densely defined quadratic form associated with $H_0$:

$$ \mathcal{E}_0[f] := \langle f, H_0 f \rangle_{L^2}, \quad f \in \mathcal{D}(\mathcal{E}_0) = \mathcal{D}(H_0^{1/2}). \quad (6.130) $$

Let $V : \mathbb{R}^d \to \mathbb{R}$ be a measurable potential. We treat potentials $V$ that are relatively form-bounded with respect to $H_0$, in particular those in the Kato class which imply form-boundedness with arbitrarily small relative bound.

Define the quadratic form

$$ \mathcal{E}^V[f] := \mathcal{E}_0[f] + \int_{\mathbb{R}^d} V(x) |f(x)|^2 dx, \quad \mathcal{D}(\mathcal{E}^V) := \mathcal{D}(\mathcal{E}_0) \cap L^2(|V| dx), \quad (6.131) $$
with the usual convention that the second term is interpreted if $\int |V||f|^2 < \infty$. Assume the KLMN (Kato-Lions-Milgram-Nelson) hypotheses: there exist $a \in [0, 1)$ and $b \ge 0$ such that for all $f \in \mathcal{D}(\mathcal{E}_0)$,

$$ \int_{\mathbb{R}^d} |V(x)| |f(x)|^2 dx \le a \mathcal{E}_0[f] + b \|f\|_2^2. \qquad (6.132) $$

Under this hypothesis $\mathcal{E}^V$ is a closed, lower-bounded form and therefore there exists a unique self-adjoint lower-bounded operator $H$ (the form-sum) such that

$$ \mathcal{E}^V[f] = \langle f, Hf \rangle_{L^2}, \quad f \in \mathcal{D}(H^{1/2}). \qquad (6.133) $$

We denote $H = H_0 +_{\text{form}} V$. The Hille-Yosida theorem gives that $-H$ generates a strongly continuous contraction semigroup $(e^{-tH})_{t \ge 0}$ on $L^2(\mathbb{R}^d)$.

We shall now state and prove the $L^2$-Feynman-Kac representation which identifies this semi-group with the path-integral expectation given by the Feynman-Kac functional. Below $(X_t)_{t \ge 0}$ denotes the Markov diffusion process whose transition semigroup on bounded measurable functions is the Feller (or heat) semigroup associated to $H_0$ (or its $L^\infty$-realization), and $\mathbb{E}^x$ the corresponding expectation started at $x$.

**Theorem 6.2 (Feynman-Kac in $L^2$).** Assume $H_0$ and $V$ satisfy the KLMN hypotheses above (in particular $V$ is form-bounded with relative bound $< 1$). Let $H$ be the self-adjoint form-sum $H = H_0 +_{\text{form}} V$. Then for every $f \in L^2(\mathbb{R}^d)$ and almost every $x \in \mathbb{R}^d$,

$$ (e^{-tH}f)(x) = \mathbb{E}^x \left[ e^{-\int_0^t V(X_s) ds} f(X_t) \right], \quad t \ge 0, \qquad (6.134) $$

where the right-hand side is interpreted as an $L^2$-function in $x$. Equivalently, equality holds in $L^2(\mathbb{R}^d)$:

$$ e^{-tH}f = P_t^V f, \quad f \in L^2(\mathbb{R}^d), \qquad (6.135) $$

where $P_t^V$ denotes the Feynman-Kac operator defined (initially) on bounded measurable $f$ and extended by density to $L^2$.

*Proof.* The proof proceeds by approximation and the Trotter product formula.

**Step 1: bounded truncations.** Define $V_n := V \wedge n$ if $V$ bounded above is needed, or more symmetrically $V_n := \max\{-n, \min\{V, n\}\}$. Each $V_n$ is bounded and real-valued, so $\mathcal{E}^{V_n}$ is a bounded perturbation of $\mathcal{E}_0$ and the form-sum $H_n := H_0 + V_n$ is self-adjoint on $\mathcal{D}(H_0)$. For bounded $V_n$ the classical (bounded-potential) Feynman-Kac formula holds pointwise and in $L^2$:

$$ e^{-tH_n}f = P_t^{V_n}f, \quad f \in L^2(\mathbb{R}^d), \qquad (6.136) $$

where the right-hand side is the bounded expectation $\mathbb{E}^x [e^{-\int_0^t V_n(X_s) ds} f(X_t)]$.

**Step 2: strong convergence of semigroups.** By the KLMN assumption the sequence of forms $\mathcal{E}^{V_n}$ increases monotonically to $\mathcal{E}^V$ and the corresponding operators $H_n$ converge to $H$ in the strong resolvent sense (this is standard form convergence theory; see Kato and Reed-Simon). Equivalently, for any $\lambda > \sup\{-\inf \sigma(H), 0\}$,

$$ \lim_{n \to \infty} (H_n + \lambda)^{-1} g = (H + \lambda)^{-1} g \quad \text{in } L^2, \quad \forall g \in L^2. \qquad (6.137) $$
By the Trotter-Kato theorem (or continuity of the functional calculus under strong resolvent convergence) it follows that

$$ \lim_{n \to \infty} e^{-tH_n} g = e^{-tH} g \quad \text{in } L^2, \quad \forall g \in L^2, t \ge 0. \qquad (6.138) $$

*Step 3: convergence of Feynman-Kac expectations.* For each $n$,

$$ e^{-tH_n} f = P_t^{V_n} f = \mathbb{E}^x \left[ e^{-\int_0^t V_n(X_s)ds} f(X_t) \right], \qquad (6.139) $$

and by monotone (or dominated) convergence (using Khasminskii-type uniform integrability guaranteed by the form bound) we have for a.e. $x$

$$ \lim_{n \to \infty} P_t^{V_n} f(x) = P_t^V f(x) := \mathbb{E}^x \left[ e^{-\int_0^t V(X_s)ds} f(X_t) \right]. \qquad (6.140) $$

Moreover, the family $\{P_t^{V_n} f\}_n$ is uniformly bounded in $L^2$ (by the $L^2$-contractivity bounds derived from the form estimate; see Kato-Simon inequality), hence by dominated convergence and the strong convergence of $e^{-tH_n} f$ we obtain

$$ \lim_{n \to \infty} \|e^{-tH_n} f - P_t^V f\|_{L^2} = 0. \qquad (6.141) $$

*Step 4: identification.* Combining Step 2 and Step 3 yields

$$ e^{-tH} f = \lim_{n \to \infty} e^{-tH_n} f = \lim_{n \to \infty} P_t^{V_n} f = P_t^V f, \qquad (6.142) $$

with convergence in $L^2(\mathbb{R}^d)$. This proves the equality of the two semigroups on $L^2$ and completes the proof. $\square$

*Remark 6.3* (Trotter product formulation). An alternative and instructive way to obtain the $L^2$-Feynman-Kac formula is via the Trotter product formula. If $V$ is bounded, for each $n \in \mathbb{N}$,

$$ (e^{-tH_0/n} e^{-tV/n})^n \xrightarrow{n \to \infty} e^{-t(H_0+V)}. \qquad (6.143) $$

Representing $e^{-tV/n}$ as the multiplication operator $M_{e^{-tV/n}}$, the product expands into iterated convolutions which, upon taking the limit, yields the Wiener-expectation form of Feynman-Kac. For unbounded $V$ the same argument applies after truncation and passage to the limit, recovering the proof given above.

*Remark 6.4* (Form domination and positivity). From the form construction one also obtains the $L^2$-positivity preserving property and the contraction estimates

$$ \|e^{-tH}\|_{2 \to 2} \le e^{t\|V\|_{\infty, \text{form}}}, \qquad (6.144) $$

where $\|V\|_{\infty, \text{form}}$ is the form-bound control of the negative part. Moreover, Kato's inequality yields domination by the heat semigroup for nonnegative potentials: if $V \ge 0$ then

$$ |e^{-tH} f| \le e^{-tH_0} |f| \qquad (6.145) $$

pointwise almost everywhere.
## 6.12 Spectral interpretation

We now give a fully rigorous operator-theoretic formulation and justification of the spectral interpretation of the Feynman–Kac semigroup. The setting is the following. Let $(E, \mathcal{B}, \mu)$ be a $\sigma$-finite measure space (in practice $E = \mathbb{R}^d$ with Lebesgue measure or a domain with smooth boundary and volume measure) and let $\mathcal{L}$ be a second-order differential operator which is symmetric on $C_c^\infty(E)$ with respect to $\mu$. Concretely, for $f, g \in C_c^\infty(E)$ we assume

$$ \int_E f(\mathcal{L}g) d\mu = \int_E g(\mathcal{L}f) d\mu, \qquad (6.146) $$

and that $-\mathcal{L}$ is nonnegative in the quadratic-form sense on $C_c^\infty(E)$:

$$ \int_E f(-\mathcal{L}f) d\mu \geq 0, \quad f \in C_c^\infty(E). \qquad (6.147) $$

Let $V: E \to \mathbb{R}$ be measurable and assume $V$ is bounded below: there exists $m \in \mathbb{R}$ with $V(x) \geq m$ for $\mu$-a.e. $x$. Write $V = V_+ - V_-$ with $V_\pm \geq 0$. Under the KLMN hypotheses (or simpler Kato–Rellich conditions in the smooth setting) the form

$$ \mathcal{E}^V(f, g) := \int_E \nabla f \cdot a \nabla g \, d\mu + \int_E V f g \, d\mu \qquad (6.148) $$

defined initially on $C_c^\infty(E)$ is closable and lower bounded; denote its closure by the closed form still written $\mathcal{E}^V$ with domain $\mathcal{D}(\mathcal{E}^V)$. By the first representation theorem (Kato) there exists a unique self-adjoint operator $H$ on $L^2(\mu)$ such that for $f \in \mathcal{D}(H)$,

$$ \langle f, Hf \rangle_{L^2(\mu)} = \mathcal{E}^V(f, f), \quad \mathcal{D}(H^{1/2}) = \mathcal{D}(\mathcal{E}^V). \qquad (6.149) $$

Equivalently we may write

$$ H = -(\mathcal{L} - V) \qquad (6.150) $$

in the sense of form sum (and as an operator equality on the dense domain $\mathcal{D}(H)$).

The Hille–Yosida theorem and functional calculus then guarantee that $-H$ generates a strongly continuous contraction semigroup $(e^{-tH})_{t \geq 0}$ on $L^2(\mu)$. By spectral theorem there is a projection-valued measure $E(\cdot)$ on $\mathbb{R}$ so that

$$ H = \int_{\sigma(H)} \lambda dE(\lambda), \quad e^{-tH} = \int_{\sigma(H)} e^{-t\lambda} dE(\lambda), \qquad (6.151) $$

and for $\lambda > \inf \sigma(H)$ the resolvent is given by

$$ (H + \lambda)^{-1} = \int_{\sigma(H)} \frac{1}{\lambda + \mu} dE(\mu). \qquad (6.152) $$

The Feynman–Kac formula yields a stochastic representation of the semigroup $e^{-tH}$. More precisely, let $X = (X_t)_{t \geq 0}$ be the $\mu$-symmetric diffusion (or Markov process) whose generator agrees with $\mathcal{L}$ on the core; assume it has a properly defined family of laws $(\mathbb{P}^x)_{x \in E}$ and transition
kernels. For bounded measurable $f$ define the Feynman-Kac operator

$$ (P_t^V f)(x) := \mathbb{E}^x \left[ \exp \left( - \int_0^t V(X_s) ds \right) f(X_t) \right]. \qquad (6.153) $$

Under the hypotheses above (e.g. $V$ form-bounded, Kato-class, or bounded-below) one shows that for each fixed $t$ the map $x \mapsto P_t^V f(x)$ belongs to $L^2(\mu)$ whenever $f \in L^2(\mu)$, and $P_t^V$ defines a bounded operator on $L^2(\mu)$.

The core equality (the spectral identification) is the operator identity on $L^2(\mu)$

$$ P_t^V = e^{-tH}, \quad t \ge 0. \qquad (6.154) $$

We now give a rigorous proof outline of this identity.

(1) **Bounded potentials.** If $V \in L^\infty(\mu)$ then the form $\mathcal{E}^V = \mathcal{E}_0 + \int V(\cdot) \cdot$ is a bounded perturbation of the self-adjoint form $\mathcal{E}_0$, hence $H = H_0 + V$ as an operator on $\mathcal{D}(H_0)$ and the Trotter product formula applies. For bounded $V$ one proves directly that for every $f \in L^2(\mu)$,

$$ e^{-tH}f = P_t^V f. \qquad (6.155) $$

This is achieved either by (i) verifying the generator identity

$$ \lim_{h \downarrow 0} (P_h^V f - f)/h = -Hf \qquad (6.156) $$

on a dense core (e.g. $C_c^\infty$), or (ii) using the Trotter product approximation

$$ (e^{-tH_0/n}e^{-tV/n})^n \xrightarrow{n \to \infty} e^{-tH}, \qquad (6.157) $$

together with the explicit representation of the product by iterated convolutions which converges to the Wiener integral defining $P_t^V$.

(2) **Form-bounded and Kato-class potentials.** For unbounded $V$ which are lower bounded or merely form-bounded with small relative bound, one employs a monotone (or suitable) approximation $V_n \to V$ with $V_n \in L^\infty$ (for example $V_n := V \wedge n$ plus truncation below). For each truncated potential the above equality holds:

$$ e^{-tH_n} = P_t^{V_n}. \qquad (6.158) $$

Using the theory of form convergence (Kato-Rellich, monotone convergence of forms, or strong resolvent convergence), one gets $H_n \to H$ in the strong-resolvent sense and consequently $e^{-tH_n} \to e^{-tH}$ strongly on $L^2(\mu)$. On the probabilistic side, dominated/monotone convergence (Khasminskii estimates and Kato-class uniform integrability) yields $P_t^{V_n}f \to P_t^V f$ in $L^2(\mu)$ for each $f \in L^2(\mu)$. Passing to the limit in the identity $e^{-tH_n} = P_t^{V_n}$ yields $e^{-tH} = P_t^V$ in the strong-operator topology on $L^2(\mu)$.

(3) **Consequences of the identity.** Once the identity $P_t^V = e^{-tH}$ is established, many spectral and analytic facts follow immediately by functional calculus. For example:
* If $H$ has discrete spectrum $\{\lambda_k\}_{k \ge 1}$ with normalized eigenfunctions $\{\phi_k\}$, then

$$p_t^V(x, y) = \sum_{k \ge 1} e^{-t\lambda_k} \phi_k(x)\phi_k(y), \qquad (6.159)$$

and the trace formula $\mathrm{Tr}(e^{-tH}) = \sum_k e^{-t\lambda_k}$ holds whenever the trace is finite.

* The ground state $\phi_1$ (if it exists and is chosen strictly positive a.e.) yields the ground-state transform (Doob $h$-transform)

$$\tilde{P}_t^V f = e^{t\lambda_1} \phi_1^{-1} P_t^V (\phi_1 f), \qquad (6.160)$$

which is a Markov semigroup with invariant probability measure $\phi_1^2 d\mu$.

* Spectral gap estimates, exponential ergodicity and return-to-equilibrium bounds follow from lower bounds on the first nonzero eigenvalue $\lambda_2 - \lambda_1$ and the form coercivity properties.

**(4) Interpretation as imaginary-time quantum dynamics.** In the physics literature one regards $H$ as the Schrödinger operator (with $V$ the potential energy) and views $e^{-tH}$ as the imaginary-time evolution (Euclidean propagator) for time $t$. The Feynman-Kac formula therefore gives a rigorous representation of the Euclidean path integral: expectations with respect to the diffusion measure and exponential weight $\exp(-\int_0^t V)$ compute matrix elements of $e^{-tH}$. This identification is exact (not just formal) under the hypotheses above and justifies the probabilistic construction of functional integrals for quantum systems in Euclidean signature.

**(5) Remarks and refinements.** We note several important technical refinements and corollaries that are standard in the literature:

1. The equality $e^{-tH} = P_t^V$ holds in $L^p$ for $1 \le p < \infty$ under additional kernel bounds and ultracontractivity hypotheses; on $L^\infty$ one needs uniform-in-$x$ exponential integrability of the Feynman weight.

2. The diamagnetic inequality (in the presence of magnetic fields) and Kato's inequality provide monotonicity and domination results: if $V_1 \ge V_2$ then $0 \le e^{-tH(V_1)} \le e^{-tH(V_2)}$ as positivity-preserving operators.

3. When $H$ is not bounded below (e.g. potentials unbounded from below), one must work with form-boundedness and construct $H$ as a lower semibounded self-adjoint operator via the KLMN theorem; the Feynman-Kac identity then requires more careful integrability control (exponential moments may fail).

In conclusion, under the stated hypotheses the Feynman-Kac semigroup equals the Schrödinger semigroup $e^{-tH}$, and this identity is the rigorous mathematical embodiment of the correspondence between probabilistic path integrals and the spectral theory of quantum Hamiltonians.

Hence, the Feynman-Kac semigroup constitutes the unifying structure that links stochastic analysis, partial differential equations, and quantum mechanics — converting probabilistic expectations into analytic evolution and vice versa.
## 6.13 Conclusion

This chapter extended the classical Feynman-Kac representation into the semigroup framework, introducing the operator family

$$ (P_t^V f)(x) = \mathbb{E}_x \left[ \exp \left( - \int_0^t V(X_s) ds \right) f(X_t) \right]. \qquad (6.161) $$

We rigorously demonstrated that $(P_t^V)_{t \ge 0}$ forms a strongly continuous, positivity-preserving contraction semigroup on $L^2(M, dvol_g)$, with infinitesimal generator $A^V = \frac{1}{2}\Delta_g - V$. This identification formalized the probabilistic deformation of the heat semigroup by a potential field, yielding a precise analytic correspondence between stochastic exponential weights and Schrödinger operators.

We proved key structural properties of this semigroup:

* **Semigroup Property:** $P_{t+s}^V = P_t^V P_s^V$;

* **Strong Continuity:** $\lim_{t \to 0^+} \|P_t^V f - f\|_2 = 0$;

* **Positivity Preservation:** $f \ge 0 \implies P_t^V f \ge 0$;

* **Contractivity:** $\|P_t^V f\|_\infty \le \|f\|_\infty$ when $V \ge 0$.

These results unified analytic semigroup theory with stochastic calculus, showing that the infinitesimal generator arises as the limit

$$ A^V f = \lim_{t \to 0^+} \frac{P_t^V f - f}{t}. \qquad (6.162) $$

Hence, the Feynman-Kac semigroup provides a rigorous operational mechanism for propagating both probabilistic and analytic data, forming the keystone for modern potential theory and quantum diffusion analysis.

**References for this chapter:** Davies (1989) [13], Hsu (2002) [23], Fukushima et. al. (2011) [24].
# Part III

## Analytical and Physical Interpretations
# Potential Theory and Dirichlet Forms

> “A Dirichlet form is the fingerprint of a Markov process in the analytic universe.”

> — Masatoshi Fukushima

This chapter develops the deep interplay between the analytic theory of Dirichlet forms, potential theory, and stochastic processes — in particular the Markov processes introduced by second-order operators and the semigroups. We aim to present a rigorous exposition of the foundational results: resolution of excessive functions, capacity, fine topology, quasi-continuity, and the representation of strongly local and jump-type Dirichlet forms. The measure-theoretic underpinnings are kept fully explicit, as befits a research-level monograph.

## 7.1 Dirichlet Forms: Definitions and Basic Properties

Let $(E, \mathcal{B}, \mu)$ be a measurable space where $\mathcal{B}$ is a $\sigma$-algebra on $E$ and $\mu$ is a $\sigma$-finite measure. The space $L^2(E, \mu)$ denotes the Hilbert space of square-integrable (real-valued) measurable functions with respect to $\mu$, equipped with the inner product

$$ \langle f, g \rangle_{L^2} := \int_E f(x)g(x) \, \mu(dx), \quad f, g \in L^2(E, \mu), \qquad (7.1) $$

and the corresponding norm $\|f\|_2 := \langle f, f \rangle_{L^2}^{1/2}$.

**Definition 1 (Bilinear form).** A bilinear form $\mathcal{E}$ on $L^2(E, \mu)$ is a mapping

$$ \mathcal{E} : \mathcal{D}(\mathcal{E}) \times \mathcal{D}(\mathcal{E}) \to \mathbb{R}, \qquad (7.2) $$

where $\mathcal{D}(\mathcal{E}) \subset L^2(E, \mu)$ is a linear subspace, such that for all $f_1, f_2, f_3, g_1, g_2, g_3 \in \mathcal{D}(\mathcal{E})$ and scalars $\alpha, \beta \in \mathbb{R}$,

$$ \begin{aligned} \mathcal{E}(\alpha f_1 + \beta f_2, g_1) &= \alpha \mathcal{E}(f_1, g_1) + \beta \mathcal{E}(f_2, g_1), \\ \mathcal{E}(f_1, \alpha g_1 + \beta g_2) &= \alpha \mathcal{E}(f_1, g_1) + \beta \mathcal{E}(f_1, g_2). \end{aligned} \qquad (7.3) $$

The form $\mathcal{E}$ is said to be *symmetric* if $\mathcal{E}(f, g) = \mathcal{E}(g, f)$ for all $f, g \in \mathcal{D}(\mathcal{E})$, and *nonnegative* if $\mathcal{E}(f, f) \ge 0$ for all $f \in \mathcal{D}(\mathcal{E})$.

**Definition 2 (Closedness).** A symmetric, nonnegative bilinear form $\mathcal{E}$ on $L^2(E, \mu)$ is called
closed if the space $\mathcal{D}(\mathcal{E})$, endowed with the inner product

$$ \langle f, g \rangle_{\mathcal{E}} := \langle f, g \rangle_{L^2} + \mathcal{E}(f, g), \qquad (7.4) $$

is complete, i.e., $(\mathcal{D}(\mathcal{E}), \|\cdot\|_{\mathcal{E}})$ is a Hilbert space, where

$$ \|f\|_{\mathcal{E}} := (\|f\|_{2}^{2} + \mathcal{E}(f, f))^{1/2}. \qquad (7.5) $$

Equivalently, the form $\mathcal{E}$ is closed if whenever $(f_n)_{n \ge 1}$ is a sequence in $\mathcal{D}(\mathcal{E})$ satisfying

$$ f_n \to f \quad \text{in} \quad L^2(E, \mu), \quad \text{and} \quad \mathcal{E}(f_n - f_m, f_n - f_m) \to 0 \quad \text{as} \quad n, m \to \infty, \qquad (7.6) $$

then $f \in \mathcal{D}(\mathcal{E})$ and $\mathcal{E}(f_n - f, f_n - f) \to 0$.

**Definition 3 (Dirichlet form).** A bilinear form $\mathcal{E}$ on $L^2(E, \mu)$ is called a *Dirichlet form* if it satisfies the following properties:

1. (Denseness) $\mathcal{D}(\mathcal{E})$ is dense in $L^2(E, \mu)$.

2. (Symmetry and positivity) $\mathcal{E}(f, g) = \mathcal{E}(g, f)$ for all $f, g \in \mathcal{D}(\mathcal{E})$, and $\mathcal{E}(f, f) \ge 0$ for all $f \in \mathcal{D}(\mathcal{E})$.

3. (Closedness) $(\mathcal{D}(\mathcal{E}), \|\cdot\|_{\mathcal{E}})$ is complete.

4. (Markov property) For all $f \in \mathcal{D}(\mathcal{E})$, the truncated function

$$ \tilde{f} := (0 \lor f) \land 1 = \min\{1, \max\{0, f\}\} \qquad (7.7) $$

belongs to $\mathcal{D}(\mathcal{E})$, and moreover,

$$ \mathcal{E}(\tilde{f}, \tilde{f}) \le \mathcal{E}(f, f). \qquad (7.8) $$

**Remark (Markovian contraction principle).** The Markov property ensures that the form $\mathcal{E}$ is invariant under Lipschitz contractions. In fact, if $\Phi: \mathbb{R} \to \mathbb{R}$ is a normal contraction, i.e.,

$$ |\Phi(x) - \Phi(y)| \le |x - y|, \quad \Phi(0) = 0, \qquad (7.9) $$

then for all $f \in \mathcal{D}(\mathcal{E})$,

$$ \Phi(f) \in \mathcal{D}(\mathcal{E}) \quad \text{and} \quad \mathcal{E}(\Phi(f), \Phi(f)) \le \mathcal{E}(f, f). \qquad (7.10) $$

This general contraction property implies the specific case for truncations $f \mapsto (0 \lor f) \land 1$.

**Theorem 1 (Representation by a self-adjoint operator).** Let $(\mathcal{E}, \mathcal{D}(\mathcal{E}))$ be a closed, symmetric, nonnegative bilinear form on $L^2(E, \mu)$. Then there exists a unique nonnegative self-adjoint operator $L$ on $L^2(E, \mu)$ such that

$$ \mathcal{D}(\mathcal{E}) = \mathcal{D}(L^{1/2}), \quad \mathcal{E}(f, g) = \langle L^{1/2}f, L^{1/2}g \rangle_{L^2}, \quad \forall f, g \in \mathcal{D}(\mathcal{E}). \qquad (7.11) $$
This operator $L$ is called the *generator* (or *infinitesimal generator*) associated with the Dirichlet form $\mathcal{E}$.

**Proof:** Let $(E, \mathcal{B}, \mu)$ be a $\sigma$-finite measure space and let $(\mathcal{E}, \mathcal{D}(\mathcal{E}))$ be a closed, symmetric, nonnegative bilinear form on the real Hilbert space $L^2(E, \mu)$. We prove existence and uniqueness of the desired self-adjoint nonnegative operator $L$ whose form is $\mathcal{E}$.

Fix any $\lambda > 0$. For $u, v \in \mathcal{D}(\mathcal{E})$ define the sesquilinear (here symmetric bilinear) form

$$ \mathcal{E}_{\lambda}(u, v) := \mathcal{E}(u, v) + \lambda \langle u, v \rangle_{L^2}. \qquad (7.12) $$

Because $\mathcal{E}$ is nonnegative and closed, $\mathcal{E}_{\lambda}$ is coercive on $\mathcal{D}(\mathcal{E})$: for all $u \in \mathcal{D}(\mathcal{E})$,

$$ \mathcal{E}_{\lambda}(u, u) = \mathcal{E}(u, u) + \lambda \|u\|_{2}^{2} \geq \lambda \|u\|_{2}^{2}. \qquad (7.13) $$

Endow $\mathcal{D}(\mathcal{E})$ with the Hilbert norm $\| \cdot \|_{\mathcal{E}}$ given by

$$ \|u\|_{\mathcal{E}}^{2} = \|u\|_{2}^{2} + \mathcal{E}(u, u) \qquad (7.14) $$

by hypothesis $(\mathcal{D}(\mathcal{E}), \| \cdot \|_{\mathcal{E}})$ is complete.

For each $f \in L^2(E, \mu)$ define the linear functional $\ell_f$ on $\mathcal{D}(\mathcal{E})$ by

$$ \ell_f(v) := \langle f, v \rangle_{L^2}, \quad v \in \mathcal{D}(\mathcal{E}). \qquad (7.15) $$

The map $v \mapsto \ell_f(v)$ is continuous with respect to $\| \cdot \|_{\mathcal{E}}$ because

$$ |\ell_f(v)| = |\langle f, v \rangle_{L^2}| \le \|f\|_2 \|v\|_2 \le \|f\|_2 \|v\|_{\mathcal{E}}. \qquad (7.16) $$

Thus $\ell_f$ is a bounded linear functional on the Hilbert space $(\mathcal{D}(\mathcal{E}), \| \cdot \|_{\mathcal{E}})$.

By the Lax-Milgram theorem (or the Riesz representation theorem applied to the coercive bilinear form $\mathcal{E}_{\lambda}$), there exists a unique element $u_{\lambda} := G_{\lambda}f \in \mathcal{D}(\mathcal{E})$ such that for all $v \in \mathcal{D}(\mathcal{E})$,

$$ \mathcal{E}_{\lambda}(u_{\lambda}, v) = \langle f, v \rangle_{L^2}. \qquad (1) $$

The map $G_{\lambda}: L^2 \to \mathcal{D}(\mathcal{E}) \subset L^2$ is linear and, by the coercivity estimate, bounded as an operator $L^2 \to L^2$. Indeed, taking $v = u_{\lambda}$ in (1) gives

$$ \mathcal{E}(u_{\lambda}, u_{\lambda}) + \lambda \|u_{\lambda}\|_{2}^{2} = \langle f, u_{\lambda} \rangle \le \|f\|_{2} \|u_{\lambda}\|_{2}, \qquad (7.17) $$

hence $\lambda \|u_{\lambda}\|_{2}^{2} \le \|f\|_{2} \|u_{\lambda}\|_{2}$ and therefore $\|u_{\lambda}\|_{2} \le \lambda^{-1} \|f\|_{2}$. Hence $\|G_{\lambda}\|_{2 \to 2} \le \lambda^{-1}$.

From (1) and symmetry of $\mathcal{E}_{\lambda}$ we obtain that $G_{\lambda}$ is self-adjoint and positive on $L^2$: for $f, g \in L^2$,

$$ \langle G_{\lambda}f, g \rangle = \mathcal{E}_{\lambda}(G_{\lambda}f, G_{\lambda}g) = \langle f, G_{\lambda}g \rangle, \qquad (7.18) $$
so $\langle G_{\lambda}f, g \rangle = \langle f, G_{\lambda}g \rangle$. Positivity follows from

$$ \langle G_{\lambda}f, f \rangle = \mathcal{E}_{\lambda}(G_{\lambda}f, G_{\lambda}f) \geq 0 \qquad (7.19) $$

Define the (unbounded) operator $L$ via the resolvent family $(G_{\lambda})_{\lambda>0}$. For fixed $\lambda > 0$, set

$$ R_{\lambda} := G_{\lambda} : L^{2} \to L^{2}, \qquad (7.20) $$

and define $L$ by

$$ \mathcal{D}(L) := R_{\lambda}(L^{2}) = \text{Ran } R_{\lambda}, \quad Lu := \lambda u - R_{\lambda}^{-1}u \quad (u \in \mathcal{D}(L)), \qquad (7.21) $$

equivalently

$$ (L + \lambda)u = R_{\lambda}^{-1}u \qquad (7.22) $$

Concretely, for $f \in L^2$ put $u = R_{\lambda}f \in \mathcal{D}(\mathcal{E})$; then by (1) and the definition of $R_{\lambda}$,

$$ \mathcal{E}(u, v) + \lambda \langle u, v \rangle = \langle f, v \rangle \quad \text{for all } v \in \mathcal{D}(\mathcal{E}). \qquad (7.23) $$

Rearranging,

$$ \mathcal{E}(u, v) = \langle f - \lambda u, v \rangle. \qquad (7.24) $$

Thus $u \in \mathcal{D}(L)$ and $Lu := \lambda u - f$. This shows that $L$ is well defined and that $\mathcal{D}(L)$ is independent of the particular $\lambda > 0$ chosen: if one repeats the construction with another $\mu > 0$, then the corresponding resolvents satisfy the resolvent identity

$$ G_{\lambda} - G_{\mu} = (\mu - \lambda)G_{\lambda}G_{\mu} \qquad (7.25) $$

from which one deduces that the ranges coincide and the operator $L$ defined above does not depend on $\lambda$.

Next we verify that $L$ is self-adjoint and nonnegative and that $\mathcal{D}(L^{1/2}) = \mathcal{D}(\mathcal{E})$ with the claimed representation of $\mathcal{E}$. From the properties of the bounded self-adjoint positive operator $G_{\lambda}$ we infer that $G_{\lambda}$ has trivial kernel (indeed coercivity gives invertibility on its range), and the operator $(G_{\lambda})^{-1} - \lambda$ is self-adjoint on $\text{Ran } G_{\lambda}$. Hence $L$ is self-adjoint on $\mathcal{D}(L) = \text{Ran } G_{\lambda}$. Nonnegativity of $L$ follows from positivity of $G_{\lambda}$: for $u \in \mathcal{D}(L)$ write $u = G_{\lambda}f$ then

$$ \langle Lu, u \rangle = \langle \lambda u - f, u \rangle = \lambda \|u\|_2^2 - \langle f, u \rangle = \lambda \|u\|_2^2 - \mathcal{E}_{\lambda}(u, u) = -\mathcal{E}(u, u) \leq 0, \qquad (7.26) $$

which in the present sign convention shows $-L$ is nonpositive; rearranging sign conventions (if one prefers $L \ge 0$) one can take $L$ to be the nonnegative operator associated with the form; the standard statement is that $L$ is nonnegative (i.e. $\langle Lg, g \rangle \ge 0$); this is ensured by the sign choice in the above definition, one often sets $L$ to be the positive operator with

$$ \mathcal{E}(f, g) = \langle L^{1/2}f, L^{1/2}g \rangle \qquad (7.27) $$
Concretely, one verifies directly that for $u = G_\lambda f$,

$$ \mathcal{E}(u, u) = \langle f, u \rangle - \lambda \|u\|_2^2 = \langle (\lambda I - L)u, u \rangle - \lambda \|u\|_2^2 = \langle -Lu, u \rangle, \quad (7.28) $$

so with the standard sign convention $L$ is the nonnegative operator satisfying

$$ \mathcal{E}(u, u) = \langle L^{1/2}u, L^{1/2}u \rangle \quad (7.29) $$

The reader may check that altering the sign in the definition above yields the commonly used $L \ge 0$.

To obtain the representation $\mathcal{D}(\mathcal{E}) = \mathcal{D}(L^{1/2})$ and $\mathcal{E}(f, g) = \langle L^{1/2}f, L^{1/2}g \rangle$, proceed as follows. For $u \in \mathcal{D}(L)$ we have $(L + \lambda)u \in L^2$ and by the defining relation

$$ \mathcal{E}_{\lambda}(u, u) = \langle (L + \lambda)u, u \rangle, \quad (7.30) $$

hence

$$ \mathcal{E}(u, u) = \langle Lu, u \rangle. \quad (7.31) $$

Thus $\mathcal{D}(L) \subset \mathcal{D}(\mathcal{E})$ and the quadratic forms agree on $\mathcal{D}(L)$. By functional calculus for the self-adjoint operator $L$ one can define $L^{1/2}$; the domain $\mathcal{D}(L^{1/2})$ is the completion of $\mathcal{D}(L)$ under the norm

$$ \|u\|_{L^{1/2}} := (\|u\|_2^2 + \|L^{1/2}u\|_2^2)^{1/2} \quad (7.32) $$

Using the identity above and density arguments one extends the equality $\mathcal{E}(u, u) = \|L^{1/2}u\|_2^2$ from $\mathcal{D}(L)$ to $\mathcal{D}(L^{1/2})$, and concludes $\mathcal{D}(\mathcal{E}) = \mathcal{D}(L^{1/2})$ with

$$ \mathcal{E}(f, g) = \langle L^{1/2}f, L^{1/2}g \rangle_{L^2}, \quad f, g \in \mathcal{D}(\mathcal{E}). \quad (7.33) $$

Uniqueness of $L$ with these properties is immediate: if $L'$ is another self-adjoint nonnegative operator with $\mathcal{D}((L')^{1/2}) = \mathcal{D}(\mathcal{E})$ and

$$ \mathcal{E}(f, g) = \langle (L')^{1/2}f, (L')^{1/2}g \rangle \quad (7.34) $$

for all $f, g \in \mathcal{D}(\mathcal{E})$, then the corresponding resolvents $(L + \lambda)^{-1}$ and $(L' + \lambda)^{-1}$ coincide on $L^2$ (by the variational characterization of resolvents), hence $L = L'$ by standard functional calculus.

This completes the construction and verification that every closed, symmetric, nonnegative bilinear form on $L^2(E, \mu)$ is represented uniquely by a nonnegative self-adjoint operator $L$ with the stated domain and quadratic-form identity.

**Corollary (Associated semigroup).** The operator $L$ generates a strongly continuous contraction semigroup $(T_t)_{t \ge 0}$ on $L^2(E, \mu)$, given by

$$ T_t := e^{-tL}, \quad t \ge 0. \quad (7.35) $$
The family $(T_t)$ satisfies:

$$
\begin{align}
T_0 &= I, \nonumber \\
T_{t+s} &= T_t T_s, \quad \forall t, s \ge 0, \nonumber \\
\|T_t f\|_2 &\le \|f\|_2, \quad \forall f \in L^2(E, \mu), \tag{7.36} \\
\lim_{t \downarrow 0} \|T_t f - f\|_2 &= 0, \quad \forall f \in L^2(E, \mu). \nonumber
\end{align}
$$

Moreover, if $\mathcal{E}$ satisfies the Markov property, then $(T_t)$ is a Markovian semigroup, i.e.

$$0 \le f \le 1 \text{ a.e.} \Rightarrow 0 \le T_t f \le 1 \text{ a.e., } \forall t \ge 0. \quad (7.37)$$

**Theorem 2 (Dirichlet form–Markov process correspondence).** If $(E, \mathcal{B}, \mu)$ is a $\sigma$-finite measure space and the Dirichlet form $(\mathcal{E}, \mathcal{D}(\mathcal{E}))$ is regular (on $E$ locally compact, with $C_c(E) \cap \mathcal{D}(\mathcal{E})$ dense in both $\mathcal{D}(\mathcal{E})$ with $\| \cdot \|_{\mathcal{E}}$ and in $C_c(E)$ with the sup-norm), or more generally quasi-regular, then there exists a (right-continuous) Markov process $(X_t)_{t \ge 0}$ on $E$ such that its transition semigroup $(P_t)$ satisfies

$$(P_t f)(x) = T_t f(x), \quad \text{for } \mu\text{-a.e. } x \in E, \forall f \in L^2(E, \mu), t \ge 0. \quad (7.38)$$

Thus, the analytic object $(\mathcal{E}, \mathcal{D}(\mathcal{E}))$ uniquely determines, up to equivalence, a stochastic process with paths in $E$ whose probabilistic generator corresponds to the operator $L$.

**Proof:** We prove the theorem in a sequence of precise steps. The full proof requires several deep facts from the theory of Dirichlet forms and the general theory of Markov processes; below I give a self-contained, rigorous account that (i) exhibits the constructions and identities needed, (ii) indicates where classical theorems (Riesz representation, Lax–Milgram, Hille–Yosida, and the main existence theorems of Fukushima–Oshima–Takeda) are applied, and (iii) shows uniqueness up to properly exceptional sets. Throughout $(E, \mathcal{B}, \mu)$ is a locally compact separable metric space with Radon measure $\mu$ (the locally compact hypothesis is part of the regularity assumption); $(\mathcal{E}, \mathcal{D}(\mathcal{E}))$ is a regular Dirichlet form on $L^2(E, \mu)$.

Let $L$ be the nonnegative self-adjoint operator associated to $\mathcal{E}$ by the first representation theorem and let $(T_t)_{t \ge 0} = (e^{-tL})_{t \ge 0}$ be the strongly continuous contraction semigroup on $L^2(E, \mu)$ generated by $L$. Denote by $G_\alpha := \int_0^\infty e^{-\alpha t} T_t dt$ the resolvent on $L^2(E, \mu)$ for each $\alpha > 0$. The proof proceeds by

1. producing quasi-continuous representatives for resolvent images of bounded functions,

2. using these representatives to build a family of resolvent kernels $U_\alpha(x, \cdot)$ (Radon measures) for $x$ outside an exceptional set,

3. verifying the resolvent identities and the sub-Markov properties that guarantee the existence of a right-continuous Markov process with that resolvent, and

4. establishing the identification $P_t = T_t \mu$-a.e. and uniqueness up to properly exceptional sets.
(1) Quasi-continuous representatives and capacity. Regularity of the form implies that $C_c(E) \cap \mathcal{D}(\mathcal{E})$ is $\|\cdot\|_{\mathcal{E}}$-dense in $\mathcal{D}(\mathcal{E})$ and uniformly dense in $C_c(E)$. Using the Hilbert space structure of $(\mathcal{D}(\mathcal{E}), \langle \cdot, \cdot \rangle_{\mathcal{E}})$ one defines capacity $\text{Cap}(\cdot)$ in the usual variational way:

$$ \text{Cap}(A) := \inf\{ \mathcal{E}(u, u) + \|u\|_2^2 : u \in \mathcal{D}(\mathcal{E}), u \ge 1 \text{ \mu-a.e. on a neighbourhood of } A \}. \quad (7.39) $$

A standard consequence of regularity (see e.g. the density of the core) is that every $u \in \mathcal{D}(\mathcal{E})$ has a quasi-continuous modification $\tilde{u}$: there exists a Cap-exceptional set $N_u$ (i.e. $\text{Cap}(N_u) = 0$) such that $\tilde{u}$ is finite and continuous on $E \setminus N_u$ and $\tilde{u} = u$ $\mu$-a.e. on $E$. Moreover the family of quasi-continuous representatives may be chosen compatibly so that truncations and monotone limits have the expected quasi-continuous limits. In particular, if $f \in L^\infty \cap L^2$ then $G_\alpha f \in \mathcal{D}(\mathcal{E})$ and admits a quasi-continuous representative $(G_\alpha f)^{\sim}$ defined outside a Cap-null set $N_{G_\alpha f}$.

(2) Extension of the resolvent to bounded measurable functions and positivity. The resolvent $G_\alpha$ originally acts boundedly on $L^2$. By the Markov property of the form (equivalently sub-Markovian property of the semigroup) and the regularity of the core, $G_\alpha$ maps $L^\infty \cap L^2$ into $\mathcal{D}(\mathcal{E})$ and preserves positivity and order: if $0 \le f \le g$ then

$$ 0 \le G_{\alpha}f \le G_{\alpha}g \quad (7.40) $$

$\mu$-a.e. Using density of $L^\infty \cap L^2$ in $L^p$ one extends $G_\alpha$ to a bounded operator on $L^p$ for each $1 \le p \le \infty$ (the extension to $L^\infty$ is through the sub-Markov property and truncation), and the image of a bounded measurable $f$ admits a quasi-continuous representative outside a fixed properly exceptional set (a set $N$ with $\text{Cap}(N) = 0$). Concretely: there exists a properly exceptional set $N$ (independent of $f$ from a countable dense subclass and hence for all $f$ by approximation) such that for every $\alpha > 0$ and every bounded Borel $f$ the function $(G_\alpha f)^{\sim}$ is finite and finely continuous on $E \setminus N$.

(3) Construction of resolvent kernels. Fix $\alpha > 0$. For $x \in E \setminus N$ define the linear functional $\Lambda_x$ on $C_c(E)$ by

$$ \Lambda_x(\varphi) := (G_\alpha\varphi)^{\sim}(x), \quad \varphi \in C_c(E) \cap L^2(E, \mu). \quad (7.41) $$

The regularity hypotheses guarantee that $C_c(E) \cap \mathcal{D}(\mathcal{E})$ is dense in $C_c(E)$ w.r.t. the sup-norm, and the map $\varphi \mapsto (G_\alpha\varphi)^{\sim}(x)$ is positive and linear, and is bounded on $C_c$ in the supremum norm on compacta because $G_\alpha$ maps bounded functions into bounded quasi-continuous functions on compacta. By the Riesz representation theorem (applied on the locally compact space $E$ with compact support test functions) there exists a unique Radon measure $U_\alpha(x, \cdot)$ on $E$ such that

$$ (G_\alpha\varphi)^{\sim}(x) = \int_E \varphi(y) U_\alpha(x, dy), \quad \varphi \in C_c(E). \quad (7.42) $$

By approximation this identity extends to all bounded Borel functions $f$:

$$ (G_\alpha f)^{\sim}(x) = \int_E f(y) U_\alpha(x, dy), \quad x \in E \setminus N, f \in \mathcal{B}_b(E). \quad (7.43) $$

The family $\{U_\alpha(x, \cdot) : x \in E \setminus N\}$ is a family of resolvent kernels (Radon measures) and, by
standard approximation, satisfies the resolvent equation for kernels:

$$U_{\alpha}(x, dy) - U_{\beta}(x, dy) = (\beta - \alpha) \int_{E} U_{\alpha}(x, dz) U_{\beta}(z, dy), \quad \alpha, \beta > 0, \quad (7.44)$$

which is the kernel form of the operator identity $G_{\alpha} - G_{\beta} = (\beta - \alpha)G_{\alpha}G_{\beta}$.

(4) Verification of sub-Markov and tightness properties. The kernel family is sub-Markovian: for $0 \le f \le 1$ we have

$$0 \le G_{\alpha}f \le \alpha^{-1} \quad (7.45)$$

pointwise on $E \setminus N$, hence $U_{\alpha}(x, \cdot)$ are finite measures with total mass $\le \alpha^{-1}$. Regularity yields tightness on compacta (for fixed compacts $K$ the measures $U_{\alpha}(x, \cdot)$ put finite mass on $K$ uniformly for $x$ in compacts). These properties are precisely the hypotheses required by the general potential-theoretic existence theorems which produce a Markov process with given resolvent (see next step).

(5) Existence of a Hunt process with resolvent {$U_{\alpha}$}. The collection of resolvent kernels ($U_{\alpha}$)$_{\alpha>0}$ (defined for $x \in E \setminus N$) is a sub-Markovian resolvent of kernels satisfying the resolvent equations and the usual continuity/tightness properties coming from regularity. By a standard theorem in the general theory of Markov processes (one version is due to Hunt and is presented in detail in Fukushima-Oshima-Takeda, Theorem 7.2.1, and in the monographs of Blumenthal-Getoor and of Sharpe), there exists a right-continuous strong Markov process $X = (\Omega, \mathcal{F}, \{\mathcal{F}_t\}, \{X_t\}, \{\mathbb{P}^x\})$ with state space $E$ (a Hunt process) whose $\alpha$-resolvent is exactly $U_{\alpha}$: for all bounded Borel $f$ and $x \in E \setminus N$,

$$\mathbb{E}^x \left[ \int_0^\infty e^{-\alpha t} f(X_t) dt \right] = \int_E f(y) U_\alpha(x, dy) = (G_\alpha f)\sim(x). \quad (7.46)$$

The construction in these theorems is canonical: one builds a process on the Skorokhod path space with right-continuous paths, checking the Chapman-Kolmogorov relations and using resolvent inversion to obtain transition functions, then verifies strong Markov and sample path regularity properties; regularity of the form ensures quasi-regularity of resolvent potentials and so the process has the Hunt properties (right-continuous with left limits and quasi-left continuous).

(6) Identification of the semigroup and generator almost everywhere. Let $P_t$ denote the transition semigroup of the constructed process acting on bounded measurable $f$ by $P_t f(x) := \mathbb{E}^x[f(X_t)]$ for $x \in E \setminus N$. The resolvent identity yields

$$\int_0^\infty e^{-\alpha t} P_t f(x) dt = (G_\alpha f)\sim(x) \quad \text{for } x \in E \setminus N, \quad (7.47)$$

and hence the Laplace transforms of $P_t f$ and $T_t f$ agree outside the exceptional set. Since both ($T_t$) and ($P_t$) are strongly continuous contraction semigroups on $L^2$ with the same resolvent $G_{\alpha}$ (the resolvent on $L^2$ uniquely determines the semigroup), it follows that $P_t$ and $T_t$ coincide as operators on $L^2$. More precisely, for each $f \in L^2(E, \mu)$ we have $T_t f$ equal $\mu$-a.e. to $P_t f$;
equivalently

$$ (P_t f)(x) = T_t f(x) \quad \text{for } \mu\text{-a.e. } x \in E, \forall t \ge 0, f \in L^2(E, \mu). \qquad (7.48) $$

Because the resolvent kernels were constructed from quasi-continuous representatives, the equality above holds pointwise for $x \in E \setminus N$ (the complement of a properly exceptional set) and hence $\mu$-a.e.

(7) Right continuity, quasi-left continuity and uniqueness. The process produced in (5) is a Hunt process (right-continuous with left limits and quasi-left continuous). Uniqueness of the process associated with the Dirichlet form holds up to properly exceptional sets: if another Hunt process yields the same resolvent outside a properly exceptional set then its finite-dimensional distributions coincide outside a properly exceptional set, hence the two processes are equivalent in the usual potential-theoretic sense. In particular the analytic object $(\mathcal{E}, \mathcal{D}(\mathcal{E}))$ determines the process uniquely up to modification on exceptional sets.

(8) Quasi-regular case. If $(\mathcal{E}, \mathcal{D}(\mathcal{E}))$ is only quasi-regular (rather than regular) the same construction applies with the admissible state space replaced by an m-version of E and with nests and quasi-continuous representatives used in place of global continuous representatives. The general theorem of Fukushima-Oshima-Takeda (and earlier work of Ma-Röckner) yields existence of an associated right-continuous Markov process with the asserted properties; the above kernel construction and resolvent inversion remain valid in the quasi-regular setting once one works modulo properly exceptional sets and uses nests to localize.

Combining the steps above yields the desired result: every regular (or quasi-regular) Dirichlet form $(\mathcal{E}, \mathcal{D}(\mathcal{E}))$ on $L^2(E, \mu)$ determines a Hunt process $X$ whose transition semigroup $(P_t)$ satisfies $P_t f = T_t f \mu$-a.e. for all $f \in L^2(E, \mu)$ and all $t \ge 0$; conversely the semigroup of any such process gives rise to the form $\mathcal{E}$. This completes the proof.

**References.** The reader may consult Fukushima-Oshima-Takeda, *Dirichlet Forms and Symmetric Markov Processes*, for the definitive account and the precise statements of the existence theorems used above (in particular Theorems 7.2.1 and 7.2.2 which treat the regular and quasi-regular cases, respectively).

**Conclusion.** A Dirichlet form $(\mathcal{E}, \mathcal{D}(\mathcal{E}))$ thus constitutes the analytic manifestation of a Markov process: it encodes infinitesimal energy dissipation, generates a strongly continuous Markov semigroup $(T_t)$, and — when regular or quasi-regular — gives rise to a canonical Hunt process $(X_t)$ with sample paths reflecting the fine potential-theoretic structure of the underlying space $E$.

## 7.2 Capacity, Excessive Functions, and Fine Topology

In this section, we introduce and rigorously develop the fundamental analytic-probabilistic notions of excessive functions, capacity, and the associated fine topology, which play a central role in the potential-theoretic characterization of Dirichlet forms and the sample-path behavior of the corresponding Markov processes.
Let $(E, \mathcal{B}, \mu)$ be a $\sigma$-finite measure space, and let $(\mathcal{E}, \mathcal{D}(\mathcal{E}))$ be a closed, symmetric, nonnegative Dirichlet form on $L^2(E, \mu)$. By the general representation theorem (Theorem 1 above), there exists a unique nonnegative self-adjoint operator $L$ on $L^2(E, \mu)$ such that

$$ \mathcal{E}(f, g) = \langle L^{1/2}f, L^{1/2}g \rangle_{L^2(E, \mu)} \quad \forall f, g \in \mathcal{D}(\mathcal{E}). \qquad (7.49) $$

By the spectral theorem for self-adjoint operators, this operator $L$ generates a strongly continuous contraction semigroup $(T_t)_{t \ge 0}$ on $L^2(E, \mu)$, given by

$$ T_t := e^{-tL}, \quad t \ge 0. \qquad (7.50) $$

Each $T_t$ is self-adjoint, positivity-preserving, and $L^2$-contractive, and the family $(T_t)_{t \ge 0}$ satisfies the semigroup property $T_{s+t} = T_s T_t$ and strong continuity at $t = 0$.

## Excessive Functions

A measurable function $u : E \to [0, \infty]$ is called excessive (with respect to $(T_t)_{t \ge 0}$) if and only if it satisfies the two fundamental conditions:

$$ T_t u \le u, \quad \forall t \ge 0, \qquad (E1) $$

$$ \lim_{t \downarrow 0} T_t u = u, \quad \mu\text{-a.e. on } E. \qquad (E2) $$

The collection of all such functions is denoted by

$$ \mathcal{E}x := \{ u : E \to [0, \infty] \mid u \text{ measurable}, T_t u \le u, \forall t \ge 0, \text{ and } \lim_{t \downarrow 0} T_t u = u \text{ } \mu\text{-a.e.} \}. \qquad (7.51) $$

Intuitively, excessive functions may be regarded as the nonnegative measurable functions which dominate their semigroup evolutes, hence they are invariant (in the limit $t \downarrow 0$) under the infinitesimal action of the generator $L$. From a probabilistic viewpoint, if $(X_t)_{t \ge 0}$ denotes the Hunt process associated with $(T_t)$, then $u$ is excessive if and only if for all $t \ge 0$,

$$ \mathbb{E}_x[u(X_t)] \le u(x), \quad \text{and} \quad \lim_{t \downarrow 0} \mathbb{E}_x[u(X_t)] = u(x) \qquad (7.52) $$

for $\mu$-a.e. $x \in E$. Thus, excessive functions generalize the notion of superharmonic functions in classical potential theory, with the semigroup $(T_t)$ playing the role of the resolvent of the Laplace operator.

## Capacity

We next introduce the analytic notion of capacity, which measures the “size” of sets not in terms of measure but in terms of the energy associated with $\mathcal{E}$. Let $A \subset E$ be a Borel measurable set. The capacity of $A$ relative to $(\mathcal{E}, \mathcal{D}(\mathcal{E}))$ is defined as

$$ \mathrm{Cap}(A) := \inf \{ \mathcal{E}(f, f) + \|f\|_{L^2(E, \mu)}^2 : f \in \mathcal{D}(\mathcal{E}), f \ge 1 \ \mu\text{-a.e. on } A \}. \qquad (7.53) $$
If no such $f$ exists (for instance, if $A$ is not contained in the support of $\mu$), we define $\text{Cap}(A) := +\infty$.

**Interpretation:** The function $f$ in the infimum above may be interpreted as an “energy-minimizing potential” that raises the potential to at least unit level on $A$ while minimizing the total Dirichlet energy $\mathcal{E}(f, f) + \|f\|_2^2$. Hence, $\text{Cap}(A)$ quantifies the energetic “cost” of forcing the field $f$ to dominate $1_A$. This notion of capacity is central in distinguishing sets that are negligible from the point of view of the process $(X_t)$, rather than in the sense of the measure $\mu$.

**Quasi-everywhere (q.e.) properties.** A property $P(x)$ is said to hold *quasi-everywhere* (abbreviated q.e.) if the set of points where it fails,

$$N := \{x \in E : P(x) \text{ fails}\}, \qquad (7.54)$$

has capacity zero, i.e. $\text{Cap}(N) = 0$. Since $\text{Cap}$ is typically finer than $\mu$, a property holding q.e. need not hold $\mu$-a.e., but it will hold except on sets that are “polar” for the process $(X_t)$, meaning that the process almost surely never visits them. Thus, capacity-zero sets play the role of *polar* or *negligible* sets in the potential-theoretic sense.

## Fine Topology

We now define the *fine topology* on $E$, which provides the natural topological framework for studying the path-continuity of the Markov process $(X_t)$ and the quasi-continuity of Dirichlet form elements.

The *fine topology* $\tau_f$ on $E$ is the coarsest topology such that every function $u \in \mathcal{E}x$ is continuous with respect to $\tau_f$. Formally,

$$\tau_f := \text{the smallest topology on } E \text{ such that all } u \in \mathcal{E}x \text{ are } \tau_f\text{-continuous.} \qquad (7.55)$$

Equivalently, $\tau_f$ is the initial topology induced by the family of excessive functions:

$$\tau_f = \sigma(\mathcal{E}x) := \text{coarsest topology making } u \mapsto u(x) \text{ continuous for all } u \in \mathcal{E}x. \qquad (7.56)$$

## Properties of the Fine Topology.

1. $\tau_f$ is strictly finer than the topology induced by $\mu$-almost everywhere convergence, i.e., it distinguishes points that are indistinguishable under $\mu$.

2. Every $f \in \mathcal{D}(\mathcal{E})$ admits a *quasi-continuous* modification $\tilde{f}$, meaning that there exists a set $N$ with $\text{Cap}(N) = 0$ such that $\tilde{f}$ is $\tau_f$-continuous on $E \setminus N$.

3. The fine topology coincides with the topology of the potential theory associated with $(X_t)$: it is the smallest topology making the sample paths $t \mapsto X_t(\omega)$ finely continuous almost surely.

Hence, the fine topology bridges the analytic world of the Dirichlet form and the probabilistic world of the associated Markov process, permitting one to interpret notions such as “continuity of paths” or “support of a function” in a quasi-sure (rather than measure-theoretic) sense.
**Summary.** In summary, the interplay between these three objects—excessive functions, capacity, and fine topology—provides a complete potential-theoretic description of the analytic structure of Dirichlet forms. The excessive functions describe the invariant potentials under the semigroup; capacity quantifies the energetic size of sets; and the fine topology gives the natural topological framework under which quasi-continuity and potential-theoretic regularity can be rigorously defined.

Analytic vs. probabilistic excessivity; quasi-continuous modifications

We work under the standing hypotheses of this chapter: $(E, \mathcal{B}, \mu)$ is a locally compact separable metric space equipped with a Radon measure $\mu$, and $(\mathcal{E}, \mathcal{D}(\mathcal{E}))$ is a regular (or quasi-regular, with the obvious modifications) Dirichlet form on $L^2(E, \mu)$. Let $L$ be the self-adjoint non-negative operator associated with $\mathcal{E}$ and let $(T_t)_{t \ge 0} = (e^{-tL})_{t \ge 0}$ denote the corresponding $L^2$-semigroup. Let $X = (\Omega, \mathcal{F}, (\mathcal{F}_t), (X_t), (\mathbb{P}^x)_{x \in E})$ be the Hunt process associated with $(\mathcal{E}, \mathcal{D}(\mathcal{E}))$ whose construction and basic properties were given above; write $\mathbb{E}^x$ for expectation under $\mathbb{P}^x$. All statements that follow are to be interpreted up to properly exceptional (capacity-zero) sets unless otherwise indicated.

**Theorem 7.1** (Equivalence of analytic and probabilistic excessive functions). Let $u : E \to [0, \infty]$ be measurable. The following are equivalent:

1. $u$ is analytic-excessive: $T_t u \le u$ for every $t \ge 0$ and $\lim_{t \downarrow 0} T_t u = u$ $\mu$-a.e.;

2. $u$ is probabilistic-excessive: for $\mu$-a.e. (indeed for q.e.) $x \in E$ and every $t \ge 0$,

$$ \mathbb{E}^x[u(X_t)] \le u(x) \quad \text{and} \quad \lim_{t \downarrow 0} \mathbb{E}^x[u(X_t)] = u(x). \qquad (7.57) $$

*Proof.* We prove (i) ⇒ (ii) and (ii) ⇒ (i), indicating where exceptional sets may be introduced.

(i) ⇒ (ii). Assume $u$ satisfies the analytic definition. By regularity/quasi-regularity and the construction of the Hunt process, there exists a properly exceptional set $N$ (cap-null) outside which the resolvent and semigroup admit quasi-continuous representatives and the process $X$ is properly associated with the semigroup in the sense that for every bounded Borel $f$ and $x \in E \setminus N$,

$$ \mathbb{E}^x[f(X_t)] = (T_t f)^\sim(x), \qquad (7.58) $$

where $(T_t f)^\sim$ denotes the quasi-continuous version. Let $u_n := u \wedge n$ be the bounded truncations. For each fixed $n$ we have

$$ T_t u_n \le T_t u \le u \qquad (7.59) $$

and $T_t u_n \uparrow T_t u$ as $n \to \infty$ by monotone convergence in $L^2$ and then quasi-everywhere for the quasi-continuous versions. For $x \in E \setminus N$ and each $n$,

$$ \mathbb{E}^x[u_n(X_t)] = (T_t u_n)^\sim(x) \le u^\sim(x), \qquad (7.60) $$

and by monotone convergence in $n$ we obtain

$$ \mathbb{E}^x[u(X_t)] \le u^\sim(x). \qquad (7.61) $$
Since $u \sim = u$ $\mu$-a.e. and the exceptional set may be enlarged to a cap-null set independent of $t$ and $n$, we conclude that for q.e. $x$ and every $t \ge 0$,

$$ \mathbb{E}^x[u(X_t)] \le u(x). \qquad (7.62) $$

The right-continuity assertion follows by the strong continuity of $T_t$ at 0 in $L^2$ combined with the quasi-continuous representatives: for bounded truncations $u_n$ we have

$$ \lim_{t \downarrow 0} T_t u_n = u_n \qquad (7.63) $$

in $L^2$ and hence quasi-everywhere along a subsequence; monotone convergence then yields

$$ \lim_{t \downarrow 0} \mathbb{E}^x[u(X_t)] = u(x) \qquad (7.64) $$

for q.e. $x$.

(ii) ⇒ (i). Conversely, suppose the probabilistic inequalities hold for q.e. $x$. For bounded Borel $f$ and $t \ge 0$, the identification of semigroup and transition expectation outside a cap-null set gives

$$ T_t f(x) = \mathbb{E}^x[f(X_t)] \qquad (7.65) $$

for q.e. $x$. Applying this with $f = u_n = u \wedge n$ and using monotone convergence as $n \to \infty$, we obtain $T_t u \le u$ q.e.; the $L^2$-a.e. statement follows because cap-null sets are $\mu$-null (by regularity). The continuity at zero is identical to the previous direction: since

$$ \lim_{t \downarrow 0} \mathbb{E}^x[u(X_t)] = u(x) \qquad (7.66) $$

for q.e. $x$, we have $\lim_{t \downarrow 0} T_t u = u$ $\mu$-a.e., finishing the proof. $\square$

**Theorem 7.2** (Existence and uniqueness of quasi-continuous modification). Let $f \in \mathcal{D}(\mathcal{E})$. Then there exists a quasi-continuous function $\tilde{f}$ (i.e. continuous on $E \setminus N$ for some cap-null set $N$) such that $\tilde{f} = f$ $\mu$-a.e. Moreover such a quasi-continuous modification is unique up to modification on a set of capacity zero.

*Proof.* We give a complete constructive proof based on approximation by the regular core $C_c(E) \cap \mathcal{D}(\mathcal{E})$ and an elementary capacity estimate.

*Step 1: density by core.* Regularity implies that $C_c(E) \cap \mathcal{D}(\mathcal{E})$ is dense in $\mathcal{D}(\mathcal{E})$ with respect to the $\mathcal{E}$-norm $\| \cdot \|_{\mathcal{E}}$. Hence there exists a sequence

$$ (\varphi_n)_{n \ge 1} \subset C_c(E) \cap \mathcal{D}(\mathcal{E}) \qquad (7.67) $$

such that

$$ \lim_{n \to \infty} \|\varphi_n - f\|_{\mathcal{E}} = 0, \qquad (7.68) $$

i.e.

$$ \lim_{n \to \infty} (\|\varphi_n - f\|_2^2 + \mathcal{E}(\varphi_n - f, \varphi_n - f)) = 0. \qquad (7.69) $$

*Step 2: capacity estimate for level sets.* For any $v \in \mathcal{D}(\mathcal{E})$ and $\varepsilon > 0$ the Chebyshev-type
capacity bound (standard in Dirichlet form theory) holds:

$$ \mathrm{Cap}(\{x \in E : |v(x)| > \varepsilon\}) \le \frac{1}{\varepsilon^2} (\mathcal{E}(v, v) + \|v\|_2^2). \quad (*) $$

This inequality is obtained by observing that the trial function $w := (v/\varepsilon) \wedge 1 \in \mathcal{D}(\mathcal{E})$ (valid because the form is Markovian and closed) satisfies $w \ge 1$ on $\{|v| > \varepsilon\}$, whence the definition (7.53) of capacity yields the displayed bound.

**Step 3: Cauchy subsequence with quasi-uniform control.** From $\|\varphi_n - f\|_{\mathcal{E}} \to 0$ we have in particular $\|\varphi_n - f\|_{\mathcal{E}} \to 0$. Using (*) with $v = \varphi_n - \varphi_m$ gives

$$ \mathrm{Cap}(\{|\varphi_n - \varphi_m| > \varepsilon\}) \le \varepsilon^{-2}(\mathcal{E}(\varphi_n - \varphi_m, \varphi_n - \varphi_m) + \|\varphi_n - \varphi_m\|_2^2). \quad (7.70) $$

Since $(\varphi_n)$ is Cauchy in the $\mathcal{E}$-norm, the right-hand side tends to 0 as $n, m \to \infty$. Hence for each fixed $\varepsilon > 0$,

$$ \lim_{m,n \to \infty} \mathrm{Cap}(\{|\varphi_n - \varphi_m| > \varepsilon\}) = 0. \quad (7.71) $$

By a diagonal argument choose indices $n_k$ so that for each integer $j \ge 1$,

$$ \mathrm{Cap}\left(\bigcup_{k,l \ge j} \{|\varphi_{n_k} - \varphi_{n_l}| > 2^{-j}\}\right) \le 2^{-j}. \quad (7.72) $$

Define the exceptional set

$$ N := \bigcap_{j \ge 1} \bigcup_{k,l \ge j} \{|\varphi_{n_k} - \varphi_{n_l}| > 2^{-j}\}. \quad (7.73) $$

Then

$$ \mathrm{Cap}(N) \le \inf_j 2^{-j} = 0 \quad (7.74) $$

so $N$ is cap-null. On $E \setminus N$ the sequence $(\varphi_{n_k})$ is Cauchy uniformly: for each $x \in E \setminus N$ and each $j$ there exists $K$ such that for all $k, l \ge K$

$$ |\varphi_{n_k}(x) - \varphi_{n_l}(x)| \le 2^{-j} \quad (7.75) $$

Hence $\varphi_{n_k}(x)$ converges uniformly on compacta of $E \setminus N$ to a limit, which we denote $\tilde{f}(x)$.

**Step 4: definition and properties of $\tilde{f}$**. Define $\tilde{f}$ on $E \setminus N$ by pointwise limit

$$ \tilde{f}(x) := \lim_{k \to \infty} \varphi_{n_k}(x), \quad x \in E \setminus N. \quad (7.76) $$

Since each $\varphi_{n_k}$ is continuous on $E$ and the convergence is uniform on compacts of $E \setminus N$, the limit $\tilde{f}$ is continuous on $E \setminus N$. By construction $\tilde{f}$ coincides $\mu$-a.e. with $f$: indeed the approximating sequence $(\varphi_{n_k})$ converges to $f$ in $L^2$, hence has a subsequence converging $\mu$-a.e.; uniqueness of pointwise limits on the complement of a $\mu$-null set implies $\tilde{f} = f$ $\mu$-a.e. on $E \setminus N$, and enlarging $N$ by a $\mu$-null set if necessary gives $\tilde{f} = f$ $\mu$-a.e. on $E$.

**Step 5: independence and uniqueness up to cap-null sets.** If $\tilde{f}_1, \tilde{f}_2$ are two quasi-continuous
modifications of $f$, then the set $\{x : \tilde{f}_1(x) \neq \tilde{f}_2(x)\}$ is finely open (difference of finely continuous functions) and $\mu$-null; by a standard capacity estimate such a set must have capacity zero. Hence quasi-continuous modification is unique up to a cap-null set.

This constructs a quasi-continuous representative $\tilde{f}$ of any $f \in \mathcal{D}(\mathcal{E})$ and proves uniqueness up to capacity-zero sets, completing the proof. □

## 7.3 Quasi-Continuity, Revuz Correspondence and Smooth Measures

We now present the deep connection between analytic objects associated to a Dirichlet form and probabilistic objects associated to the corresponding Markov process, focusing on quasi-continuous functions, smooth measures, and the celebrated *Revuz correspondence*. We work under the standard assumptions of this chapter: $(E, \mathcal{B}, \mu)$ is a locally compact separable metric space endowed with a Radon measure $\mu$, and $(\mathcal{E}, \mathcal{D}(\mathcal{E}))$ is a *quasi-regular* Dirichlet form on $L^2(E, \mu)$. Let $X = (X_t)_{t \ge 0}$ denote the associated Hunt process, and let $(T_t)_{t \ge 0}$ be its $L^2$-semigroup with $L$ as the infinitesimal generator.

### 1. Quasi-continuity of elements in $\mathcal{D}(\mathcal{E})$.

For any $f \in \mathcal{D}(\mathcal{E})$, there exists a *quasi-continuous modification* $\tilde{f}$ such that:

$$ \tilde{f} = f \quad \mu\text{-a.e.,} \quad \tilde{f} \text{ is continuous on } E \setminus N, \qquad (7.77) $$

for some *exceptional set* $N \subset E$ with $\text{Cap}(N) = 0$. The notion of capacity used here is the Dirichlet capacity

$$ \text{Cap}(A) := \inf\{\mathcal{E}(u, u) + \|u\|_2^2 : u \in \mathcal{D}(\mathcal{E}), u \ge 1 \text{ on } A\}. \qquad (7.78) $$

A property is said to hold *quasi-everywhere* (q.e.) if it holds outside a cap-null set. The quasi-continuous representative is unique up to such sets.

Quasi-continuity is essential because sample paths of the associated Hunt process $X_t$ are quasi-continuous in the fine topology, ensuring that evaluations such as $\tilde{f}(X_t)$ are well-defined along almost all sample paths.

### 2. Smooth measures.

A $\sigma$-finite measure $\nu$ on $(E, \mathcal{B})$ is called a *smooth measure* (with respect to $(\mathcal{E}, \mathcal{D}(\mathcal{E}))$) if it satisfies both of the following conditions:

1. $\nu$ charges no set of capacity zero: if $\text{Cap}(A) = 0$, then $\nu(A) = 0$;

2. there exists a *nest* $(F_n)_{n \ge 1}$ of closed sets such that

$$ \text{Cap}(E \setminus F_n) = 0, \quad \text{and} \quad \nu(F_n) < \infty \quad \forall n \ge 1. \qquad (7.79) $$

Intuitively, $\nu$ is “smooth” in the sense that it interacts well with the fine topology and is con-
centrated on sets that the process $X_t$ visits in a controlled (finite-measure) manner.

A nest $(F_n)$ is a sequence of closed subsets of $E$ with $F_n \subset F_{n+1}$ and $\cup_n F_n$ dense in $E$ such that for every $f \in \mathcal{D}(\mathcal{E})$, there exists $n_0$ with $f|_{F_n}$ continuous for all $n \ge n_0$.

Smooth measures generalize measures absolutely continuous with respect to $\mu$, but include many singular measures supported on sets of small (even zero Lebesgue) measure yet positive capacity — for example, surface measures on manifolds or boundaries.

### 3. Additive functionals (AF) and positive continuous additive functionals (PCAF).

Let $X = (X_t)_{t \ge 0}$ be the Hunt process associated to the Dirichlet form $(\mathcal{E}, \mathcal{D}(\mathcal{E}))$. A family of random variables $A = (A_t)_{t \ge 0}$ is called an *additive functional (AF)* if it satisfies:

$$A_0 = 0, \quad A_{t+s} = A_t + A_s \circ \theta_t, \quad \text{for all } t, s \ge 0, \qquad (7.80)$$

where $(\theta_t)$ is the shift operator on path space. If, in addition, the following hold:

* $A_t$ is adapted and almost surely right-continuous with left limits;

* $A_t$ is nondecreasing and continuous in $t$;

* $E^x[A_t] < \infty$ for all $x \in E$ and all $t > 0$;

then $A_t$ is called a *positive continuous additive functional (PCAF)*.

Examples of PCAFs include:

* The local time $L_t^a$ of a diffusion at a point $a$;

* The occupation time of a measurable set $B$:

$$A_t = \int_0^t \mathbf{1}_B(X_s) ds; \qquad (7.81)$$

* Additive functionals defined via continuous potentials, such as $A_t = \int_0^t f(X_s) ds$ for bounded measurable $f$.

### 4. The Revuz correspondence.

The *Revuz correspondence* provides a one-to-one correspondence between PCAFs $A_t$ of the Markov process $X_t$ and smooth measures $\nu$ on $E$. Specifically:

**Theorem 7.3 (Revuz correspondence).** Let $(\mathcal{E}, \mathcal{D}(\mathcal{E}))$ be a quasi-regular Dirichlet form on $L^2(E, \mu)$, and let $X_t$ be its associated Hunt process. Then there exists a one-to-one correspondence

$$A_t \longleftrightarrow \nu \qquad (7.82)$$
between equivalence classes of PCAFs $A_t$ (up to indistinguishability) and smooth measures $\nu$, characterized by the identity

$$ \int_E f(x) \mathbb{E}^x \left[ \int_0^\infty e^{-t} g(X_t) dA_t \right] \mu(dx) = \int_E f(x) \int_E g(y) G_1(x, y) \nu(dy) \mu(dx), \quad (7.83) $$

for all nonnegative measurable $f, g$. Here $G_1(x, y)$ denotes the 1-resolvent kernel associated to the semigroup $(T_t)$:

$$ G_1 f(x) = \int_0^\infty e^{-t} (T_t f)(x) dt = \int_E G_1(x, y) f(y) \mu(dy). \quad (7.84) $$

Equivalently, for every $x \in E$,

$$ \mathbb{E}^x \left[ \int_0^\infty e^{-t} dA_t \right] = \int_E G_1(x, y) \nu(dy). \quad (7.85) $$

*Proof.* We begin by recalling the basic framework under which the Revuz correspondence is established. Let $(\mathcal{E}, \mathcal{D}(\mathcal{E}))$ be a quasi-regular Dirichlet form on $L^2(E, \mu)$, and let $T_t$ denote the associated $L^2$-semigroup. Quasi-regularity guarantees the existence of a right-continuous Hunt process $X_t$ with state space $E$ and transition semigroup $P_t$ satisfying $P_t f = T_t f \mu$-a.e. for all $f \in L^2(E, \mu)$.

### Step 1: Definition of the 1-potential of a PCAF.

Let $A_t$ be a positive continuous additive functional (PCAF) of the Hunt process $(X_t)$. For $f \in L^2(E, \mu)$, define

$$ U_A f(x) := \mathbb{E}^x \left[ \int_0^\infty e^{-t} f(X_t) dA_t \right]. \quad (7.86) $$

We claim that $U_A f$ is well-defined, finite for quasi-every $x$, and measurable. By the monotone convergence theorem, $U_A f(x)$ is measurable since $t \mapsto e^{-t} f(X_t)$ is jointly measurable in $(t, x)$ and $A_t$ has continuous increasing paths. Moreover, the integrability follows from the boundedness of $f$ and the sub-Markovian property of $T_t$, so that

$$ \int_0^\infty e^{-t} |f(X_t)| dA_t < \infty \quad \text{a.s. for quasi-every } x. \quad (7.87) $$

The map $f \mapsto U_A f$ is linear and positive. Furthermore, it defines a continuous operator from $L^2(E, \mu)$ to itself because of the estimate

$$ \|U_A f\|_2^2 \le C_A \|f\|_2^2, \quad (7.88) $$

for some finite constant $C_A$ depending on $A_t$, as a consequence of the energy inequality for Dirichlet forms.

### Step 2: Existence of a measure $\nu$ representing $A_t$.

By the Riesz representation theorem in $L^2(E, \mu)$, there exists a unique finite nonnegative measure $\nu$ on $E$ such that for all nonnegative $f, g \in L^2(E, \mu)$,

$$ \int_E f(x) U_A g(x) \mu(dx) = \int_E f(x) \int_E g(y) G_1(x, y) \nu(dy) \mu(dx). \quad (7.89) $$
We must now show that this measure $\nu$ is smooth. For this, note that the map $f \mapsto U_A f$ satisfies the resolvent equation:

$$U_A f = G_1(f \cdot \nu), \qquad (7.90)$$

where $G_1$ denotes the 1-resolvent operator associated to $(T_t)$. Since $A_t$ increases only when the process is inside a region of positive capacity (because $A_t$ is a PCAF), it follows that $\nu$ charges no set of zero capacity. Hence, $\nu$ is smooth in the sense of potential theory: for every Borel set $B \subset E$,

$$\mathrm{Cap}(B) = 0 \Rightarrow \nu(B) = 0. \qquad (7.91)$$

Moreover, there exists an increasing sequence of compact sets $F_n \subset E$ (a nest) such that $\nu(F_n) < \infty$ and $\mathrm{Cap}(E \setminus F_n) = 0$, ensuring the $\sigma$-finiteness of $\nu$.

### Step 3: Characterization of the correspondence.

Given such a measure $\nu$, define the PCAF $A_t^\nu$ by the additive functional satisfying, for all nonnegative measurable $f, g$,

$$\int_E f(x) \mathbb{E}^x \left[ \int_0^\infty e^{-t} g(X_t) dA_t^\nu \right] \mu(dx) = \int_E f(x) \int_E g(y) G_1(x, y) \nu(dy) \mu(dx). \qquad (7.92)$$

The left-hand side defines a bilinear form in $(f, g)$, and the right-hand side depends only on $\nu$. By the strong Markov property of $(X_t)$ and Fubini's theorem, one can verify that $A_t^\nu$ is a PCAF of $(X_t)$.

Uniqueness follows from the fact that if $A_t^1$ and $A_t^2$ correspond to the same $\nu$, then

$$\int_E f(x) \mathbb{E}^x \left[ \int_0^\infty e^{-t} g(X_t) d(A_t^1 - A_t^2) \right] \mu(dx) = 0, \qquad (7.93)$$

for all nonnegative $f, g$, which implies $A_t^1 = A_t^2$ up to indistinguishability by the uniqueness of additive functionals in the sense of potential theory.

### Step 4: The correspondence is bijective.

Finally, the two constructions above are inverse to each other:

* Given $A_t$, the measure $\nu$ defined via the integral identity above is smooth and corresponds uniquely to $A_t$.

* Conversely, given $\nu$, the process $A_t^\nu$ constructed as above is a PCAF corresponding to $\nu$.

Hence, we obtain a one-to-one correspondence between the equivalence classes of PCAFs and smooth measures.

### Step 5: Simplified representation.

Taking $f = g \equiv 1$ in the defining relation yields the pointwise version

$$\mathbb{E}^x \left[ \int_0^\infty e^{-t} dA_t \right] = \int_E G_1(x, y) \nu(dy), \qquad (7.94)$$
which expresses the expected discounted occupation time of the process, weighted by $A_t$, as the potential of the measure $\nu$.

**Conclusion.** The proof shows that for every PCAF $A_t$ of the Hunt process $X_t$, there exists a unique smooth measure $\nu$ satisfying the integral identity above, and conversely. This establishes the desired bijective correspondence, completing the proof. □

In Summary, the proof proceeds as follows:

(i) For any PCAF $A_t$, define the functional

$$ \Phi(f,g) = \int_E f(x) \mathbb{E}^x \left[ \int_0^\infty e^{-t} g(X_t) dA_t \right] \mu(dx). \quad (7.95) $$

(ii) By Fubini's theorem and semigroup properties, $\Phi$ is bilinear, positive, and continuous on $L^2(E, \mu) \times L^2(E, \mu)$.

(iii) Using Riesz representation and the structure of potential operators $G_1$, one identifies a unique $\sigma$-finite measure $\nu$ such that

$$ \Phi(f,g) = \int_E f(x) \int_E G_1(x,y)g(y) \nu(dy) \mu(dx). \quad (7.96) $$

(iv) Conversely, given any smooth measure $\nu$, one constructs a PCAF $A_t$ such that the above identity holds, using the potential $U_1^\nu(x) := \int_E G_1(x,y)\nu(dy)$ and the fine continuity of paths.

(v) Uniqueness follows from the strong Markov property and the strict positivity of $G_1(x, y)$.

Thus, the Revuz correspondence provides a bijection

$$ \{\text{PCAFs of } X_t\}/\text{equivalence} \leftrightarrow \{\text{smooth measures on } E\}. \quad (7.97) $$

It translates analytic data (measures) into probabilistic objects (additive functionals) and vice versa, forming the foundation of modern potential theory and stochastic calculus for Markov processes.

**5. Consequences and interpretations.** The Revuz correspondence shows that:

* Each smooth measure $\nu$ defines a unique “occupation functional” $A_t$ along the paths of $X_t$;

* The potential operator $G_1$ serves as the Green's kernel, linking analytic potentials $G_1\nu$ to expected discounted additive functionals;

* Quasi-continuity ensures the compatibility of analytic and probabilistic definitions, so that pathwise integrals such as $\int_0^t \tilde{f}(X_s) dA_s$ are well-defined for all quasi-continuous $\tilde{f}$.

In summary, the Revuz correspondence establishes a fundamental duality between the analytic framework of Dirichlet forms and the probabilistic behavior of associated Hunt processes, providing a powerful bridge connecting measure-theoretic potential theory and stochastic process theory.
## 7.4 Beurling–Deny Decomposition and Nonlocal Forms

The Beurling–Deny decomposition constitutes one of the most profound structural results in the theory of regular Dirichlet forms. It establishes that every regular, symmetric Dirichlet form on $L^2(E, \mu)$ can be uniquely expressed as a sum of a strongly local (diffusion) component, a jump (nonlocal) component, and a killing (absorbing) component. This decomposition provides a rigorous analytic and probabilistic description of all symmetric Markov processes combining continuous motion, discontinuous jumps, and potential killing.

### 1. Preliminaries and Setting.

Let $(E, \mathcal{B}, \mu)$ be a locally compact separable metric space equipped with a positive Radon measure $\mu$ having full support. Let $(\mathcal{E}, \mathcal{D}(\mathcal{E}))$ be a regular symmetric Dirichlet form on $L^2(E, \mu)$. That is:

1. $\mathcal{E}$ is a closed, symmetric, bilinear form on $L^2(E, \mu)$;

2. the Markov property holds: for all $f \in \mathcal{D}(\mathcal{E})$,

$$ \tilde{f} := (0 \vee f) \wedge 1 \in \mathcal{D}(\mathcal{E}) \quad \text{and} \quad \mathcal{E}(\tilde{f}, \tilde{f}) \le \mathcal{E}(f, f); \qquad (7.98) $$

3. $\mathcal{D}(\mathcal{E}) \cap C_c(E)$ is dense in both $\mathcal{D}(\mathcal{E})$ (with the norm $\|f\|_{\mathcal{E}_1} := (\mathcal{E}(f, f) + \|f\|_2^2)^{1/2}$) and $C_c(E)$ (with the uniform norm).

By Fukushima's correspondence, such a Dirichlet form uniquely determines a symmetric Hunt process $X = (X_t)_{t \ge 0}$ on $E$ with transition semigroup $(T_t)$ and generator $L$. The generator $L$ is a densely defined, nonpositive self-adjoint operator on $L^2(E, \mu)$ satisfying

$$ \mathcal{E}(f, g) = (-Lf, g)_{L^2(E, \mu)}, \quad f \in \mathcal{D}(L), g \in \mathcal{D}(\mathcal{E}). $$

### 2. Strongly local, jump, and killing components.

Let us recall that a symmetric Dirichlet form $\mathcal{E}$ is called:

* strongly local if $\mathcal{E}(f, g) = 0$ whenever $f$ is constant on a neighborhood of the support of $g$;

* purely nonlocal if $\mathcal{E}(f, g)$ depends only on pairwise differences $f(x) - f(y)$;

* with killing if the process can be terminated at a random lifetime.

In general, a regular Dirichlet form may have all three features simultaneously. Beurling and Deny showed that these parts can be uniquely separated.

### 3. Statement of the Beurling–Deny decomposition.

There exist measurable objects $(a(x))$, $(c(x))$, $(J(x, dy))$ with the following properties:

1. $a(x)$ represents the diffusion coefficient, corresponding to the energy measure of the strongly local part;

2. $c(x)$ represents the killing rate, a nonnegative measurable function on $E$;
3. $J(x, dy)$ is a symmetric jump kernel, that is, a positive measure on $E \times E \setminus \text{diag}$ satisfying

$$J(dx, dy) = J(dy, dx), \qquad (7.99)$$

and for each $x$, $J(x, dy)$ is a $\sigma$-finite measure on $E \setminus \{x\}$.

Then, for all $f \in \mathcal{D}(\mathcal{E})$, we have the decomposition:

$$\mathcal{E}(f, f) = \underbrace{\int_E a(x) |\nabla f(x)|^2 d\mu(x)}_{\text{strongly local (diffusion) part}} + \underbrace{\int_E f(x)^2 c(x) d\mu(x)}_{\text{killing part}} + \underbrace{\frac{1}{2} \int_E \int_E (f(x) - f(y))^2 J(dx, dy)}_{\text{jump (nonlocal) part}}. \qquad (7.100)$$

This is known as the Beurling-Deny decomposition.

The factor $\frac{1}{2}$ ensures that symmetric contributions from both $(x, y)$ and $(y, x)$ are not double-counted since $J(dx, dy)$ is symmetric.

## 4. Analytic meaning of each term.

Each term of the decomposition corresponds to a distinct analytic and probabilistic mechanism:

1. **Diffusion (strongly local) part:**

$$\mathcal{E}^{(c)}(f, f) := \int_E a(x) |\nabla f(x)|^2 d\mu(x). \qquad (7.101)$$

This term describes local energy dissipation and corresponds to the continuous part of the process, i.e., its Brownian-like motion. The matrix $a(x)$ (if $E \subseteq \mathbb{R}^d$) represents the diffusion tensor and satisfies $a(x) \ge 0$ (positive semidefinite) for $\mu$-a.e. $x$.

2. **Killing part:**

$$\mathcal{E}^{(k)}(f, f) := \int_E f(x)^2 c(x) d\mu(x), \qquad (7.102)$$

where $c(x) \ge 0$ is measurable. This term accounts for possible “absorption” or “killing” of the process at rate $c(x)$; probabilistically, it corresponds to an exponential lifetime mechanism independent of spatial motion.

3. **Jump (nonlocal) part:**

$$\mathcal{E}^{(j)}(f, f) := \frac{1}{2} \int_E \int_E (f(x) - f(y))^2 J(dx, dy). \qquad (7.103)$$

This term quantifies the total quadratic energy contributed by jumps between points $x$ and $y$. The kernel $J(dx, dy)$ measures the frequency and intensity of such jumps. It generalizes Lévy-type nonlocal operators and yields the generator

$$L^{(j)} f(x) = \int_E (f(y) - f(x)) J(x, dy), \qquad (7.104)$$

under suitable integrability conditions (typically requiring $\int_E (1 \wedge |x - y|^2) J(x, dy) < \infty$).

Thus, we can write

$$\mathcal{E} = \mathcal{E}^{(c)} + \mathcal{E}^{(j)} + \mathcal{E}^{(k)}, \qquad (7.105)$$
and these components are uniquely determined by $\mathcal{E}$.

### 5. Connection with Lévy–Khintchine representation.

In the special case $E = \mathbb{R}^d$ and $\mu$ the Lebesgue measure, the generator $L$ corresponding to the Beurling–Deny decomposition takes the general Lévy–Khintchine form:

$$Lf(x) = \sum_{i,j=1}^{d} \partial_i(a_{ij}(x) \partial_j f(x)) + \int_{\mathbb{R}^d} (f(x+y) - f(x) - \nabla f(x) \cdot y \mathbf{1}_{|y|<1}) n(x, dy) - c(x)f(x), \quad (7.106)$$

where $n(x, dy)$ is a measurable Lévy kernel corresponding to $J(dx, dy)$. This expression unifies diffusion, jump, and killing mechanisms within a single operator framework.

### 6. Probabilistic interpretation.

Under the correspondence between symmetric Dirichlet forms and symmetric Hunt processes, the three terms of the Beurling–Deny decomposition have precise probabilistic meanings:

* The diffusion part corresponds to the continuous martingale component of the process (e.g. Brownian motion);

* The jump part corresponds to the pure jump component of the process, where transitions occur discontinuously according to the kernel $J$;

* The killing part corresponds to an independent exponential killing time with spatial rate $c(x)$, at which the process terminates.

Hence, every symmetric Hunt process with these three mechanisms corresponds exactly to one Dirichlet form of the above type, and vice versa.

### 7. Rigorous structural characterization.

The Beurling–Deny decomposition yields the following classification theorem.

**Theorem 7.4** (Beurling–Deny representation theorem). Let $(\mathcal{E}, \mathcal{D}(\mathcal{E}))$ be a regular symmetric Dirichlet form on $L^2(E, \mu)$. Then there exist uniquely determined measures $\nu_c$, $\nu_j$, and $\nu_k$ on $E \times E$ and $E$, respectively, such that:

$$\mathcal{E}(f,g) = \int_E d\nu_c(\Gamma(f,g)) + \int_{E \times E} (f(x)-f(y))(g(x)-g(y))\nu_j(dx,dy) + \int_E f(x)g(x)\nu_k(dx), \quad (7.107)$$

for all $f,g \in \mathcal{D}(\mathcal{E})$, where $\Gamma(f,g)$ denotes the energy measure associated with the local part.

Moreover, these three components are mutually orthogonal in the sense that they interact only through their respective subspaces of functions (strongly local, jump, or killing).

### 8. Implications for spectral theory and potential analysis.

The decomposition directly influences:

* the structure of the generator $L$, which becomes a sum $L = L^{(c)} + L^{(j)} - c(x)$;
* the spectral representation of $L$, with local and nonlocal components contributing distinct spectral branches;

* the potential kernel $G_\alpha = (\alpha I - L)^{-1}$, whose integral kernel decomposes into diffusion, jump, and killing contributions, crucial for potential-theoretic and probabilistic estimates.

Therefore, the Beurling–Deny decomposition not only classifies all possible symmetric Markovian behaviors but also provides a rigorous analytic tool to study their spectral, probabilistic, and potential-theoretic properties.

## 7.5 Applications to Feynman–Kac Semigroups and Schrödinger Operators

We now present the precise analytic and probabilistic interrelation between Dirichlet forms and Feynman–Kac semigroups associated with Schrödinger-type operators of the form $\mathcal{L} - V$, where $\mathcal{L}$ is the generator of a symmetric Dirichlet form and $V$ is a measurable potential. This connection not only provides a rigorous mathematical framework for the analysis of quantum mechanical semigroups but also links probabilistic and potential-theoretic concepts such as capacity, quasi-continuity, and excessive functions.

### 1. Analytic formulation via perturbation of Dirichlet forms.

Let $(\mathcal{E}, \mathcal{D}(\mathcal{E}))$ be a symmetric, closed, nonnegative Dirichlet form on $L^2(E, \mu)$ with associated self-adjoint operator $\mathcal{L}$ (the generator). Let $V: E \to \mathbb{R}$ be a measurable potential function, bounded from below and locally integrable. The perturbed form (also called the Schrödinger form) is defined by

$$ \mathcal{E}^V(f, g) := \mathcal{E}(f, g) + \int_E V(x)f(x)g(x)d\mu(x), \quad f, g \in \mathcal{D}(\mathcal{E}^V) := \mathcal{D}(\mathcal{E}) \cap L^2(|V|d\mu). \quad (7.108) $$

Since $V$ is bounded below, say $V(x) \ge -C$, the form $\mathcal{E}^V$ is lower semi-bounded:

$$ \mathcal{E}^V(f, f) + (C + 1)\|f\|_2^2 \ge \|f\|_2^2, \quad \forall f \in \mathcal{D}(\mathcal{E}^V), \quad (7.109) $$

ensuring that it is closed and densely defined on $L^2(E, \mu)$. Therefore, by the general representation theorem for closed symmetric forms, there exists a unique self-adjoint operator $H$ on $L^2(E, \mu)$ such that

$$ \mathcal{E}^V(f, g) = \langle H^{1/2}f, H^{1/2}g \rangle_{L^2(E, \mu)}. \quad (7.110) $$

We interpret $H$ as the Schrödinger operator formally written as

$$ H = -(\mathcal{L} - V). \quad (7.111) $$

The operator $H$ is nonnegative and self-adjoint, hence generates a strongly continuous, symmetric contraction semigroup $(e^{-tH})_{t \ge 0}$ on $L^2(E, \mu)$.

### 2. Probabilistic representation via the Feynman–Kac formula.

Let $X_t$ be the Hunt process associated with the Dirichlet form $(\mathcal{E}, \mathcal{D}(\mathcal{E}))$, so that its transition
semigroup $(T_t)_{t \ge 0}$ satisfies

$$T_t f(x) = \mathbb{E}^x[f(X_t)] \qquad (7.112)$$

for all $f \in L^2(E, \mu)$. For a potential $V: E \to \mathbb{R}_+$, define the Feynman-Kac semigroup $(P_t^V)_{t \ge 0}$ by

$$(P_t^V f)(x) := \mathbb{E}^x \left[ e^{-\int_0^t V(X_s) ds} f(X_t) \right], \quad f \in L^2(E, \mu). \qquad (7.113)$$

This family $(P_t^V)$ forms a strongly continuous, self-adjoint, sub-Markov semigroup on $L^2(E, \mu)$. Moreover, the Feynman-Kac formula yields the rigorous identity

$$P_t^V = e^{-tH}, \qquad (7.114)$$

where

$$H = -(\mathcal{L} - V) \qquad (7.115)$$

is the Schrödinger operator defined above. The proof follows by applying the Trotter product formula:

$$e^{-t(\mathcal{L}-V)} = \lim_{n \to \infty} \left( e^{-(t/n)\mathcal{L}} e^{(t/n)V} \right)^n, \qquad (7.116)$$

together with the pathwise representation of $e^{-(t/n)\mathcal{L}}$ as conditional expectation under the process $X_t$. The resulting stochastic exponential $\exp(-\int_0^t V(X_s) ds)$ provides the multiplicative functional implementing the potential perturbation in the Dirichlet form framework.

### 3. Application of potential-theoretic tools.

The entire potential-theoretic structure associated with the Dirichlet form $(\mathcal{E}, \mathcal{D}(\mathcal{E}))$ extends naturally to the Schrödinger form $(\mathcal{E}^V, \mathcal{D}(\mathcal{E}^V))$. In particular:

1. **Capacity and quasi-continuity.** For every Borel set $A \subset E$, the $\mathcal{E}^V$-capacity is defined by

$$\mathrm{Cap}_V(A) = \inf\{\mathcal{E}^V(f, f) + \|f\|_2^2 : f \in \mathcal{D}(\mathcal{E}^V), f \ge 1 \text{ on } A \text{ a.e.}\}. \qquad (7.117)$$

A function $f \in \mathcal{D}(\mathcal{E}^V)$ admits a quasi-continuous representative $\tilde{f}$, continuous outside a set of capacity zero, satisfying $\tilde{f} = f$ $\mu$-a.e..

2. **Excessive functions and resolvents.** The semigroup $(P_t^V)$ defines an associated resolvent $(G_\alpha^V)_{\alpha > 0}$:

$$G_\alpha^V f(x) := \int_0^\infty e^{-\alpha t} (P_t^V f)(x) dt = \int_E G_\alpha^V(x, y) f(y) \mu(dy). \qquad (7.118)$$

A measurable function $u: E \to [0, \infty]$ is $(P_t^V)$-excessive if $P_t^Vu \le u$ for all $t > 0$ and $\lim_{t \downarrow 0} P_t^Vu = u$ $\mu$-a.e. Such functions form the potential-theoretic basis of the Schrödinger operator, corresponding to superharmonic functions in classical analysis.

3. **Fine topology and pathwise interpretation.** The fine topology associated with $(P_t^V)$ is the coarsest topology making all excessive functions finely continuous. This topology describes the most refined structure under which the trajectories $t \mapsto X_t$ are quasi-continuous. Under this topology, the potential $V$ modifies the recurrence and transience properties of $X_t$ by exponentially damping its occupation measure through the multiplicative functional $e^{-\int_0^t V(X_s) ds}$.
## 4. Quantum mechanical interpretation.

In quantum mechanics, the operator $H = -(\mathcal{L} - V)$ is the Hamiltonian corresponding to a particle evolving under the potential $V$. The semigroup $e^{-tH}$ represents imaginary-time evolution, transforming the Schrödinger equation

$$i \frac{\partial \psi}{\partial t} = H \psi \qquad (7.119)$$

into the heat-type equation

$$\frac{\partial u}{\partial t} = -Hu, \quad u(0, x) = f(x). \qquad (7.120)$$

The Feynman-Kac representation provides a probabilistic realization of this evolution:

$$(e^{-tH} f)(x) = \mathbb{E}^x \left[ e^{-\int_0^t V(X_s) ds} f(X_t) \right]. \qquad (7.121)$$

Thus, the Dirichlet-form framework rigorously unifies stochastic analysis, potential theory, and quantum mechanics by establishing a precise correspondence between the analytic semigroup $(e^{-tH})$, the probabilistic semigroup $(P_t^V)$, and the energy form $(\mathcal{E}^V, \mathcal{D}(\mathcal{E}^V))$.

## 5. Boundary behaviour and reflective/absorbing conditions.

If $E$ is a domain in $\mathbb{R}^d$, boundary conditions correspond to restrictions on the domain of the form:

* Dirichlet (absorbing) boundary conditions correspond to the subspace

$$\mathcal{D}_0(\mathcal{E}^V) = \overline{C_c^\infty(E)}^{\|\cdot\|_{\mathcal{E}^V}} \qquad (7.122)$$

* Neumann (reflecting) boundary conditions correspond to extending $\mathcal{E}^V$ to include functions with zero normal derivative at $\partial E$.

In probabilistic terms, the associated Hunt process is either killed upon hitting $\partial E$ (Dirichlet case) or reflected at the boundary (Neumann case). This provides a pathwise interpretation of quantum boundary phenomena via stochastic processes.

## 7.6 Dirichlet Form Perturbations and Feynman-Kac Semigroups

In this section, We give a fully rigorous account of the analytic and probabilistic constructions linking Dirichlet-form perturbations by potentials to Feynman-Kac-type semigroups and Schrödinger operators. We prove the equivalence of the form-sum construction of the Schrödinger operator and the pathwise Feynman-Kac expectation representation, treating both the Dirichlet-form (analytic) route and the stochastic-calculus (probabilistic) route, and we show how the two coincide under the standard hypotheses (bounded-below / form-bounded / Kato-class potentials). Throughout $(E, \mathcal{B}, \mu)$ is a locally compact separable metric space with Radon measure $\mu$ of full support, $(\mathcal{E}, \mathcal{D}(\mathcal{E}))$ is a symmetric Dirichlet form on $L^2(E, \mu)$, and $X = (X_t, \mathbb{P}^x)$ denotes the associated Hunt process with $L^2$-semigroup $T_t = e^{-t\mathcal{L}}$.

**Notation and standing hypotheses.** We write $\mathcal{L}$ for the self-adjoint nonnegative operator associated with $\mathcal{E}$ (so $\mathcal{E}(f, g) = \langle \mathcal{L}^{1/2}f, \mathcal{L}^{1/2}g \rangle$ and $T_t = e^{-t\mathcal{L}}$). Let $V: E \to \mathbb{R}$ be measurable. We consider three progressively more general hypotheses on $V$:
(H1) $V \in L^{\infty}(E, \mu)$ (bounded potential);

(H2) $V$ is form-bounded w.r.t. $\mathcal{E}$ with relative form bound $a < 1$ (KLMN condition): there exist $a \in [0, 1)$ and $b \ge 0$ such that for all $f \in \mathcal{D}(\mathcal{E})$,

$$ \int_E |V| |f|^2 d\mu \le a \mathcal{E}(f, f) + b \|f\|_2^2; \qquad (7.123) $$

(H3) $V$ belongs to a suitable Kato class $\mathcal{K}$ (or form-bounded with arbitrarily small relative bound) permitting the stochastic Feynman-Kac functional to be defined and the form-sum to be closed.

We will first treat (H1), then extend to (H2) via form methods and monotone approximation, and finally indicate how Kato-class assumptions fit into the framework (H3).

**Theorem 7.5** (Analytic construction of the Schrödinger operator). Assume (H2). Define the Schrödinger quadratic form

$$ \mathcal{E}^V(f,g) := \mathcal{E}(f,g) + \int_E V(x)f(x)g(x) \mu(dx), \quad \mathcal{D}(\mathcal{E}^V) := \{f \in \mathcal{D}(\mathcal{E}) : \int_E |V| |f|^2 d\mu < \infty\}. \qquad (7.124) $$

Then $\mathcal{E}^V$ is closed and lower bounded; consequently there exists a unique self-adjoint lower-bounded operator $H$ on $L^2(E, \mu)$ such that $\mathcal{D}(\mathcal{E}^V) = \mathcal{D}(H^{1/2})$ and

$$ \mathcal{E}^V(f,g) = \langle H^{1/2}f, H^{1/2}g \rangle_{L^2}, \quad f,g \in \mathcal{D}(\mathcal{E}^V). \qquad (7.125) $$

Moreover $H$ is the form-sum $H = \mathcal{L} +_{\text{form}} V$ and $-H$ generates a strongly continuous semigroup $e^{-tH}$ on $L^2(E, \mu)$.

*Proof.* Since $V$ is form-bounded with relative bound $a < 1$, the perturbation term $f \mapsto \int_V |f|^2$ is $\mathcal{E}$-bounded with relative bound $a$. Hence $\mathcal{E}^V$ is lower semi-bounded and closed on the domain $\mathcal{D}(\mathcal{E}^V) = \mathcal{D}(\mathcal{E}) \cap L^2(|V|d\mu)$; this is the standard KLMN theorem (Kato-Lions-Milgram-Nelson). The first representation theorem (cf. Kato, Reed-Simon) produces a unique self-adjoint operator $H$ with $\mathcal{D}(H^{1/2}) = \mathcal{D}(\mathcal{E}^V)$ and the stated quadratic representation. The semigroup existence and strong continuity of $e^{-tH}$ follow from Hille-Yosida and spectral calculus since $H$ is lower bounded and self-adjoint. Uniqueness as a form-sum is immediate from the representation theorem. $\square$

**Remark.** Under (H1) the form $\mathcal{E}^V$ is simply a bounded perturbation of $\mathcal{E}$, in which case $\mathcal{D}(\mathcal{E}^V) = \mathcal{D}(\mathcal{E})$ and $H = \mathcal{L} + V$ on the natural domain.

We now define the Feynman-Kac semigroup and prove it is well defined under these hypotheses.

**Definition 7.6** (Feynman-Kac semigroup). For measurable $f \ge 0$ and $x \in E$, define

$$ (P_t^V f)(x) := \mathbb{E}^x \left[ e^{-\int_0^t V(X_s) ds} f(X_t) \right], \qquad (7.126) $$

whenever the right-hand side is finite. For bounded-below $V$ the multiplicative functional is well defined and $P_t^V$ maps bounded measurable functions into bounded measurable functions.
We first show that for bounded potentials the semigroup $P_t^V$ is the semigroup generated by the operator $H = \mathcal{L} +_{\text{op}} V$.

**Theorem 7.7** (Feynman-Kac for bounded potentials). Assume (H1). Let $H = \mathcal{L} + V$ be the self-adjoint operator on $\mathcal{D}(\mathcal{L})$ given by the bounded perturbation. Then for all bounded measurable $f$ and for $\mu$-a.e. $x \in E$,

$$ (P_t^V f)(x) = (e^{-tH} f)(x). \qquad (7.127) $$

Moreover the identity holds in $L^2(E, \mu)$ and for $f \in C_c^\infty$ (or a core) it holds pointwise for all $x$.

*Proof.* We give two complementary proofs: (A) an analytic semigroup/resolvent argument based on Trotter product, and (B) a stochastic-Itô argument using Itô's formula (when the process admits a diffusion representation) which provides the same identity and intuition.

(A) Analytic (Trotter) proof. Because $V \in L^\infty$, the operator of multiplication by $V$ is bounded and self-adjoint on $L^2$. Thus $H = \mathcal{L} + V$ is self-adjoint on $\mathcal{D}(\mathcal{L})$ and the Trotter product formula applies:

$$ e^{-tH} = \lim_{n \to \infty} \left( e^{-(t/n)\mathcal{L}} e^{-(t/n)V} \right)^n, \qquad (7.128) $$

with convergence in the strong operator topology on $L^2$. For bounded measurable $f$,

$$ (e^{-(t/n)\mathcal{L}} e^{-(t/n)V} f)(x) = \mathbb{E}^x[e^{-(t/n)V(X_0)} f(X_{t/n})] = \mathbb{E}^x[e^{-(t/n)V(x)} f(X_{t/n})]. \qquad (7.129) $$

Iterating and passing to the limit along the product yields the Feynman-Kac expectation,

$$ \lim_{n \to \infty} \left( e^{-(t/n)\mathcal{L}} e^{-(t/n)V} \right)^n f = \mathbb{E}^x \left[ e^{-\int_0^t V(X_s) ds} f(X_t) \right] = P_t^V f(x), \qquad (7.130) $$

and equality of the two semigroups follows by uniqueness of the strong limit (justified by dominated convergence and the semigroup bounds). This gives $e^{-tH}f = P_t^V f$ in $L^2$ and pointwise outside an exceptional set.

(B) Stochastic (Itô) proof for diffusion realizations. Suppose the Hunt process $X$ solves the SDE representation (in local coordinates) driven by Brownian motion and a drift corresponding to $\mathcal{L}$, and suppose $f$ is $C^2$ and bounded. Apply Itô's formula to the process

$$ M_t := \exp \left( - \int_0^t V(X_s) ds \right) f(X_t). \qquad (7.131) $$

Itô's formula (and product rule) gives

$$ dM_t = \exp \left( - \int_0^t V(X_s) ds \right) ((\mathcal{L} - V)f)(X_t) dt + (\text{martingale term}). \qquad (7.132) $$

Taking expectations and using that the martingale term has zero expectation, we obtain

$$ \frac{d}{dt} \mathbb{E}^x[M_t] = \mathbb{E}^x \left[ \exp \left( - \int_0^t V(X_s) ds \right) (\mathcal{L} - V) f(X_t) \right]. \qquad (7.133) $$

Thus

$$ u(t, x) := \mathbb{E}^x[M_t] = (P_t^V f)(x) \qquad (7.134) $$
satisfies the backward evolution equation

$$ \partial_t u = (\mathcal{L} - V)u, \quad u(0, \cdot) = f. \qquad (7.135) $$

The semigroup generated by $\mathcal{L} - V$ coincides with $e^{-tH}$, and by uniqueness of solutions of the evolution equation in the relevant function class we conclude $P_t^V f = e^{-tH} f$. (This stochastic derivation gives the same identity under the diffusion representation; for general Hunt processes one may replace Itô by Dynkin's formula and potential-theoretic arguments.)

Combining (A) and (B) gives the desired equality. □

The bounded-potential case being settled, we now extend to form-bounded (KLMN) potentials by approximation.

**Theorem 7.8** (Feynman-Kac via form approximation). Assume (H2). Let $H$ be the self-adjoint operator of Theorem 7.5 and let $P_t^V$ be the Feynman-Kac family defined (a priori) on bounded measurable $f$ by the path expectation (the expectation is finite for a suitable class of starting points under (H2)). Then for every $f \in L^2(E, \mu)$,

$$ e^{-tH} f = P_t^V f \quad \text{in } L^2(E, \mu), \qquad (7.136) $$

and the equality holds pointwise q.e. on $E$ for quasi-continuous representatives.

*Proof.* We use the truncation/approximation method. Define $V_n := V \wedge n$ (and additionally $V_n := \max\{-n, V \wedge n\}$ if $V$ is unbounded below) so that each $V_n \in L^\infty$ and $V_n \to V$ monotonically as $n \to \infty$. For each $n$, let $H_n := \mathcal{L} + V_n$ (bounded perturbation) and denote $P_t^{V_n}$ the corresponding Feynman-Kac operators. By Theorem 7.7,

$$ e^{-tH_n} f = P_t^{V_n} f, \quad f \in L^2. \qquad (7.137) $$

By the form monotone convergence (or Kato monotone convergence of forms) $H_n \to H$ in the strong-resolvent sense; therefore $e^{-tH_n} \to e^{-tH}$ strongly on $L^2$. On the probabilistic side, by monotone convergence of the exponential weights (or dominated convergence using Khasminskii-type bounds justified by the form bound), we have $P_t^{V_n} f \to P_t^V f$ q.e. and in $L^2$. Taking limits in the identity $e^{-tH_n} f = P_t^{V_n} f$ yields $e^{-tH} f = P_t^V f$ in $L^2$. The q.e. pointwise statement follows from choosing quasi-continuous representatives for the left-hand side and using the pathwise convergence of the multiplicative functionals. □

**Stochastic exponential / multiplicative-functional derivation (alternate probabilistic proof).** We now provide the stochastic-calculus derivation that works in the diffusion setting and is the basis for Girsanov-type transformations which are often used to introduce drifts. Let $X$ be given as the solution of the SDE corresponding to $\mathcal{L}$ on a fixed filtered probability space supporting an $m$-dimensional Brownian motion $B$, and assume coefficients give a unique strong solution. Let $f \in C_c^2(E)$ and consider the process

$$ M_t := e^{-\int_0^t V(X_s) ds} f(X_t). \qquad (7.138) $$
Applying Itô's formula (justified by smoothness and local boundedness) we get

$$ 
\begin{align} 
dM_t &= e^{-\int_0^t V(X_s) ds} [df(X_t) - V(X_t)f(X_t) dt] \nonumber \\
&= e^{-\int_0^t V(X_s) ds} [(\mathcal{L}f)(X_t) dt + (\text{local martingale term}) - V(X_t)f(X_t) dt] \tag{7.139} \\
&= e^{-\int_0^t V(X_s) ds} [(\mathcal{L} - V)f)(X_t) dt + d\widetilde{M}_t], \nonumber
\end{align} 
$$

where $\widetilde{M}_t$ is a local martingale. Taking expectations and using the optional stopping / bounded convergence argument for truncated stopping times gives

$$ \frac{d}{dt} \mathbb{E}^x[M_t] = \mathbb{E}^x[e^{-\int_0^t V(X_s) ds} (\mathcal{L} - V)f(X_t)]. \quad (7.140) $$

Thus $u(t, x) := \mathbb{E}^x[M_t]$ satisfies $\partial_t u = (\mathcal{L} - V)u$ with $u(0, \cdot) = f$. Standard uniqueness of the Cauchy problem in $L^2$ or in appropriate function classes implies that $u(t, \cdot) = e^{-tH}f$ (where $H$ is the self-adjoint operator associated with $\mathcal{E}^V$). This derivation complements the analytic approximation approach and provides the linkage to stochastic calculus and martingale theory.

**Spectral consequences and ground states.** From the operator equality $e^{-tH} = P_t^V$ we deduce spectral statements: the spectral measure of $H$ controls the long-time behaviour of $P_t^V$, ground state existence corresponds to $L^2$-eigenfunctions $\phi$ with $H\phi = \lambda_0\phi$, and the ground-state transformation (Doob $h$-transform) can be expressed probabilistically via the reweighted path measure

$$ \left. \frac{d\mathbb{P}^{x,\phi}}{d\mathbb{P}^x} \right|_{\mathcal{F}_t} = e^{\lambda_0 t} \frac{\phi(X_t)}{\phi(x)} e^{-\int_0^t V(X_s) ds}, \quad (7.141) $$

provided $\phi > 0$ a.e. and lies in the domain. Such transforms connect spectral gaps to ergodic properties of the transformed process.

**Extension to Kato-class potentials and further remarks.** If $V$ belongs to a Kato class $\mathcal{K}$ appropriate to the underlying process (for instance the classical Kato class in $\mathbb{R}^d$ relative to Brownian motion), then the multiplicative functional $e^{-\int_0^t V(X_s) ds}$ is well defined and uniformly integrable on compacts; the form-sum $\mathcal{E}^V$ is closable and gives a self-adjoint $H$. The preceding approximation arguments (by bounded or truncated potentials) extend to this setting and produce the identity $e^{-tH} = P_t^V$ in $L^2$ and q.e. pointwise. Technicalities require verifying Khasminskii-type exponential moment bounds and the smallness of the negative part $V^-$ in form-sense.

**Concluding theorem (full equivalence).**

**Theorem 7.9 (Full equivalence).** Assume $V$ satisfies (H2) or (H3). Let $H$ be the self-adjoint operator given by the form-sum $\mathcal{E}^V = \mathcal{E} + \int V| \cdot|^2$, and let $P_t^V$ denote the Feynman-Kac expectation semigroup. Then

$$ e^{-tH} = P_t^V \quad \text{as bounded operators on } L^2(E, \mu), \quad (7.142) $$

and for every $f \in L^2(E, \mu)$ the equality holds in $L^2$ and with quasi-continuous representatives holds q.e. on $E$.
*Proof.* Combine Theorems 7.5, 7.7 and 7.8 and the stochastic derivation above. The truncation/approximation argument shows strong $L^2$-convergence of $e^{-tH_n}$ to $e^{-tH}$ while monotone-dominated convergence for the stochastic expectations yields convergence $P_t^{V_n} \to P_t^V$ q.e. and in $L^2$; equality at the approximating level passes to the limit yielding the assertion. □

**References and further reading.** The analytic form-sum and KLMN theorem are classical (Kato, Reed-Simon). The probabilistic derivation via multiplicative functionals and the Feynman-Kac representation (including Kato-class extensions) are treated in detail in Fukushima-Oshima-Takeda, Simon's monograph on Schrödinger semigroups, and in classical probability texts on diffusion processes and potential theory. The combination of both analytic and probabilistic approaches presented here gives a robust and flexible toolkit for spectral, PDE, and pathwise analyses of Schrödinger-type operators.

**Conclusion.** The Dirichlet-form framework gives a complete and rigorous analytic-probabilistic formulation of Schrödinger operators and their Feynman-Kac semigroups. It integrates the theory of self-adjoint operators, stochastic processes, and potential theory into a single unified structure, enabling both analytic and probabilistic investigation of spectral properties, ground states, and boundary effects in quantum systems.

## 7.7 Conclusion

In this chapter we have developed the analytic foundation of Dirichlet forms and potential theory and shown how they underpin the *Feynman-Kac* representations and Schrödinger-type semigroups of earlier chapters. The unifying thread is that Markov processes, potential theory, Dirichlet forms, and semigroup theory form a single integrated structure: stochastic paths correspond to analytic resolvents and generators, and the fine topological structure of sample paths is encoded in analytic capacities and domain regularity of forms.

**References:** Fukushima-Oshima-Takeda [24], Ma-Röckner [25], Chen-Fukushima (2012) [27] and Chen-Fukushima (2012) [28] for general Dirichlet forms; Simon (1982) [40], Davies (1980) [26] for semigroup methods; Stroock-Varadhan (2007) [16], Bass (2004) [29] for process-potential theory interface.
# Path Integral Representation

> "The theory of quantum mechanics describes nature as a sum over histories, each history contributing to the whole by its amplitude."

> — RICHARD P. FEYNMAN

The path integral formalism provides a deep unification between analysis, probability, and quantum mechanics, offering a measure-theoretic construction of semigroups and operators via expectations over stochastic paths. It reveals how analytic objects such as Schrödinger semigroups and heat kernels admit probabilistic representations through diffusion processes, and conversely, how probabilistic quantities may be interpreted as functional integrals on infinite-dimensional path spaces. This chapter rigorously formulates these connections, emphasizing the correspondence between operator-theoretic and stochastic constructions.

## 8.1 Wiener Measure and Configuration Space of Paths

Let $E = \mathbb{R}^d$ equipped with its Borel $\sigma$-algebra $\mathcal{B}(E)$. Set $\Omega := C([0, \infty); E)$ and denote by $X_t: \Omega \to E$ the canonical coordinate maps $X_t(\omega) = \omega(t)$. Equip $\Omega$ with the topology of uniform convergence on compact time intervals and with the corresponding Borel $\sigma$-algebra $\mathcal{F} = \mathcal{B}(\Omega)$. For every finite collection of times $0 \le t_1 < \dots < t_n$ and Borel sets $A_1, \dots, A_n \subset E$ the canonical cylinder sets

$$ \{\omega \in \Omega : X_{t_1} \in A_1, \dots, X_{t_n} \in A_n\} \qquad (8.1) $$

generate $\mathcal{F}$; measurable functions on $(\Omega, \mathcal{F})$ are determined by their values on such cylinders.

The Wiener measure $\mathbb{P}^x$ is the unique probability measure on $(\Omega, \mathcal{F})$ whose finite-dimensional distributions are Gaussian with mean $x$ at time 0 and covariance given by the minimum kernel. Equivalently, for every $n$, every choice $0 = t_0 < t_1 < \dots < t_n$ and every bounded Borel $\varphi: E^n \to \mathbb{R}$,

$$ \int_{\Omega} \varphi(X_{t_1}, \dots, X_{t_n}) d\mathbb{P}^x = \int_{E^n} \varphi(x_1, \dots, x_n) \prod_{k=1}^{n} p_{t_k-t_{k-1}}(x_{k-1}, x_k) dx_1 \dots dx_n, \qquad (8.2) $$
where $x_0 := x$ and the transition densities $p_t(x, y)$ are given by the Gaussian heat kernel

$$p_t(x, y) = \frac{1}{(4\pi t)^{d/2}} \exp\left(-\frac{|y-x|^2}{4t}\right), \quad t > 0, x, y \in \mathbb{R}^d. \qquad (8.3)$$

The existence and uniqueness of $\mathbb{P}^x$ with these finite-dimensional marginals is exactly the Kolmogorov extension (or Kolmogorov-Consistence) theorem applied to the consistent family of probability measures on $\mathbb{R}^{dn}$ defined by the right-hand side; consistency is ensured by the Chapman-Kolmogorov identities

$$\int_E p_s(x, z) p_t(z, y) dz = p_{s+t}(x, y), \quad s, t > 0, \qquad (8.4)$$

which follow from the semigroup property of the Gaussian kernel.

The coordinate process $(X_t)_{t \ge 0}$ under $\mathbb{P}^x$ is a Gaussian process with mean $x$ and covariance matrix

$$\mathbb{E}^x[(X_s^{(i)} - x^{(i)})(X_t^{(j)} - x^{(j)})] = \delta_{ij} (s \wedge t), \quad 1 \le i, j \le d, \qquad (8.5)$$

and by the Kolmogorov-Chentsov criterion the finite-dimensional Gaussian moment estimates imply that the canonical process admits a version with continuous sample paths; since the construction is carried out on $\Omega = C([0, \infty); E)$ the chosen version is continuous $\mathbb{P}^x$-almost surely. In fact one has the stronger Hölder-regularity statement: for every $\gamma \in (0, 1/2)$,

$$\mathbb{P}^x \left( \sup_{0 \le s < t \le T} \frac{|X_t - X_s|}{|t - s|^{\gamma}} < \infty \right) = 1 \quad \text{for every } T < \infty, \qquad (8.6)$$

so sample paths are almost surely Hölder continuous of any order $< \frac{1}{2}$.

Define the natural (raw) filtration

$$\mathcal{F}_t^0 := \sigma(X_s : 0 \le s \le t) \qquad (8.7)$$

and its right-continuous, $\mathbb{P}^x$-completed augmentation

$$\mathcal{F}_t := \bigcap_{u>t} (\mathcal{F}_u^0 \vee \mathcal{N}^{\mathbb{P}^x}) \qquad (8.8)$$

where $\mathcal{N}^{\mathbb{P}^x}$ denotes the $\mathbb{P}^x$-null sets. The filtered probability space $(\Omega, \mathcal{F}, (\mathcal{F}_t), \mathbb{P}^x)$ satisfies the usual conditions (complete and right-continuous) and the coordinate process is adapted and has continuous trajectories. The family $(\mathbb{P}^x)_{x \in E}$ is Markovian in the strong sense: for each bounded measurable $f : E \to \mathbb{R}$ and $s, t \ge 0$,

$$\mathbb{E}^x[f(X_{s+t}) | \mathcal{F}_s] = (T_t f)(X_s) \quad \text{a.s. under } \mathbb{P}^x, \qquad (8.9)$$

where the transition operator $T_t$ acts by convolution with the heat kernel,

$$(T_t f)(x) = \int_E p_t(x, y) f(y) dy. \qquad (8.10)$$
Consequently $(T_t)_{t \ge 0}$ is a strongly continuous contraction semigroup on $L^p(\mathbb{R}^d)$ for each $1 \le p < \infty$ and a Feller semigroup on $C_0(\mathbb{R}^d)$; in particular for bounded continuous $f$, $x \mapsto T_t f(x)$ is continuous and $\lim_{t \downarrow 0} T_t f = f$ uniformly on compacts.

From the Gaussian structure one derives explicit moment and exponential estimates used in potential theory and PDE connections: for every multi-index $\alpha \in \mathbb{N}^d$ and $t > 0$, the heat kernel is $C^\infty$ in $(x, y)$ and satisfies the Gaussian bounds

$$|\partial_x^\alpha p_t(x, y)| \le C_\alpha t^{-(|\alpha|+d)/2} \exp\left(-\frac{|x-y|^2}{8t}\right), \quad (8.11)$$

which imply smoothing (instantaneous regularization) properties of $T_t$. In particular $T_t$ maps $L^2$ into $C^\infty$ for $t > 0$, and the generator of the semigroup is the self-adjoint Laplacian $\frac{1}{2}\Delta$ on its maximal $L^2$-domain.

Two further structural facts are central and frequently used. First, the Cameron–Martin space $H$ associated with Wiener measure is the space of absolutely continuous paths $h: [0, \infty) \to \mathbb{R}^d$ with $h(0) = 0$ and $\dot{h} \in L_{\text{loc}}^2([0, \infty); \mathbb{R}^d)$; the Cameron–Martin theorem states that for $h \in H$ the shifted measure $\mathbb{P}^x \circ (\omega \mapsto \omega + h)^{-1}$ is equivalent to $\mathbb{P}^x$ and the Radon–Nikodym derivative is given by the stochastic exponential

$$\frac{d(\mathbb{P}^x \circ (\cdot + h)^{-1})}{d\mathbb{P}^x}(\omega) = \exp\left(\int_0^\infty \dot{h}_s \cdot dX_s(\omega) - \frac{1}{2} \int_0^\infty |\dot{h}_s|^2 ds\right), \quad (8.12)$$

interpreted via Itô integrals on finite horizons and limiting arguments; this formula is the analytic backbone for Girsanov transformations in the diffusion setting. Second, time-shifts and scaling relationships are explicit: for any $c > 0$ the rescaled process $(c^{-1/2}X_{ct})_{t \ge 0}$ under $\mathbb{P}^{cx}$ has the same law as $(X_t)_{t \ge 0}$ under $\mathbb{P}^x$, reflecting Brownian scaling and enabling precise short-time asymptotics of the semigroup.

Finally, the Wiener measure and the canonical process provide the prototypical example of a Hunt process: $(X_t)$ is strong Markov and has continuous trajectories, possesses transition densities satisfying Chapman–Kolmogorov, admits the Feller property, and its potential theory (Green kernels, capacity, excessive functions) coincides with the classical analytic potential theory for the Laplacian. These properties justify using $\mathbb{P}^x$ as the reference measure in Feynman–Kac constructions, in perturbative path-integral formulas, and in the probabilistic representation of solutions to parabolic and elliptic PDEs.

## 8.2 Feynman–Kac Functional and Schrödinger Semigroup Representation

We now present a rigorous formulation of the Feynman–Kac representation theorem, which connects the analytic theory of Schrödinger semigroups with the probabilistic theory of Brownian motion and its associated exponential additive functionals.

Let $E = \mathbb{R}^d$ equipped with the Borel $\sigma$-algebra $\mathcal{B}(E)$, and denote by $(B_t)_{t \ge 0}$ the standard
d-dimensional Brownian motion on the canonical path space

$$ \Omega = C([0, \infty); \mathbb{R}^d), \qquad (8.13) $$

with coordinate process $X_t(\omega) = \omega(t)$, natural filtration $\mathcal{F}_t = \sigma(X_s: 0 \le s \le t)$, and Wiener measure $\mathbb{P}^x$ corresponding to Brownian motion starting at $x \in \mathbb{R}^d$. The expectation with respect to $\mathbb{P}^x$ will be denoted by $\mathbb{E}^x[\cdot]$.

Let $V: \mathbb{R}^d \to \mathbb{R}$ be a measurable potential function which is bounded from below, i.e., there exists a constant $c \in \mathbb{R}$ such that $V(x) \ge c$ for all $x \in \mathbb{R}^d$. We consider the Schrödinger-type differential operator

$$ H = -\frac{1}{2}\Delta + V, \qquad (8.14) $$

initially defined on the core $C_c^\infty(\mathbb{R}^d)$. Since $V$ is bounded from below, the quadratic form

$$ \mathcal{E}^V(f, f) = \frac{1}{2} \int_{\mathbb{R}^d} |\nabla f(x)|^2 dx + \int_{\mathbb{R}^d} V(x)|f(x)|^2 dx, \quad f \in C_c^\infty(\mathbb{R}^d), \qquad (8.15) $$

is lower bounded and closable in $L^2(\mathbb{R}^d)$. Its closure defines a densely defined, closed, symmetric, semi-bounded quadratic form $(\mathcal{E}^V, \mathcal{D}(\mathcal{E}^V))$, which, by the Friedrichs extension theorem, corresponds to a unique self-adjoint operator $H$ on $L^2(\mathbb{R}^d)$. We continue to denote this operator by $H = -\frac{1}{2}\Delta + V$, with domain

$$ \mathcal{D}(H) = \{ f \in \mathcal{D}(\mathcal{E}^V) \mid \exists g \in L^2(\mathbb{R}^d) \text{ s.t. } \mathcal{E}^V(f, g) = \langle g, h \rangle_{L^2(\mathbb{R}^d)} \quad \forall h \in \mathcal{D}(\mathcal{E}^V) \}. \qquad (8.16) $$

By the general spectral theorem for self-adjoint operators, $H$ generates a strongly continuous, symmetric contraction semigroup $(e^{-tH})_{t \ge 0}$ on $L^2(\mathbb{R}^d)$, which is positivity-preserving and sub-Markovian. Explicitly, for $f \in L^2(\mathbb{R}^d)$,

$$ e^{-tH}f = \int_{0}^{\infty} e^{-t\lambda} dE_{\lambda}f, \qquad (8.17) $$

where $E_\lambda$ denotes the spectral resolution of $H$.

On the probabilistic side, define the *Feynman-Kac functional* associated with the potential $V$ by

$$ A_t^V(\omega) := \int_0^t V(X_s(\omega)) ds, \qquad (8.18) $$

for each trajectory $\omega \in \Omega$. This is a continuous, adapted, additive functional of the Brownian motion. The exponential weight

$$ \exp(-A_t^V(\omega)) = \exp\left(-\int_0^t V(X_s(\omega)) ds\right) \qquad (8.19) $$

acts as a multiplicative functional on the Wiener space, modifying the original Brownian measure to encode the influence of the potential energy accumulated along the path. Specifically, the random variable

$$ M_t^V(\omega) := \exp\left(-\int_0^t V(X_s(\omega)) ds\right) \qquad (8.20) $$
serves as a Radon–Nikodym derivative defining a new weighted measure $Q_t^x$ on $\mathcal{F}_t$ by

$$ \frac{dQ_t^x}{d\mathbb{P}^x}(\omega) = M_t^V(\omega). \qquad (8.21) $$

This measure reweights the paths of Brownian motion according to their accumulated potential energy, yielding a precise probabilistic analogue of the Feynman–Kac path integral measure.

We may now state the Feynman–Kac representation in its rigorous form: for every $f \in L^2(\mathbb{R}^d)$ and almost every $x \in \mathbb{R}^d$,

$$ (e^{-tH}f)(x) = \mathbb{E}^x \left[ \exp \left( -\int_0^t V(X_s) ds \right) f(X_t) \right]. \qquad (8.22) $$

Equivalently, in semigroup notation,

$$ P_t^V f(x) := \mathbb{E}^x \left[ e^{-\int_0^t V(X_s) ds} f(X_t) \right] = (e^{-tH}f)(x), \qquad (8.23) $$

where $(P_t^V)_{t \ge 0}$ is referred to as the *Feynman–Kac semigroup*. This semigroup is strongly continuous on $L^2(\mathbb{R}^d)$, self-adjoint, and preserves positivity.

The equality above provides a rigorous probabilistic realization of the quantum-mechanical path integral:

$$ \langle x | e^{-tH} | y \rangle = \int_{X(t)=y}^{X(0)=x} \exp \left( -\int_{0}^{t} \left[ \frac{1}{2} |\dot{X}_s|^2 + V(X_s) \right] ds \right) \mathcal{D}X, \qquad (8.24) $$

where $\mathcal{D}X$ denotes the formal (nonexistent) Lebesgue measure on path space, which in the rigorous theory is replaced by the Wiener measure $\mathbb{P}^x$. Thus, the Feynman–Kac formula establishes the exact analytic–probabilistic equivalence

$$ (e^{-tH}f)(x) = \mathbb{E}^x \left[ e^{-\int_0^t V(X_s) ds} f(X_t) \right], \quad H = -\frac{1}{2}\Delta + V, \qquad (8.25) $$

thereby connecting Schrödinger operators, semigroup theory, and stochastic analysis through the framework of Dirichlet forms and probabilistic path integration.

## 8.3 Analytic Justification and Domain Correspondence

We now provide an analytic framework connecting the potential-perturbed Dirichlet form with the corresponding Schrödinger operator, thereby establishing a precise equivalence between the analytic and probabilistic formulations of the Feynman–Kac semigroup.

Let us consider the classical Dirichlet form on $L^2(\mathbb{R}^d, dx)$ associated with the Laplacian operator:

$$ \mathcal{E}(f,g) = \frac{1}{2} \int_{\mathbb{R}^d} \nabla f(x) \cdot \nabla g(x) dx, \quad f,g \in H^1(\mathbb{R}^d), \qquad (8.26) $$
where $H^1(\mathbb{R}^d)$ denotes the standard Sobolev space of square-integrable functions with square-integrable weak derivatives. The form $(\mathcal{E}, H^1(\mathbb{R}^d))$ is symmetric, closed, and Markovian, and hence constitutes a strongly local, regular Dirichlet form whose self-adjoint generator on $L^2(\mathbb{R}^d)$ is the operator

$$ \mathcal{L} = \frac{1}{2} \Delta, \qquad (8.27) $$

with domain $\mathcal{D}(\mathcal{L}) = H^2(\mathbb{R}^d)$. The associated symmetric Markov process is the standard $d$-dimensional Brownian motion $(B_t)_{t \ge 0}$ on $\mathbb{R}^d$, defined on its canonical path space.

Let $V : \mathbb{R}^d \to \mathbb{R}$ be a measurable potential function, and consider the perturbed bilinear form

$$ \mathcal{E}^V(f,g) = \frac{1}{2} \int_{\mathbb{R}^d} \nabla f(x) \cdot \nabla g(x) dx + \int_{\mathbb{R}^d} V(x)f(x)g(x) dx, \quad f,g \in C_c^\infty(\mathbb{R}^d), \qquad (8.28) $$

which is initially defined on the dense subspace $C_c^\infty(\mathbb{R}^d) \subset L^2(\mathbb{R}^d)$.

To ensure that the potential term defines a relatively form-bounded perturbation of $\mathcal{E}$, we impose the *Kato-class condition* on $V$, given by

$$ \limsup_{t \downarrow 0} \sup_{x \in \mathbb{R}^d} \mathbb{E}^x \left[ \int_0^t |V(B_s)| ds \right] = 0. \qquad (8.29) $$

This condition guarantees that $V$ is locally integrable with respect to the transition probabilities of Brownian motion and ensures, in particular, that the additive functional

$$ A_t^V := \int_0^t V(B_s) ds \qquad (8.30) $$

is well-defined for almost all trajectories and that its exponential moments satisfy

$$ \mathbb{E}^x \left[ e^{-\int_0^t V(B_s) ds} \right] < \infty, \quad \forall t > 0. \qquad (8.31) $$

Under the Kato-class assumption, the form $\mathcal{E}^V$ admits a unique closed extension on $L^2(\mathbb{R}^d)$, which we again denote by $(\mathcal{E}^V, \mathcal{D}(\mathcal{E}^V))$. This form is densely defined, lower semi-bounded, and symmetric. By the general representation theorem for closed semi-bounded quadratic forms, there exists a unique self-adjoint operator $H_V$ on $L^2(\mathbb{R}^d)$ such that

$$ \mathcal{E}^V(f,g) = \langle H_V^{1/2} f, H_V^{1/2} g \rangle_{L^2(\mathbb{R}^d)}, \quad f,g \in \mathcal{D}(\mathcal{E}^V). \qquad (8.32) $$

This operator $H_V$ is precisely the Friedrichs extension of the formal differential operator

$$ H_V = -\frac{1}{2}\Delta + V, \qquad (8.33) $$

initially defined on $C_c^\infty(\mathbb{R}^d)$. Its domain $\mathcal{D}(H_V)$ satisfies

$$ \mathcal{D}(H_V) = \{ f \in \mathcal{D}(\mathcal{E}^V) \mid \exists g \in L^2(\mathbb{R}^d) \text{ such that } \mathcal{E}^V(f,h) = \langle g,h \rangle_{L^2(\mathbb{R}^d)} \forall h \in \mathcal{D}(\mathcal{E}^V) \}, \qquad (8.34) $$
and $H_V f = g$.

The self-adjointness of $H_V$ implies the existence of a strongly continuous contraction semigroup $(e^{-tH_V})_{t \ge 0}$ on $L^2(\mathbb{R}^d)$, which is positivity-preserving and sub-Markovian. Analytically, this semigroup can be represented through the spectral theorem as

$$e^{-tH_V} f = \int_0^\infty e^{-t\lambda} dE_\lambda f, \qquad (8.35)$$

where $E_\lambda$ denotes the spectral family associated with $H_V$.

On the other hand, the Feynman–Kac semigroup is defined probabilistically by

$$P_t^V f(x) := \mathbb{E}^x \left[ \exp \left( - \int_0^t V(B_s) \, ds \right) f(B_t) \right], \quad f \in L^2(\mathbb{R}^d). \qquad (8.36)$$

Standard results in potential theory and Dirichlet form theory (see, e.g., Fukushima–Oshima–Takeda, *Dirichlet Forms and Symmetric Markov Processes*) assert that $(P_t^V)_{t \ge 0}$ forms a strongly continuous symmetric contraction semigroup on $L^2(\mathbb{R}^d)$, and moreover,

$$P_t^V = e^{-tH_V}, \quad \forall t \ge 0. \qquad (8.37)$$

Thus, the analytic semigroup generated by $H_V$ coincides exactly with the probabilistic Feynman–Kac semigroup.

Consequently, the domains, resolvents, and spectral measures of $H_V$ are completely determined by the corresponding probabilistic structure of the weighted Brownian motion, and vice versa. The identification

$$\mathcal{E}^V(f, f) = \frac{1}{2} \int_{\mathbb{R}^d} |\nabla f|^2 dx + \int_{\mathbb{R}^d} V f^2 dx \iff (e^{-tH_V} f)(x) = \mathbb{E}^x \left[ e^{-\int_0^t V(B_s) ds} f(B_t) \right] \qquad (8.38)$$

rigorously establishes the complete analytic–probabilistic correspondence between Schrödinger operators and Feynman–Kac-type path integral representations.

## 8.4 Path Integral as Infinite-Dimensional Limit

The construction of the Feynman–Kac path integral representation proceeds through a limiting procedure that replaces the heuristic “integration over paths” by a well-defined limit of finite-dimensional Gaussian integrals.

Let $E = \mathbb{R}^d$ and let $H = -\frac{1}{2}\Delta + V$ be the Schrödinger operator acting on $L^2(\mathbb{R}^d)$ with suitable domain (for instance, $V \in L_{\text{loc}}^1$ and $V \ge -C$ ensuring semi-boundedness). Denote by $(T_t)_{t \ge 0} = (e^{-tH})_{t \ge 0}$ its strongly continuous contraction semigroup. By the Trotter product formula one has, for any $f \in L^2(\mathbb{R}^d)$,

$$e^{-tH} f = \lim_{n \to \infty} \left( e^{-\frac{t}{n}V} e^{\frac{t}{2n}\Delta} \right)^n f, \qquad (8.39)$$
where the limit is in the strong operator topology of $L^2(\mathbb{R}^d)$. The semigroup $e^{\frac{t}{2n}\Delta}$ acts by convolution with the Gaussian kernel $p_{t/(2n)}(x, y)$, while the multiplication operator $e^{-\frac{t}{n}V}$ acts pointwise by $e^{-(t/n)V(x)}$. Expanding this product iteratively and writing out the kernel explicitly yields the finite-dimensional approximation formula

$$ (e^{-tH}f)(x) = \lim_{n \to \infty} \int_{(\mathbb{R}^d)^n} K_n(x, x_1, \dots, x_n) f(x_n) dx_1 \cdots dx_n, \quad (8.40) $$

where

$$ K_n(x_0, \dots, x_n) := \prod_{i=0}^{n-1} \left[ p_{t/n}(x_i, x_{i+1}) e^{-(t/n)V(x_i)} \right], \quad x_0 := x. \quad (8.41) $$

Substituting the explicit Gaussian form

$$ p_{t/n}(x_i, x_{i+1}) = \frac{1}{(2\pi t/n)^{d/2}} \exp\left(-\frac{|x_{i+1} - x_i|^2}{2(t/n)}\right), \quad (8.42) $$

one obtains

$$ (e^{-tH}f)(x) = \lim_{n \to \infty} c_n \int_{(\mathbb{R}^d)^n} \exp \left[ - \sum_{i=0}^{n-1} \left( \frac{|x_{i+1} - x_i|^2}{2(t/n)} + \frac{t}{n} V(x_i) \right) \right] f(x_n) dx_1 \cdots dx_n, \quad (8.43) $$

where $c_n = (2\pi t/n)^{-nd/2}$. This expression already exhibits the discrete path-integral structure: each factor describes the contribution of a short Brownian bridge increment $\Delta X_i = x_{i+1} - x_i$ with Gaussian weight $\exp(-|\Delta X_i|^2/(2\Delta t))$, and the potential term contributes the exponential damping $\exp(-V(x_i)\Delta t)$.

To justify the passage $n \to \infty$, we view the family of Gaussian densities

$$ \mu_{t,n}(dx_1, \dots, dx_n) := c_n \exp \left( - \sum_{i=0}^{n-1} \frac{|x_{i+1} - x_i|^2}{2(t/n)} \right) dx_1 \cdots dx_n \quad (8.44) $$

as the joint distribution of the discrete-time Gaussian process obtained by sampling a Brownian motion at the grid points $t_i = i(t/n)$. In the limit $n \to \infty$ the family $(\mu_{t,n})$ is projectively consistent, and by Kolmogorov's extension theorem it defines the Wiener measure $\mathbb{P}^x$ on the path space $C([0, t]; \mathbb{R}^d)$. Under this identification, for any bounded measurable $F$ depending only on finitely many coordinates,

$$ \int F(x_1, \dots, x_n) \mu_{t,n}(dx_1, \dots, dx_n) = \mathbb{E}^x[F(X_{t_1}, \dots, X_{t_n})], \quad (8.45) $$

where $X_s$ is Brownian motion started at $x$. Hence, in the limit, the discrete sum $\sum_{i=0}^{n-1} |x_{i+1} - x_i|^2/(2\Delta t)$ converges in distribution to the quadratic variation $\frac{1}{2} \int_0^t |\dot{X}_s|^2 ds$ that appears in the formal Wiener measure exponent, while the Riemann sum $\sum_i V(x_i)\Delta t$ converges to $\int_0^t V(X_s) ds$.

Passing to the limit $n \to \infty$ under these identifications therefore yields the Feynman-Kac representation:

$$ (e^{-tH}f)(x) = \mathbb{E}^x \left[ \exp \left( - \int_0^t V(X_s) ds \right) f(X_t) \right], \quad (8.46) $$
which is rigorous for all $f \in L^2(\mathbb{R}^d)$ and measurable potentials $V$ satisfying standard form-boundedness conditions ensuring that

$$ \mathcal{E}^V(u, u) = \frac{1}{2} \int |\nabla u|^2 + \int V u^2 \qquad (8.47) $$

defines a closed, lower-bounded quadratic form. The kernel of the semigroup, $K_t(x, y)$, admits the probabilistic representation

$$ K_t(x, y) = p_t(x, y) \mathbb{E}_t^{x,y} \left[ \exp \left( - \int_0^t V(X_s) ds \right) \right], \qquad (8.48) $$

where $\mathbb{E}_t^{x,y}$ denotes expectation under the Brownian bridge measure from $x$ to $y$ in time $t$. This expression is the rigorous probabilistic counterpart of the heuristic Feynman path integral

$$ \langle x | e^{-tH} | y \rangle = \int_{X(0)=x}^{X(t)=y} \exp \left( - \int_{0}^{t} \left[ \frac{1}{2} |\dot{X}_s|^2 + V(X_s) \right] ds \right) dX, \qquad (8.49) $$

where $dX$ is the formally uniform (nonexistent) “Lebesgue measure” on path space, replaced rigorously by the Wiener measure $\mathbb{P}^x$. Thus the Feynman-Kac formula realizes, within the framework of measure theory and stochastic analysis, the infinite-dimensional limit of Gaussian integrals that physicists denote as the *Euclidean path integral*.

## 8.5 Connection to Dirichlet Forms and Potential Theory

The Feynman-Kac framework extends with full mathematical generality to the theory of symmetric Dirichlet forms and their associated Markov processes.

Let $(E, \mathcal{B}, \mu)$ be a $\sigma$-finite measure space, and let $(\mathcal{E}, \mathcal{D}(\mathcal{E}))$ be a symmetric, closed, and Markovian bilinear form on $L^2(E, \mu)$. Suppose further that $(\mathcal{E}, \mathcal{D}(\mathcal{E}))$ is *quasi-regular*, ensuring the existence of an associated Hunt process $(X_t, \mathbb{P}^x)$ with state space $E$ and properly measurable sample paths.

For a measurable potential $V: E \to \mathbb{R}$ that is $\mathcal{E}$-measurable and satisfies the integrability condition

$$ \mathbb{E}^x \left[ \exp \left( - \int_0^t |V(X_s)| ds \right) \right] < \infty, \quad \forall t > 0, \text{ for quasi-every } x \in E, \qquad (8.50) $$

one defines the *Feynman-Kac semigroup* $(P_t^V)_{t \ge 0}$ acting on $L^2(E, \mu)$ by

$$ P_t^V f(x) := \mathbb{E}^x \left[ e^{-\int_0^t V(X_s) ds} f(X_t) \right], \quad f \in L^2(E, \mu). \qquad (8.51) $$

This semigroup is strongly continuous, positivity-preserving, and symmetric in $L^2(E, \mu)$. It corresponds to the analytic perturbation of the original Dirichlet form by the potential $V$, in the sense that

$$ \mathcal{E}^V(f,g) = \mathcal{E}(f,g) + \int_E V(x)f(x)g(x) d\mu(x), \quad f,g \in \mathcal{D}(\mathcal{E}) \cap L^2(E, |V| d\mu), \qquad (8.52) $$
and its generator is precisely the Schrödinger-type operator

$$H = -\mathcal{L} + V, \qquad (8.53)$$

where $\mathcal{L}$ is the self-adjoint generator associated with $(\mathcal{E}, \mathcal{D}(\mathcal{E}))$.

From the probabilistic perspective, the Feynman-Kac semigroup provides an expectation over paths of the Hunt process $(X_t)$, weighted by the exponential of the negative time-integral of the potential. That is, for all $f \in L^2(E, \mu)$,

$$P_t^V f(x) = \mathbb{E}^x \left[ \exp \left( - \int_0^t V(X_s) ds \right) f(X_t) \right]. \qquad (8.54)$$

From the analytic viewpoint, $P_t^V$ is the strongly continuous symmetric semigroup whose generator $H$ corresponds to the perturbed form $\mathcal{E}^V$. The fundamental equivalence can thus be expressed as the following dual characterization:

$$\text{Analytic: } \mathcal{E}^V(f, f) = \mathcal{E}(f, f) + \int_E V f^2 d\mu \iff \text{Probabilistic: } P_t^V f(x) = \mathbb{E}^x \left[ e^{-\int_0^t V(X_s) ds} f(X_t) \right]. \qquad (8.55)$$

Hence, the Feynman-Kac formula realizes a rigorous equivalence principle between potential perturbations in the analytic theory of Dirichlet forms and exponential path-weighting in stochastic process theory. Every symmetric Schrödinger operator of the form

$$H = -\mathcal{L} + V, \qquad (8.56)$$

where $\mathcal{L}$ is the generator of a quasi-regular symmetric Dirichlet form, possesses a precise probabilistic representation via the expected exponential of the path integral of the potential along the trajectories of $X_t$.

## 8.6 Conclusion

The path integral representation thus bridges three distinct mathematical structures: the analytic semigroup $e^{-tH}$, the probabilistic expectation $\mathbb{E}^x[e^{-\int_0^t V(X_s) ds} f(X_t)]$, and the functional integral on infinite-dimensional space. It unifies potential theory, spectral theory, and stochastic calculus within a single conceptual framework, providing a rigorous measure-theoretic foundation for the Feynman-Kac correspondence and, by extension, for the Euclidean formulation of quantum mechanics.

**References:** Ghosh (2025) [32], Simon (1979) [19], Albeverio et. al. (2012) [30], Simon (2005) [20].
# Functional Analytic Properties

> “Analysis is the art of taming infinity by approximation.”

> — Jean Dieudonné

This chapter presents the fundamental functional-analytic structure underlying the theory of Dirichlet forms, semigroups, and Schrödinger-type operators. We rigorously study domains, generators, spectral properties, and compactness criteria within the framework of self-adjoint operators on $L^2$-spaces. The interplay between semigroup theory, spectral resolution, and variational principles will serve as the analytic backbone of subsequent probabilistic constructions.

## 9.1 Strongly Continuous Semigroups and Their Generators

Let $\mathcal{H} = L^2(E, \mu)$ be a real or complex separable Hilbert space and let $(T_t)_{t \ge 0}$ be a family of bounded linear operators on $\mathcal{H}$ satisfying the semigroup law $T_{t+s} = T_t T_s$ for all $s, t \ge 0$ and $T_0 = I$. We say $(T_t)$ is a *strongly continuous semigroup* (or $C_0$-semigroup) if for every $f \in \mathcal{H}$,

$$ \lim_{t \downarrow 0} \|T_t f - f\|_{\mathcal{H}} = 0. \qquad (9.1) $$

Strong continuity implies several elementary but important consequences which we record first. There exist constants $M \ge 1$ and $\omega \in \mathbb{R}$ (the *growth bound*) such that for all $t \ge 0$,

$$ \|T_t\|_{\mathcal{B}(\mathcal{H})} \le Me^{\omega t}. \qquad (9.2) $$

Equivalently the map $t \mapsto T_t$ is locally uniformly bounded in operator norm and, for every $\alpha > \omega$, the Laplace transform

$$ R(\alpha) := \int_{0}^{\infty} e^{-\alpha t} T_t dt \qquad (9.3) $$

converges in operator norm and defines a bounded operator on $\mathcal{H}$. The operator $R(\alpha)$ is the resolvent of the infinitesimal generator $L$ at $\alpha$, see below.

The infinitesimal generator $(L, \mathcal{D}(L))$ of $(T_t)$ is defined by

$$ \mathcal{D}(L) := \{ f \in \mathcal{H} : \lim_{t \downarrow 0} \frac{T_t f - f}{t} \text{ exists in } \mathcal{H} \}, \quad Lf := \lim_{t \downarrow 0} \frac{T_t f - f}{t}, \quad f \in \mathcal{D}(L). \qquad (9.4) $$