$$
\begin{equation}
\begin{split}
\bar{\psi}_{-}^{(1)}(\mathbf{k}, t, e) = \bar{\psi}_{+}(\mathbf{k}, t, e = 0) ={}& \\
& (2\pi)^4 \delta(\mathbf{k}) \bar{\psi}^* \int_0^t d\tau \delta(\tau) e^{-(\mathbf{k}^2+m^2)(t-\tau)} + m^2 \bar{\psi}^* \int_0^t d\tau e^{-(\mathbf{k}^2+m^2)(t-\tau)} + \\
& \qquad + \int_0^t d\tau e^{-(\mathbf{k}^2+m^2)(t-\tau)} \{(\mathbf{k}+m)\eta_2(\mathbf{k}, \tau) + \bar{\eta}_1(\mathbf{k}, \tau)\} =
\end{split}
\tag{5.2.20}
\end{equation}
$$

$$
A_{\mu-}^{(n)}(\mathbf{k}, t, e) = A_{\mu-} \left[ \psi_{-}^{(n-1)}(\mathbf{k}, t, e), \bar{\psi}_{+}^{(n-1)}(\mathbf{k}, t, e), A_{\mu-}^{(n-1)}(\mathbf{k}, t, e), \right] = \tag{5.2.21}
$$

$$
\begin{align}
A_{\mu-}^{(1)}(\mathbf{k}, t, e = 0) = & \nonumber \\
= -(2\pi)^4 a_\mu \delta(\mathbf{k}) & \times \int_0^t \delta(\tau) e^{-(\mathbf{k}^2+m_{\text{reg}}^2)(t-\tau)} d\tau - m_{\text{reg}}^2 a_\mu \delta(\mathbf{k}) \int_0^t e^{-(\mathbf{k}^2+m_{\text{reg}}^2)(t-\tau)} d\tau + \nonumber \\
& + \int_0^t \left[ e^{-(\mathbf{k}^2+m_{\text{reg}}^2)(t-\tau)} T_{\mu\nu} + L_{\mu\nu} \right] \tag{5.2.23}
\end{align}
$$

$$
\begin{align}
A_{\mu-}^{(1)}(\mathbf{k}, t, e = 0) = & \nonumber \\
= -(2\pi)^4 a_{\mu} \delta(\mathbf{k}) & \times \int_0^t \delta(\tau) e^{-(\mathbf{k}^2+m_{\text{reg}}^2)(t-\tau)} d\tau - m_{\text{reg}}^2 a_{\mu} \delta(\mathbf{k}) \int_0^t e^{-(\mathbf{k}^2+m_{\text{reg}}^2)(t-\tau)} d\tau + \nonumber \\
& + \int_0^t \left[ e^{-(\mathbf{k}^2+m_{\text{reg}}^2)(t-\tau)} T_{\mu\nu}^{\text{reg}} + L_{\mu\nu}^{\text{reg}} \right] \tag{5.2.24}
\end{align}
$$

$$
A_{\mu+}^{(n)}(\mathbf{k}, t, e) = A_{\mu-} \left[ \psi_{-}^{(n-1)}(\mathbf{k}, t, e), \bar{\psi}_{-}^{(n-1)}(\mathbf{k}, t, e), A_{\mu+}^{(n-1)}(\mathbf{k}, t, e), \right] = \quad (5.2.25)
$$
$$
\begin{align}
A_{\mu+}^{(1)}(\mathbf{k}, t, e=0) = & \nonumber \\
& (2\pi)^4 a_{\mu} \delta(\mathbf{k}) \times \int_0^t \delta(\tau) e^{-(\mathbf{k}^2+m_{\text{reg}}^2)(t-\tau)} d\tau + m_{\text{reg}}^2 a_{\mu} \delta(\mathbf{k}) \int_0^t e^{-(\mathbf{k}^2+m_{\text{reg}}^2)(t-\tau)} d\tau + \tag{5.2.26} \\
& + \int_0^t \left[ e^{-(\mathbf{k}^2+m_{\text{reg}}^2)(t-\tau)} T_{\mu\nu}^{\text{reg}} + L_{\mu\nu}^{\text{reg}} \right]. \nonumber
\end{align}
$$

From Eqs.(5.2.19) we find directly

$$
\begin{align}
& \int_0^t d\tau e^{-(\mathbf{k}^2+m^2)(t-\tau)} \int_0^{t'} d\tau' \delta(\tau-\tau') e^{-(\mathbf{k}^2+m^2)(t'-\tau')} \langle \eta_1(\mathbf{k}, t) \bar{\psi}_+(\mathbf{k}', t, e=0) \rangle = \tag{5.2.27} \\
&= \langle \eta_1(\mathbf{k}, t) \bar{\eta}_1(\mathbf{k}', t') \rangle = (2\pi)^4 \int_0^t d\tau e^{-(\mathbf{k}^2+m^2)(t-\tau)} \int_0^{t'} d\tau' \delta(\tau-\tau') e^{-(\mathbf{k}^2+m^2)(t'-\tau')} \delta(\mathbf{k}+\mathbf{k}') \nonumber
\end{align}
$$

and

$$
\begin{align}
& \int_0^t d\tau e^{-(\mathbf{k}^2+m^2)(t-\tau)} \int_0^{t'} d\tau' \delta(\tau-\tau') e^{-(\mathbf{k}^2+m^2)(t'-\tau')} \langle \eta_1(\mathbf{k}, t) \psi_+(\mathbf{k}', t, e=0) \bar{\eta}_2(\mathbf{k}, t) \rangle = \nonumber \\
& \qquad = \langle \eta_2(\mathbf{k}, t) \bar{\eta}_2(\mathbf{k}', t') \rangle = \tag{5.2.28} \\
& \qquad (2\pi)^4 \int_0^t d\tau e^{-(\mathbf{k}^2+m^2)(t-\tau)} \int_0^{t'} d\tau' \delta(\tau-\tau') e^{-(\mathbf{k}^2+m^2)(t'-\tau')} \delta(\mathbf{k}+\mathbf{k}') \nonumber
\end{align}
$$

Therefore

$$
\begin{align}
& \int_0^t d\tau e^{-(\mathbf{k}^2+m^2)(t-\tau)} \int_0^{t'} d\tau' e^{-(\mathbf{k}^2+m^2)(t-\tau')} \langle \psi_-^{(1)}(\mathbf{k}, t, 0) \bar{\psi}_+^{(1)}(\mathbf{k}', t', 0) \rangle = \nonumber \\
&= (2\pi)^4 \psi_-^* \bar{\psi}_+^* \int_0^t d\tau e^{-(\mathbf{k}^2+m^2)(t-\tau)} \int_0^{t'} d\tau' e^{-(\mathbf{k}^2+m^2)(t'-\tau')} \delta(\mathbf{k}) \delta(\mathbf{k}') - \tag{5.2.29} \\
&\quad -(2\pi)^4 \int_0^t d\tau e^{-(\mathbf{k}^2+m^2)(t-\tau)} \int_0^{t'} d\tau' \delta(\tau-\tau') e^{-(\mathbf{k}^2+m^2)(t'-\tau')} (\mathbf{k}+\mathbf{m}) \delta(\mathbf{k}+\mathbf{k}'). \nonumber
\end{align}
$$

Then we easily find Fourier transcendental master equation to lowest order iteration

$n = 1$
$$
\begin{align}
\langle \psi_{-}^{(1)}(\mathbf{x}, t, 0) \bar{\psi}_{+}^{(1)}(\mathbf{x}', t', 0) \rangle &= \int d^4\mathbf{k} \int d^4\mathbf{k}' e^{i\pi(\mathbf{x}\cdot\mathbf{k}+\mathbf{x}'\cdot\mathbf{k}')} \langle \psi_{-}^{(1)}(\mathbf{k}, t, 0) \bar{\psi}_{+}^{(1)}(\mathbf{k}', t', 0) \rangle = \nonumber \\
&= m^4 \psi^* \bar{\psi}^* \int d^4\mathbf{k} \int_0^t d\tau e^{-(\mathbf{k}^2+m^2)(t-\tau)} \int_0^{t'} d\tau' e^{-(\mathbf{k}^2+m^2)(t'-\tau')} \int d^4\mathbf{k}' e^{i\pi(\mathbf{x}\cdot\mathbf{k}+\mathbf{x}'\cdot\mathbf{k}')} \delta(\mathbf{k})\delta(\mathbf{k}') - \nonumber \\
&\quad -(2\pi)^4 \int_0^t d\tau e^{-(\mathbf{k}^2+m^2)(t-\tau)} \int_0^{t'} d\tau' \delta(\tau-\tau') e^{-(\mathbf{k}^2+m^2)(t'-\tau')} \times \tag{5.2.30} \\
&\quad \times \int d^4\mathbf{k} \int d^4\mathbf{k}' e^{i\pi(\mathbf{x}\cdot\mathbf{k}+\mathbf{x}'\cdot\mathbf{k}')} (\mathbf{k}+m)\delta(\mathbf{k}+\mathbf{k}') = 0 \nonumber
\end{align}
$$

$$
\begin{align}
\int_0^t d\tau e^{-(\mathbf{k}^2+m^2)(t-\tau)} \int_0^{t'} d\tau' e^{-(\mathbf{k}^2+m^2)(t'-\tau')} &= e^{-2(\mathbf{k}^2+m^2)(t+t')} \int_0^t d\tau e^{(\mathbf{k}^2+m^2)\tau} \int_0^{t'} d\tau' e^{(\mathbf{k}^2+m^2)\tau'} = \nonumber \\
&= \frac{e^{-2(\mathbf{k}^2+m^2)(t+t')} [e^{(\mathbf{k}^2+m^2)t} - 1] (e^{(\mathbf{k}^2+m^2)t'} - 1)}{(\mathbf{k}^2+m^2)^2} = \nonumber \\
&= \frac{e^{-2(\mathbf{k}^2+m^2)(t+t')} [e^{2(\mathbf{k}^2+m^2)(t+t')} - e^{(\mathbf{k}^2+m^2)t} - e^{(\mathbf{k}^2+m^2)t'} + 1]}{(\mathbf{k}^2+m^2)^2} \xrightarrow[t \to \infty, t' \to \infty]{} \nonumber \\
&\qquad (\mathbf{k}^2+m^2)^{-2} \nonumber
\end{align}
$$

$$
\begin{align}
\int_0^t d\tau e^{-(\mathbf{k}^2+m^2)(t-\tau)} \int_0^{t'} d\tau' \delta(\tau - \tau') e^{-(\mathbf{k}^2+m^2)(t'-\tau')} &= \nonumber \\
&\quad \int_0^t d\tau e^{-(\mathbf{k}^2+m^2)(t-\tau)} e^{-(\mathbf{k}^2+m^2)(t'-\tau)} = \int_0^t d\tau e^{-(\mathbf{k}^2+m^2)((t+t')-2\tau)} = \nonumber \\
&= e^{-(\mathbf{k}^2+m^2)(t+t')} \int_0^t d\tau e^{2(\mathbf{k}^2+m^2)\tau} = \frac{e^{-(\mathbf{k}^2+m^2)(t+t')}}{2(\mathbf{k}^2+m^2)} [e^{2(\mathbf{k}^2+m^2)t} - 1] = \nonumber \\
&= \frac{1}{2(\mathbf{k}^2+m^2)} [e^{-(\mathbf{k}^2+m^2)(t+t')} e^{2(\mathbf{k}^2+m^2)t} - 1] = \nonumber \\
&\quad \frac{1}{2(\mathbf{k}^2+m^2)} [e^{-(\mathbf{k}^2+m^2)(t-t')} - e^{-(\mathbf{k}^2+m^2)(t+t')}] \xrightarrow[t=t' \to +\infty]{} \frac{1}{2(\mathbf{k}^2+m^2)} \nonumber
\end{align}
$$

From Eq.(5.2.30)-Eq.(5.2.32) we find directly transcendental master equation to lowest order iteration $n = 1$
$$
\begin{gather}
\lim_{t \to t' \to \infty} \langle \psi_-^{(1)}(\mathbf{x}, t, 0) \bar{\psi}_+^{(1)}(\mathbf{x}', t', 0) \rangle = \nonumber \\
\psi^* \bar{\psi} - (2\pi)^{-4} \int d^4\mathbf{k} \exp[(\mathbf{x} - \mathbf{x}') \cdot \mathbf{k}] \frac{\mathbf{k} - \mathbf{m}}{\mathbf{k}^2 + m^2} = 0. \tag{5.2.33}
\end{gather}
$$

Thus transcendental master equation corresponding to fermion propagator to
lowest order iteration $n = 1$ gives the standard Euclidean fermion propagator:

$$
\Delta(\mathbf{x} - \mathbf{x}') = (2\pi)^{-4} \int d^4\mathbf{k} \exp[(\mathbf{x} - \mathbf{x}') \cdot \mathbf{k}] \frac{\mathbf{k} - \mathbf{m}}{\mathbf{k}^2 + m^2}. \quad (5.2.34)
$$

From Eq.(5.2.24) and Eq.(5.2.26) we find directly transcendental master equation
corresponding to photon propagator to lowest order iteration $n = 1$

$$
\begin{gather}
\lim_{t \to t' \to +\infty} \langle A_{\mu}^{(1)}(\mathbf{x}, t, 0) A_{\nu}^{(1)}(\mathbf{x}', t', 0) \rangle = \nonumber \\
a_{\mu} a_{\nu} - (2\pi)^{-4} \int d^4\mathbf{k} \frac{\exp[(\mathbf{x}-\mathbf{x}') \cdot \mathbf{k}]}{\mathbf{k}^2 + m_{\text{reg}}^2} \left[ \delta_{\mu\nu} - \frac{k_{\mu} k_{\nu}}{\mathbf{k}^2 + m_{\text{reg}}^2} \right] = 0. \tag{5.2.35}
\end{gather}
$$

Thus transcendental master equation corresponding to photon propagator to
lowest order iteration $n = 1$ gives the standard Euclidean massive photon
propagator:

$$
(5.2.36)
$$

6. Double stochastic quantization of non-Abelian massive
gauge field and Faddeev-Popov ghost effects.

A non-Abelian massive gauge field in the 4-dimensional Euclidean space is
characterized by the action integral

$$
\begin{align}
S[A] &= \frac{1}{4} \int d^4 x \left[ (\mathcal{F}_{\mu\nu}^a(x) \mathcal{F}_{\mu\nu}^a(x)) - \frac{1}{2} m_{\text{reg}}^2 (A_\mu^a(x))^2 \right] = \nonumber \\
&\quad -\frac{1}{4} \int d^4 x \left[ (\mathcal{F}_{\mu\nu}^a(x) \mathcal{F}_{\mu\nu}^a(x)) + \frac{1}{2} m_{\text{reg}}^2 (A_\mu^a(x))^2 \right], \tag{6.1.1} \\
\mathcal{F}_{\mu\nu}^a(x) &= \partial_\mu A_\nu^a(x) - \partial_\nu A_\mu^a(x) - g \times f^{abc} A_\mu^b(x) A_\nu^c(x), \nonumber
\end{align}
$$

where *a*, *b* and *c* stand for the color indices. From (6.1.1) we get the double
stochastic

Langevin equation
$$
\begin{equation} \tag{6.1.2}
\begin{split}
\frac{d}{dt} A_{\mu}^{a}(k,t) = & -((k^2 + m^2)\delta_{\mu\nu} - k_{\mu}k_{\nu})A_{\nu}^{a}(k,t) + \eta_{\mu}^{a}(\omega, k, t) + \epsilon\bar{\eta}_{\mu}^{a}(\omega, k, t) + Y_{\mu}^{a}(k,t) \\
& + \frac{g}{(2\pi)^2} \int d^4k_1 d^4k_2 \delta^4(k-k_1-k_2) V_{\mu\kappa\lambda}^{abc}(k,-k_1,-k_2) A_{\kappa}^{b}(k_1,t) A_{\lambda}^{d}(k_2,t) + \\
& + \frac{g^2}{(2\pi)^4} \int d^4k_1 d^4k_2 d^4k_3 \delta^4(k-k_1-k_2-k_3) W_{\mu\nu\kappa\lambda}^{abcd} A_{\nu}^{b}(k_1,t) A_{\kappa}^{c}(k_2,t) A_{\lambda}^{d}(k_3,t),
\end{split}
\end{equation}
$$

where the Fourier transform $\hat{f}(k)$ and its inverse $\check{f}(k)$ is defined as [21]-[22]:

$$
\begin{align}
\hat{f}(k) &= (2\pi)^{-2} \int f(x) e^{-ik \cdot x} d^4 x, \tag{6.1.3} \\
\check{f}(k) &= (2\pi)^{-2} \int f(x) e^{-ik \cdot x} d^4 x, \nonumber
\end{align}
$$

and where

$$
\begin{align}
V_{\mu\kappa\lambda}^{abc}(k, k_1, k_2) &= \left(-\frac{i}{2}\right) f^{abc} \times \nonumber \\
&\quad \left[ (k-k_1)_\lambda \delta_{\mu\kappa} + (k_1-k_2)_\mu \delta_{\kappa\lambda} + (k_2-k)_\kappa \delta_{\mu\lambda} \right]. \tag{6.1.4} \\
W_{\mu\nu\kappa\lambda}^{abcd} &= -\frac{1}{6} \left[ \begin{aligned}[t]
& f^{abe} f^{cde} (\delta_{\mu\kappa} \delta_{v\lambda} - \delta_{\mu\lambda} \delta_{v\kappa}) + \\
& \qquad + f^{ace} f^{bde} (\delta_{\mu\nu} \delta_{\kappa\lambda} - \delta_{\mu\lambda} \delta_{\kappa\nu}) + \\
& \qquad + f^{ade} f^{be\nu} (\delta_{\mu\kappa} \delta_{v\lambda} - \delta_{\mu\nu} \delta_{\kappa\lambda})
\end{aligned} \right]. \nonumber
\end{align}
$$

From Eq.(6.1.2) we get

$$
\begin{equation}
\begin{split}
\frac{d}{dt} A_{\mu}^{a}(k,t) = -\delta(t)\delta^{4}(k)\chi_{\mu}^{a} - & \\
& -((k^2 + m^2)\delta_{\mu\nu} - k_{\mu}k_{\nu})A_{\nu}^{a}(k,t) + \eta_{\mu}^{a}(\omega, k, t) + \epsilon\eta_{\mu}^{a}(\omega, k, t) + Y_{\mu}^{a}(k,t),
\end{split}
\end{equation}
\begin{equation}
\begin{split}
Y_{\mu}^{a}(k,t) = & \\
& + \frac{g}{(2\pi)^2} \int d^4k_1 V_{\mu\kappa\lambda}^{abc}(k,-k_1,-(k-k_1))A_{\kappa}^{b}(k_1,t)A_{\lambda}^{c}(k-k_1,t) + \\
& + \frac{g^2}{(2\pi)^4} \int d^4k_1 d^4k_2 W_{\mu\nu\kappa\lambda}^{abcd}A_{\nu}^{b}(k_1,t)A_{\kappa}^{c}(k_2,t)A_{\lambda}^{d}(k-k_1-k_2,t),
\end{split}
\end{equation}
\begin{equation}
V_{\mu\kappa\lambda}^{abc}(k,k_1,k-k_1) = \left(-\frac{i}{2}\right) f^{abc} \times \\
\times \left[(k-k_1)_\lambda \delta_{\mu\kappa} + (k_1-(k-k_1))_\mu \delta_{\kappa\lambda} + (-k_1)_\kappa \delta_{\mu\lambda}\right]
\end{equation}
$$

We define now the replacement
$$
\begin{align*}
A_{\mu}^{a}(x,t) - \theta(t)\chi_{\mu} &= A_{\mu-}^{a}(x,t), \\
A_{\mu}^{a}(x,t) &= A_{\mu-}^{a}(x,t) + \theta(t)\chi_{\mu}^{a}, \\
A_{\mu}^{a}(k,t) &= \mathcal{F}[A_{\mu}^{a}(x,t)](k,t) = \mathcal{F}[A_{\mu-}^{a}(x,t)](k,t) + \theta(t)(2\pi)^{4}\delta^{4}(k)\chi_{\mu}^{a} \\
&= A_{\mu-}^{a}(p,t) + \theta(t)(2\pi)^{4}\chi_{\mu}^{a}, \\
A_{\mu}^{a}(x,t) + \theta(t)\chi_{\mu}^{a} &= A_{\mu+}^{a}(x,t), \\
A_{\mu}^{a}(x,t) &= A_{\mu+}^{a}(x,t) - \theta(t)\chi_{\mu}^{a}, \tag{6.1.6} \\
\frac{\partial A_{\mu}^{a}(x,t)}{\partial t} &= \frac{\partial A_{\mu-}^{a}(x,t)}{\partial t} + \delta(t)\chi_{\mu}^{a}, \\
\frac{\partial A_{\mu}^{a}(k,t)}{\partial t} &= \frac{\partial A_{\mu-}^{a}(k,t)}{\partial t} + \delta(t)(2\pi)^{4}\delta^{4}(k)\chi_{\mu}^{a}, \\
\frac{\partial A_{\mu}^{a}(x,t)}{\partial t} &= \frac{\partial A_{\mu+}^{a}(x,t)}{\partial t} - \delta(t)\chi_{\mu}^{a}, \\
\frac{\partial A_{\mu}^{a}(k,t)}{\partial t} &= \frac{\partial A_{\mu+}^{a}(k,t)}{\partial t} - \delta(t)(2\pi)^{4}\delta^{4}(k)\chi_{\mu}^{a}.
\end{align*}
$$

Eq.(6.1.5) by the replacement (6.1.6) we get

$$
\begin{align}
\frac{d}{dt} A_{\mu-}^{a}(k,t) = -\delta(t)\delta^{4}(k)\chi_{\mu}^{a} & - ((k^2 + m^2)\delta_{\mu\nu} - k_{\mu}k_{\nu})[A_{\nu-}^{a}(k,t) + (2\pi)^2 m^2 \delta^4(k)\chi_{\nu}^{a}] + \nonumber \\
& + \eta_{\mu}^{a}(\omega, k, t) + \epsilon\eta_{\mu}^{a}(\omega, k, t) + Y_{\mu-}^{a}(k,t), \nonumber \\
Y_{\mu-}^{a}(k,t) &= \frac{g}{(2\pi)^2} \int d^4k_1 V_{\mu\kappa\lambda}^{abc}(k, -k_1, -(k-k_1)) Y_{\mu-}^{a(1)}(k,t) + \nonumber \\
&\quad + \frac{g^2}{(2\pi)^4} \int d^4k_1 d^4k_2 W_{\mu\kappa\lambda}^{abcd} Y_{\mu-}^{a(2)}(k,t) = \nonumber \\
&\quad \frac{g}{(2\pi)^2} \int d^4k_1 V_{\mu\kappa\lambda}^{abc}(k, -k_1, -(k-k_1)) [A_{\kappa-}^{b}(k_1,t) + (2\pi)^2 \delta^4(k_1)\chi_{\kappa}^{b}] \times \nonumber \\
&\quad \times [A_{\lambda-}^{c}(k-k_1,t) + (2\pi)^2 \delta^4(k-k_1)\chi_{\lambda}^{c}] + \nonumber \\
&\quad + \frac{g^2}{(2\pi)^4} \int d^4k_1 d^4k_2 W_{\mu\kappa\lambda}^{abcd} [A_{\nu-}^{b}(k_1,t) + (2\pi)^2 \delta(k_1)\chi_{\nu}^{b}] \times \nonumber \\
&\quad \times [A_{\kappa-}^{c}(k_2,t) + (2\pi)^2 \delta(k_2)\chi_{\kappa}^{c}] \times [A_{\lambda-}^{d}(k-k_1-k_2,t) + (2\pi)^2 \delta(k-k_1-k_2)\chi_{\lambda}^{d}]. \tag{6.1.7}
\end{align}
$$

where
$$
\begin{align}
Y_{\mu-}^{a(1)}(k,t) &= [A_{\kappa-}^b(k_1,t) + (2\pi)^2 \delta^4(k_1) \chi_{\kappa}^b] \times [A_{\lambda-}^c(k-k_1,t) + (2\pi)^2 \delta^4(k-k_1) \chi_{\lambda}^c], \tag{6.1.8} \\
Y_{\mu-}^{a(2)}(k,t) &= [A_{\nu-}^b(k_1,t) + (2\pi)^2 \delta^4(k_1) \chi_{\nu}^b] \times [A_{\kappa-}^c(k_2,t) + (2\pi)^2 \delta^4(k_2) \chi_{\kappa}^c] \times \nonumber \\
&\qquad \times [A_{\lambda-}^d(k-k_1-k_2,t) + (2\pi)^2 \delta^4(k-k_1-k_2) \chi_{\lambda}^d]. \nonumber
\end{align}
$$

Note that

$$
\begin{align}
Y_{\mu-}^{a(1)}(k,t) &= [A_{\kappa-}^b(k_1,t) + (2\pi)^2 \delta^4(k_1) \chi_{\kappa}^b] \times [A_{\lambda-}^c(k-k_1,t) + (2\pi)^2 \delta^4(k-k_1) \chi_{\lambda}^c] \nonumber \\
&= \frac{A_{\kappa-}^b(k_1,t)A_{\lambda-}^c(k-k_1,t) + (2\pi)^2 \delta^4(k_1) \chi_{\kappa}^b A_{\lambda-}^c(k-k_1,t) + (2\pi)^2 \delta^4(k-k_1) \chi_{\lambda}^b A_{\kappa-}^c(k_1,t)}{} \tag{6.1.9} \\
&\quad + (2\pi)^4 \delta^4(k_1) \delta^4(k-k_1) \chi_{\kappa}^b \chi_{\lambda}^c. \nonumber
\end{align}
$$

$$
\begin{align}
Y_{\mu-}^{a(2)}(k,t) &= [A_{\nu-}^b(k_1,t) + (2\pi)^2 \delta^4(k_1) \chi_{\nu}^b] \times [A_{\kappa-}^c(k_2,t) + (2\pi)^2 \delta^4(k_2) \chi_{\kappa}^c] \times \nonumber \\
&\quad \times [A_{\lambda-}^d(k-k_1-k_2,t) + (2\pi)^2 \delta^4(k-k_1-k_2) \chi_{\lambda}^d] \nonumber \\
&= \left[ \underbrace{A_{\nu-}^b(k_1,t)A_{\kappa-}^c(k_2,t)}_{\phantom{A_{\nu-}^b(k_1,t)A_{\kappa-}^c(k_2,t)}} + (2\pi)^2 \delta^4(k_1) \chi_{\nu}^b A_{\kappa-}^c(k_2,t) + \right. \nonumber \\
&\quad \left. (2\pi)^2 \delta^4(k_2) \chi_{\kappa}^c A_{\nu-}^b(k_1,t) + (2\pi)^4 \delta^4(k_1) \delta^4(k_2) \chi_{\nu}^b \chi_{\kappa}^c \right] \times \nonumber \\
&\quad \times [A_{\lambda-}^d(k-k_1-k_2,t) + (2\pi)^2 \delta^4(k-k_1-k_2) \chi_{\lambda}^d] \nonumber \\
&= \left[ \underbrace{A_{\nu-}^b(k_1,t)A_{\kappa-}^c(k_2,t)}_{\phantom{A_{\nu-}^b(k_1,t)A_{\kappa-}^c(k_2,t)}} + (2\pi)^2 \delta^4(k_1) \chi_{\nu}^b A_{\kappa-}^c(k_2,t) + \right. \nonumber \\
&\quad \left. (2\pi)^2 \delta^4(k_2) \chi_{\kappa}^c A_{\nu-}^b(k_1,t) + (2\pi)^4 \delta^4(k_1) \delta^4(k_2) \chi_{\nu}^b \chi_{\kappa}^c \right] A_{\lambda-}^d(k-k_1-k_2,t) + \nonumber \\
&\quad \underbrace{[A_{\nu-}^b(k_1,t)A_{\kappa-}^c(k_2,t) + (2\pi)^2 \delta^4(k_1) \chi_{\nu}^b A_{\kappa-}^c(k_2,t) +}_{\phantom{\underbrace{[A_{\nu-}^b(k_1,t)A_{\kappa-}^c(k_2,t) + (2\pi)^2 \delta^4(k_1) \chi_{\nu}^b A_{\kappa-}^c(k_2,t) +}} (2\pi)^2 \delta^4(k-k_1-k_2) \chi_{\lambda}^d} \nonumber \\
&= \underbrace{A_{\nu-}^b(k_1,t)A_{\kappa-}^c(k_2,t)A_{\lambda}^d(k-k_1-k_2,t)}_{\phantom{A_{\nu-}^b(k_1,t)A_{\kappa-}^c(k_2,t)A_{\lambda}^d(k-k_1-k_2,t)}} + \underbrace{(2\pi)^2 \delta^4(k_1) \chi_{\nu}^b A_{\kappa-}^c(k_2,t) A_{\lambda-}^d(k-k_1-k_2,t)}_{\phantom{(2\pi)^2 \delta^4(k_1) \chi_{\nu}^b A_{\kappa-}^c(k_2,t) A_{\lambda-}^d(k-k_1-k_2,t)}} + \nonumber \\
&\quad \underbrace{(2\pi)^2 \delta^4(k_2) \chi_{\kappa}^c A_{\nu-}^b(k_1,t) A_{\lambda-}^d(k-k_1-k_2,t)}_{\phantom{(2\pi)^2 \delta^4(k_2) \chi_{\kappa}^c A_{\nu-}^b(k_1,t) A_{\lambda-}^d(k-k_1-k_2,t)}} + \underbrace{(2\pi)^4 \delta^4(k_1) \delta^4(k_2) \chi_{\nu}^b \chi_{\kappa}^c A_{\lambda-}^d(k-k_1-k_2,t)}_{\phantom{(2\pi)^4 \delta^4(k_1) \delta^4(k_2) \chi_{\nu}^b \chi_{\kappa}^c A_{\lambda-}^d(k-k_1-k_2,t)}} + \nonumber \\
&\quad \underbrace{(2\pi)^2 \delta^4(k-k_1-k_2) \chi_{\lambda}^d A_{\nu-}^b(k_1,t) A_{\kappa-}^c(k_2,t)}_{\phantom{(2\pi)^2 \delta^4(k-k_1-k_2) \chi_{\lambda}^d A_{\nu-}^b(k_1,t) A_{\kappa-}^c(k_2,t)}} + \nonumber \\
&\quad \underbrace{(2\pi)^4 \delta^4(k_1) \delta^4(k-k_1-k_2) \chi_{\lambda}^d \chi_{\kappa}^c A_{\kappa-}^c(k_1,t)}_{\phantom{(2\pi)^4 \delta^4(k_1) \delta^4(k-k_1-k_2) \chi_{\lambda}^d \chi_{\kappa}^c A_{\kappa-}^c(k_1,t)}} + \nonumber \\
&\quad \underbrace{+(2\pi)^4 \delta^4(k_2) \delta^4(k-k_1-k_2) \chi_{\lambda}^d \chi_{\kappa}^c A_{\nu-}^b(k_1,t)}_{\phantom{+(2\pi)^4 \delta^4(k_2) \delta^4(k-k_1-k_2) \chi_{\lambda}^d \chi_{\kappa}^c A_{\nu-}^b(k_1,t)}} + \nonumber \\
&\quad \underbrace{+(2\pi)^6 \delta^4(k_1) \delta^4(k_2) \delta^4(k-k_1-k_2) \chi_{\lambda}^d \chi_{\nu}^b \chi_{\kappa}^c}_{\phantom{+(2\pi)^6 \delta^4(k_1) \delta^4(k_2) \delta^4(k-k_1-k_2) \chi_{\lambda}^d \chi_{\nu}^b \chi_{\kappa}^c}} \nonumber
\end{align}
$$
$$
\begin{align}
Y_{\mu-}^a(k,t) &= \nonumber \\
&= \tilde{Y}_{\mu-}^{a(1)} + \tilde{Y}_{\mu-}^{a(2)}(k,t) = \nonumber \\
&\quad \frac{g}{(2\pi)^2} \int d^4 k_1 V_{\mu\kappa\lambda}^{abc}(k,-k_1,-(k-k_1)) Y_{\mu-}^{a(1)}(k,t) + \nonumber \\
&\quad + \frac{g^2}{(2\pi)^4} \int d^4 k_1 d^4 k_2 W_{\mu\kappa\lambda}^{abcd} Y_{\mu-}^{a(2)}(k,t), \tag{6.1.11} \\
\tilde{Y}_{\mu-}^{a(1)} &= \nonumber \\
&\quad \frac{g}{(2\pi)^2} \int d^4 k_1 V_{\mu\kappa\lambda}^{abc}(k,-k_1,-(k-k_1)) Y_{\mu-}^{a(1)}(k,t) \nonumber \\
\tilde{Y}_{\mu-}^{a(2)}(k,t) &= \nonumber \\
&\quad \frac{g^2}{(2\pi)^4} \int d^4 k_1 d^4 k_2 W_{\mu\kappa\lambda}^{abcd} Y_{\mu-}^{a(2)}(k,t) \nonumber
\end{align}
$$

From Eq.(6.1.9) we get
$$
\begin{align*}
\tilde{Y}_{\mu-}^{a(1)} &= \\
&= \frac{g}{(2\pi)^2} \int d^4 k_1 V_{\mu\kappa\lambda}^{abc}(k, -k_1, -(k-k_1)) Y_{\mu-}^{a(1)}(k,t) = \\
&\quad \frac{g}{(2\pi)^2} \int d^4 k_1 V_{\mu\kappa\lambda}^{abc}(k, -k_1, -(k-k_1)) \underline{A}_{\kappa-}^b(k_1,t) A_{\lambda-}^c(k-k_1,t) + \\
&\quad + \frac{g}{(2\pi)^2} \int d^4 k_1 V_{\mu\kappa\lambda}^{abc}(k, -k_1, -(k-k_1)) \times \\
&\quad \times [(2\pi)^2 \delta^4(k_1) \chi_{\kappa}^b A_{\lambda-}^c(k-k_1,t) + (2\pi)^2 \delta^4(k-k_1) \chi_{\lambda}^c A_{\kappa-}^b(k_1,t) \\
&\quad +(2\pi)^4 \delta^4(k_1) \delta^4(k-k_1) \chi_{\kappa}^b \chi_{\lambda}^c] = \\
&\quad \frac{g}{(2\pi)^2} \int d^4 k_1 V_{\mu\kappa\lambda}^{abc}(k, -k_1, -(k-k_1)) \underline{A}_{\kappa-}^b(k_1,t) A_{\lambda-}^c(k-k_1,t) + \\
&\quad +g \int d^4 k_1 V_{\mu\kappa\lambda}^{abc}(k, -k_1, -(k-k_1)) \delta^4(k_1) \chi_{\kappa}^b A_{\lambda-}^c(k-k_1,t) + \\
&\quad +g \int d^4 k_1 V_{\mu\kappa\lambda}^{abc}(k, -k_1, -(k-k_1)) \delta^4(k-k_1) \chi_{\lambda}^c A_{\kappa-}^b(k_1,t) + \\
&\quad +(2\pi)^2 g \int d^4 k_1 V_{\mu\kappa\lambda}^{abc}(k, -k_1, -(k-k_1)) \delta^4(k_1) \delta^4(k-k_1) \chi_{\kappa}^b \chi_{\lambda}^c = \\
&\quad \frac{g}{(2\pi)^2} \int d^4 k_1 V_{\mu\kappa\lambda}^{abc}(k, -k_1, -(k-k_1)) \underline{A}_{\kappa-}^b(k_1,t) A_{\lambda-}^c(k-k_1,t) + \\
&\quad +g \Big[ V_{\mu\kappa\lambda}^{abc}(k, 0, -k) \chi_{\kappa}^b A_{\lambda-}^c(k,t) + V_{\mu\kappa\lambda}^{abc}(k, -k, 0) \chi_{\lambda}^c A_{\kappa-}^b(k,t) \Big] + \\
&\quad +(2\pi)^2 g V_{\mu\kappa\lambda}^{abc}(k, -k, 0) \delta^4(k) \chi_{\kappa}^b \chi_{\lambda}^c.
\end{align*}
\tag{6.1.12}
$$

$$
\begin{align*}
V_{\mu\kappa\lambda}^{abc}(k, 0, -k) &= \left(-\frac{i}{2}\right) f^{abc} \times [k_{\lambda}\delta_{\mu\kappa} - k_{\mu}\delta_{\kappa\lambda}] \\
V_{\mu\kappa\lambda}^{abc}(k, -k, 0) &= \left(-\frac{i}{2}\right) f^{abc} \times
\end{align*}
$$

From Eq.(6.1.10) we get
$$
\begin{align}
\tilde{Y}_{\mu-}^{a(2)}(k,t) = & \nonumber \\
& = \frac{g^2}{(2\pi)^2} \int d^4 k_1 d^4 k_2 W_{\mu\nu\kappa\lambda}^{abcd} Y_{\mu-}^{a(2)}(k, k_1, k_2, t) = \nonumber \\
& \frac{g^2}{(2\pi)^4} \int d^4 k_1 d^4 k_2 W_{\mu\nu\kappa\lambda}^{abcd} \left[ \begin{aligned}[t]
& \underline{A_{\nu-}^b(k_1,t) A_{\kappa-}^c(k_2,t) A_{\lambda-}^d(k-k_1-k_2,t)} + \\
& + (2\pi)^2 \delta^4(k_1) \chi_{\nu}^b A_{\kappa-}^c(k_2,t) A_{\lambda-}^d(k-k_1-k_2,t) + \\
& + (2\pi)^2 \delta^4(k_2) \chi_{\kappa}^c A_{\nu-}^b(k_1,t) A_{\lambda-}^d(k-k_1-k_2,t) + \\
& + (2\pi)^2 \delta^4(k-k_1-k_2) \chi_{\lambda}^d A_{\nu-}^b(k_1,t) A_{\kappa-}^c(k_2,t) + \\
& + (2\pi)^2 \delta^4(k-k_1-k_2) \chi_{\lambda}^d A_{\nu-}^b(k_1,t) A_{\kappa-}^c(k_2,t)
\end{aligned} \right] + \tag{6.1.13} \\
& + \frac{g^2}{(2\pi)^4} \int d^4 k_1 d^4 k_2 W_{\mu\nu\kappa\lambda}^{abcd} \left[ \begin{aligned}[t]
& (2\pi)^4 \delta^4(k_1) \delta^4(k_2) \chi_{\nu}^b \chi_{\kappa}^c A_{\lambda-}^d(k-k_1-k_2,t) + \\
& + (2\pi)^4 \delta^4(k_2) \delta^4(k-k_1-k_2) \chi_{\lambda}^d \chi_{\kappa}^c A_{\kappa-}^c(k_1,t) + \\
& + (2\pi)^4 \delta^4(k_2) \delta^4(k-k_1-k_2) \chi_{\lambda}^d \chi_{\kappa}^c A_{\nu-}^b(k_1,t) + \\
& + (2\pi)^8 \delta^4(k_1) \delta^4(k_2) \delta^4(k-k_1-k_2) \chi_{\lambda}^d \chi_{\nu}^b \chi_{\kappa}^c
\end{aligned} \right] \nonumber
\end{align}
$$

From Eq.(6.11)-Eq.(6.13) we obtain linear differential master equations
corresponding to $A_{\mu-}^a(k,t)$:

$$
\begin{equation}
\begin{split}
\frac{d}{dt} A_{\mu-}^{a}(k,t) = -\delta(t)\delta^{4}(k)\chi_{\mu}^{a} & - ((k^2 + m^2)\delta_{\mu\nu} - k_{\mu}k_{\nu})[A_{\nu-}^{a}(k,t) + (2\pi)^2 m^2 \delta^{4}(k)\chi_{\nu}^{a}] + \\
& + \eta_{\mu}^{a}(\omega, k, t) + \tilde{Y}_{\mu-}^{\{L\}a(1)}(k,t) + \tilde{Y}_{\mu-}^{\{L\}a(2)}(k,t),
\end{split}
\tag{6.1.14}
\end{equation}
$$

where

$$
\begin{align}
\tilde{Y}_{\mu-}^{\{L\}a(1)}(k,t) = & \nonumber \\
& = g V_{\mu\kappa\lambda}^{abc}(k,0,-k) \chi_{\kappa}^{b} A_{\lambda-}^{c}(k,t) + (2\pi)^2 \delta^4(k-k_1) \chi_{\lambda}^{c} A_{\kappa-}^{b}(k_1,t) + \nonumber \\
& \quad + (2\pi)^2 g V_{\mu\kappa\lambda}^{abc}(k,-k,0) \delta^4(k) \chi_{\kappa}^{b} \chi_{\lambda}^{c}. \tag{6.1.15} \\
V_{\mu\kappa\lambda}^{abc}(k,0,-k) = & \nonumber \\
& \quad (-\frac{i}{2}) f^{abc} \times [k_{\lambda} \delta_{\mu\kappa} - k_{\mu} \delta_{\kappa\lambda}]
\end{align}
$$

and
$$
\begin{align*}
\tilde{Y}_{\mu-}^{\{L\}a(2)}(k,t) = & \\
& \frac{g^2}{(2\pi)^4} \int d^4k_1 d^4k_2 W_{\mu\nu\kappa\lambda}^{abcd} \Biggl[ (2\pi)^4 \delta^4(k_1)\delta^4(k_2) \chi_v^b \chi_\kappa^c A_{\lambda-}^d (k-k_1-k_2,t) + \\
& \qquad + (2\pi)^4 \delta^4(k_1)\delta^4(k-k_1-k_2) \chi_\lambda^d \chi_\kappa^c A_{\kappa-}^c (k_1,t) + \\
& \qquad + (2\pi)^4 \delta^4(k_2)\delta^4(k-k_1-k_2) \chi_\lambda^d \chi_\kappa^c A_{v-}^b (k_1,t) + \\
& \qquad + (2\pi)^8 \delta^4(k_1)\delta^4(k_2)\delta^4(k-k_1-k_2) \chi_\lambda^d \chi_v^b \chi_\kappa^c \Biggr] = \\
& g^2 W_{\mu\nu\kappa\lambda}^{abcd} [\chi_v^b \chi_\kappa^c A_{\lambda-}^d (k,t) + \chi_\lambda^d \chi_\kappa^c A_{v-}^b (k_1,t)] + g^2 (2\pi)^4 W_{\mu\nu\kappa\lambda}^{abcd} \delta^4(k) \chi_\lambda^d \chi_v^b \chi_\kappa^c. \tag{6.1.16}
\end{align*}
$$

From Eq.(6.1.5) by the replacement (6.1.6) we get

$$
\begin{align}
\frac{d}{dt} A_{\mu+}^a(k,t) &= \delta(t)\delta^4(k)\chi_\mu^a - ((k^2+m^2)\delta_{\mu\nu}-k_\mu k_\nu)[A_{v+}^a(k,t)-(2\pi)^2 m^2 \delta^4(k)\chi_v^a] + \nonumber \\
&\quad +\eta_\mu^a(\omega,k,t) + \epsilon\eta_\mu^a(\omega,k,t) + Y_{\mu+}^a(k,t), \nonumber \\
Y_{\mu+}^a(k,t) &= \frac{g}{(2\pi)^2} \int d^4k_1 V_{\mu\kappa\lambda}^{abc}(k,-k_1,-(k-k_1)) Y_{\mu+}^{a(1)}(k,t) + \nonumber \\
&\quad + \frac{g^2}{(2\pi)^4} \int d^4k_1 d^4k_2 W_{\mu\nu\kappa\lambda}^{abcd} Y_{\mu+}^{a(2)}(k,t) = \nonumber \\
&\quad -\frac{g}{(2\pi)^2} \int d^4k_1 V_{\mu\kappa\lambda}^{abc}(k,-k_1,-(k-k_1)) [A_{\kappa+}^b(k_1,t)-(2\pi)^2 \delta^4(k_1)\chi_\kappa^b] \times \tag{6.1.17} \\
&\quad \times [A_{\lambda+}^c(k-k_1,t)-(2\pi)^2 \delta^4(k-k_1)\chi_\lambda^c] + \nonumber \\
&\quad + \frac{g^2}{(2\pi)^4} \int d^4k_1 d^4k_2 W_{\mu\nu\kappa\lambda}^{abcd} [A_{v+}^b(k_1,t)-(2\pi)^2 \delta^4(k_1)\chi_v^b] \times \nonumber \\
&\quad \times [A_{\kappa+}^c(k_2,t)-(2\pi)^2 \delta^4(k_2)\chi_\kappa^c] \times [A_{\lambda+}^d(k-k_1-k_2,t)-(2\pi)^2 \delta^4(k-k_1-k_2)\chi_\lambda^d]. \nonumber
\end{align}
$$

where

$$
\begin{align}
Y_{\mu+}^{a(1)}(k,t) = & \nonumber \\
& [A_{\kappa+}^{b}(k_1,t) - (2\pi)^2 \delta^4(k_1) \chi_{\kappa}^{b}] \times [A_{\lambda+}^{c}(k-k_1,t) - (2\pi)^2 \delta^4(k-k_1) \chi_{\lambda}^{c}], \tag{6.1.18} \\
Y_{\mu+}^{a(2)}(k,t) = & \nonumber \\
& [A_{v+}^{b}(k_1,t) - (2\pi)^2 \delta^4(k_1) \chi_{v}^{b}] \times [A_{\kappa+}^{c}(k_2,t) - (2\pi)^2 \delta^4(k_2) \chi_{\kappa}^{c}] \times \nonumber \\
& \qquad \times [A_{\lambda+}^{d}(k-k_1-k_2,t) - (2\pi)^2 \delta^4(k-k_1-k_2) \chi_{\lambda}^{d}]. \nonumber
\end{align}
$$

Note that
$$
\begin{align}
Y_{\mu+}^{a(1)}(k,t) = & \left[ A_{\kappa+}^{b}(k_1,t) - (2\pi)^2 \delta^4(k_1) \chi_{\kappa}^{b} \right] \times \left[ A_{\lambda+}^{c}(k-k_1,t) - (2\pi)^2 \delta^4(k-k_1) \chi_{\lambda}^{c} \right] = \nonumber \\
& \frac{A_{\kappa+}^{b}(k_1,t) A_{\lambda+}^{c}(k-k_1,t) - (2\pi)^2 \delta^4(k_1) \chi_{\kappa}^{b} A_{\lambda+}^{c}(k-k_1,t) - (2\pi)^2 \delta^4(k-k_1) \chi_{\lambda}^{c} A_{\kappa+}^{b}(k_1,t) +}{(2\pi)^4 \delta^4(k_1) \delta^4(k-k_1) \chi_{\kappa}^{b} \chi_{\lambda}^{c}} \tag{6.1.19}
\end{align}
$$

$$
\begin{align}
Y_{\mu+}^{a(2)}(k,t) = & \left[ A_{v+}^{b}(k_1,t) - (2\pi)^2 \delta^4(k_1) \chi_v^b \right] \times \left[ A_{\kappa+}^{c}(k_2,t) - (2\pi)^2 \delta^4(k_2) \chi_\kappa^c \right] \times \nonumber \\
& \times \left[ A_{\lambda+}^{d}(k-k_1-k_2,t) - (2\pi)^2 \delta^4(k-k_1-k_2) \chi_\lambda^d \right] = \nonumber \\
& \left[ \frac{A_{v+}^{b}(k_1,t) A_{\kappa+}^{c}(k_2,t)}{(2\pi)^2 \delta^4(k_1) \chi_v^b \chi_\kappa^c} - (2\pi)^2 \delta^4(k_1) \chi_v^b A_{\kappa+}^{c}(k_2,t) - \right. \nonumber \\
& \left. - (2\pi)^2 \delta(k_2) \chi_\kappa^c A_{v+}^{b}(k_1,t) + (2\pi)^4 \delta^4(k_1) \delta^4(k_2) \chi_v^b \chi_\kappa^c \right] \times \nonumber \\
& \times \left[ A_{\lambda+}^{d}(k-k_1-k_2,t) - (2\pi)^2 \delta^4(k-k_1-k_2) \chi_\lambda^d \right] = \nonumber \\
& \left[ \frac{A_{v+}^{b}(k_1,t) A_{\kappa+}^{c}(k_2,t)}{(2\pi)^2 \delta^4(k_1) \chi_v^b A_{\kappa+}^{c}(k_2,t)} - (2\pi)^2 \delta^4(k_1) \chi_v^b A_{\kappa+}^{c}(k_2,t) - \right. \nonumber \\
& \left. - (2\pi)^2 \delta^4(k_2) \chi_\kappa^c A_{v+}^{b}(k_1,t) + (2\pi)^4 \delta^4(k_1) \delta^4(k_2) \chi_v^b \chi_\kappa^c \right] \times A_{\lambda+}^{d}(k-k_1-k_2,t) + \nonumber \\
& \left[ \frac{A_{v+}^{b}(k_1,t) A_{\kappa+}^{c}(k_2,t)}{(2\pi)^2 \delta(k_1) \chi_v^b A_{\kappa+}^{c}(k_2,t)} - (2\pi)^2 \delta(k_1) \chi_v^b A_{\kappa+}^{c}(k_2,t) - \right. \nonumber \\
& \left. - (2\pi)^2 \delta(k_2) \chi_\kappa^c A_{v+}^{b}(k_1,t) + (2\pi)^4 \delta^4(k_1) \delta^4(k_2) \chi_v^b \chi_\kappa^c \right] \times (2\pi)^2 \delta^4(k-k_1-k_2) \chi_\lambda^d = \tag{6.1.20} \\
& = A_{v+}^{b}(k_1,t) A_{\kappa+}^{c}(k_2,t) A_{\lambda+}^{d}(k-k_1-k_2,t) - \nonumber \\
& \quad - \frac{(2\pi)^2 \delta^4(k_1) \chi_v^b A_{\kappa+}^{c}(k_2,t) A_{\lambda+}^{d}(k-k_1-k_2,t)}{(2\pi)^2 \delta(k_1) \chi_v^b A_{\kappa+}^{c}(k_2,t) A_{\lambda+}^{d}(k-k_1-k_2,t)} - \nonumber \\
& \quad + \frac{(2\pi)^2 \delta(k_2) \chi_\kappa^c A_{v+}^{b}(k_1,t) A_{\lambda+}^{d}(k-k_1-k_2,t)}{(2\pi)^4 \delta(k_1) \delta(k_2) \chi_v^b \chi_\kappa^c A_{\lambda+}^{d}(k-k_1-k_2,t)} + \nonumber \\
& \quad + \frac{(2\pi)^4 \delta(k_1) \delta(k_2) \chi_v^b \chi_\kappa^c A_{\lambda+}^{d}(k-k_1-k_2,t)}{(2\pi)^2 \delta(k-k_1-k_2) \chi_\lambda^d A_{v+}^{b}(k_1,t) A_{\kappa-}^{c}(k_2,t)} - \nonumber \\
& \quad - \frac{(2\pi)^4 \delta^4(k_1) \delta^4(k-k_1-k_2) \chi_\lambda^d \chi_\kappa^c A_{\kappa+}^{c}(k_1,t)}{(2\pi)^4 \delta^4(k_1) \delta^4(k-k_1-k_2) \chi_\lambda^d \chi_\kappa^c A_{\kappa-}^{b}(k_1,t)} - \nonumber \\
& \quad - \frac{(2\pi)^4 \delta^4(k_2) \delta^4(k-k_1-k_2) \chi_\lambda^d \chi_\kappa^c A_{v+}^{b}(k_1,t)}{(2\pi)^4 \delta^4(k_1) \delta^4(k_2) \chi_v^b \chi_\kappa^c A_{\kappa+}^{c}(k_1,t)} + \nonumber \\
& \quad + \frac{(2\pi)^6 \delta^4(k_1) \delta^4(k_2) \delta^4(k-k_1-k_2) \chi_\lambda^d \chi_v^b \chi_\kappa^c}{(2\pi)^6 \delta^4(k_1) \delta^4(k_2) \delta^4(k-k_1-k_2) \chi_\lambda^d \chi_v^b \chi_\kappa^c} \nonumber
\end{align}
$$
$$
\begin{align}
Y_{\mu+}^a(k,t) &= \nonumber \\
&= \tilde{Y}_{\mu+}^{a(1)} + \tilde{Y}_{\mu+}^{a(2)}(k,t) = \nonumber \\
&\quad \frac{g}{(2\pi)^2} \int d^4 k_1 V_{\mu\kappa\lambda}^{abc}(k,-k_1,-(k-k_1)) Y_{\mu+}^{a(1)}(k,t) + \nonumber \\
&\quad + \frac{g^2}{(2\pi)^4} \int d^4 k_1 d^4 k_2 W_{\mu\kappa\lambda}^{abcd} Y_{\mu+}^{a(2)}(k,t), \tag{6.1.21} \\
\tilde{Y}_{\mu+}^{a(1)} &= \nonumber \\
&\quad \frac{g}{(2\pi)^2} \int d^4 k_1 V_{\mu\kappa\lambda}^{abc}(k,-k_1,-(k-k_1)) Y_{\mu+}^{a(1)}(k,t) \nonumber \\
\tilde{Y}_{\mu+}^{a(2)}(k,t) &= \nonumber \\
&\quad \frac{g^2}{(2\pi)^4} \int d^4 k_1 d^4 k_2 W_{\mu\kappa\lambda}^{abcd} Y_{\mu+}^{a(2)}(k,t) \nonumber
\end{align}
$$

From Eq.(6.1.19) we get
$$
\begin{align}
\tilde{Y}_{\mu+}^{a(1)} &= \nonumber \\
&= \frac{g}{(2\pi)^2} \int d^4 k_1 V_{\mu\kappa\lambda}^{abc}(k, -k_1, -(k-k_1)) Y_{\mu+}^{a(1)}(k,t) = \nonumber \\
&\quad \frac{g}{(2\pi)^2} \int d^4 k_1 V_{\mu\kappa\lambda}^{abc}(k, -k_1, -(k-k_1)) \underline{A_{\kappa+}^b(k_1,t) A_{\lambda+}^c(k-k_1,t)} + \nonumber \\
&\quad + \frac{g}{(2\pi)^2} \int d^4 k_1 V_{\mu\kappa\lambda}^{abc}(k, -k_1, -(k-k_1)) \times \nonumber \\
&\quad \times \left[ (2\pi)^2 \delta^4(k_1) \chi_k^b A_{\lambda+}^c(k-k_1,t) + (2\pi)^4 \delta^4(k_1) \delta^4(k-k_1) \chi_k^b \chi_{\lambda}^c \right] = \nonumber \\
&= \frac{g}{(2\pi)^2} \int d^4 k_1 V_{\mu\kappa\lambda}^{abc}(k, -k_1, -(k-k_1)) \underline{A_{\kappa+}^b(k_1,t) A_{\lambda+}^c(k-k_1,t)} + \nonumber \\
&\quad -g \int d^4 k_1 V_{\mu\kappa\lambda}^{abc}(k, -k_1, -(k-k_1)) \left[ \delta(k_1) \chi_k^b A_{\lambda+}^c(k-k_1,t) + \right. \nonumber \\
&\qquad \left. + \delta^4(k-k_1) \chi_{\lambda}^c A_{\kappa+}^b(k_1,t) \right] + \tag{6.1.22} \\
&\quad + (2\pi)^2 g \int d^4 k_1 V_{\mu\kappa\lambda}^{abc}(k, -k_1, -(k-k_1)) \delta^4(k_1) \delta^4(k-k_1) \chi_k^b \chi_{\lambda}^c = \nonumber \\
&= \frac{g}{(2\pi)^2} \int d^4 k_1 V_{\mu\kappa\lambda}^{abc}(k, -k_1, -(k-k_1)) \underline{A_{\kappa+}^b(k_1,t) A_{\lambda+}^c(k-k_1,t)} - \nonumber \\
&\quad -g V_{\mu\kappa\lambda}^{abc}(k, 0, -k) \left[ \chi_k^b A_{\lambda+}^c(k,t) + \chi_{\lambda}^c A_{\kappa+}^b(k,t) \right] + \nonumber \\
&\quad + (2\pi)^2 g V_{\mu\kappa\lambda}^{abc}(k, -k, 0) \delta^4(k) \chi_k^b \chi_{\lambda}^c. \nonumber
\end{align}
$$

$$
V_{\mu\kappa\lambda}^{abc}(k, 0, -k) = \left(-\frac{i}{2}\right) f^{abc} \times [k_{\lambda}\delta_{\mu\kappa} - k_{\mu}\delta_{\kappa\lambda}]
$$

From Eq.(6.1.20) we get
$$
\begin{align}
\tilde{Y}_{\mu+}^{a(2)}(k,t) = & \nonumber \\
& = \frac{g^2}{(2\pi)^2} \int d^4 k_1 d^4 k_2 W_{\mu\nu\kappa\lambda}^{abcd} Y_{\mu-}^{a(2)}(k, k_1, k_2, t) = \nonumber \\
& \frac{g^2}{(2\pi)^4} \int d^4 k_1 d^4 k_2 W_{\mu\nu\kappa\lambda}^{abcd} \left[ \begin{aligned}[t]
& A_{\nu+}^b(k_1,t) A_{\kappa+}^c(k_2,t) A_{\lambda+}^d(k-k_1-k_2,t) + \\
& \qquad + (2\pi)^2 \delta^4(k_1) \chi_v^b A_{\kappa+}^c(k_2,t) A_{\lambda+}^d(k-k_1-k_2,t) + \\
& \qquad + (2\pi)^2 \delta^4(k_2) \chi_\kappa^c A_{\nu+}^b(k_1,t) A_{\lambda+}^d(k-k_1-k_2,t) + \\
& \qquad + (2\pi)^2 \delta^4(k-k_1-k_2) \chi_\lambda^d A_{\nu+}^b(k_1,t) A_{\kappa-}^c(k_2,t) + \\
& \qquad + (2\pi)^2 \delta^4(k-k_1-k_2) \chi_\lambda^d A_{\nu+}^b(k_1,t) A_{\kappa+}^c(k_2,t)
\end{aligned} \right] + \nonumber \\
& + \frac{g^2}{(2\pi)^4} \int d^4 k_1 d^4 k_2 W_{\mu\nu\kappa\lambda}^{abcd} \left[ \begin{aligned}[t]
& (2\pi)^4 \delta(k_1) \delta(k_2) \chi_v^b \chi_\kappa^c A_{\lambda+}^d(k-k_1-k_2,t) + \\
& \qquad - (2\pi)^4 \delta^4(k_2) \delta^4(k-k_1-k_2) \chi_\lambda^d \chi_\kappa^c A_{\kappa+}^c(k_1,t) - \\
& \qquad - (2\pi)^4 \delta^4(k_2) \delta^4(k-k_1-k_2) \chi_\lambda^d \chi_\kappa^c A_{\nu+}^b(k_1,t) + \\
& \qquad + (2\pi)^8 \delta^4(k_1) \delta^4(k_2) \delta(k-k_1-k_2) \chi_\lambda^d \chi_v^b \chi_\kappa^c
\end{aligned} \right] \tag{6.1.23}
\end{align}
$$

From Eq.(6.21)-Eq.(6.23) we obtain linear differential master equations
corresponding to $A_{\mu+}^a(k,t)$:

$$
\begin{equation}
\begin{split}
\frac{d}{dt} A_{\mu+}^{a}(k,t) = \delta(t)\delta^4(k)\chi_{\mu}^{a} & - ((k^2 + m^2)\delta_{\mu\nu} - k_{\mu}k_{\nu})[A_{\nu+}^{a}(k,t) - (2\pi)^2 m^2 \delta^4(k)\chi_{\nu}^{a}] + \\
& + \eta_{\mu}^{a}(\omega, k, t) + \tilde{Y}_{\mu+}^{\{L\}a(1)}(k,t) + \tilde{Y}_{\mu+}^{\{L\}a(2)}(k,t),
\end{split}
\tag{6.1.24}
\end{equation}
$$

where

$$
\begin{align}
\tilde{Y}_{\mu+}^{\{L\}a(1)}(k,t) = & \nonumber \\
& = g V_{\mu\kappa\lambda}^{abc}(k,0,-k) [\chi_{\kappa}^{b} A_{\lambda+}^{c}(k,t) + \chi_{\lambda}^{c} A_{\kappa+}^{b}(k,t)] + \tag{6.1.25} \\
& \quad + (2\pi)^2 g V_{\mu\kappa\lambda}^{abc}(k,-k,0) \delta^4(k) \chi_{\kappa}^{b} \chi_{\lambda}^{c}. \nonumber \\
V_{\mu\kappa\lambda}^{abc}(k,0,-k) = & \left(-\frac{i}{2}\right) f^{abc} \times [k_{\lambda} \delta_{\mu\kappa} - k_{\mu} \delta_{\kappa\lambda}] \nonumber
\end{align}
$$

and
$$
\begin{equation}
\begin{split}
\tilde{Y}_{\mu+}^{\{\mathbf{L}\}a(2)}(k,t) ={}& \\
& \frac{g^2}{(2\pi)^4} \int d^4k_1 d^4k_2 W_{\mu\nu\kappa\lambda}^{abcd} \Biggl[ (2\pi)^4 \delta^4(k_1) \delta^4(k_2) \chi_v^b \chi_\kappa^c A_{\lambda-}^d (k-k_1-k_2,t) \\
& \phantom{\frac{g^2}{(2\pi)^4} \int d^4k_1 d^4k_2 W_{\mu\nu\kappa\lambda}^{abcd} \Biggl[} -(2\pi)^4 \delta^4(k_1) \delta^4(k-k_1-k_2) \chi_\lambda^d \chi_\kappa^c A_{\kappa+}^c (k_1,t) \\
& \phantom{\frac{g^2}{(2\pi)^4} \int d^4k_1 d^4k_2 W_{\mu\nu\kappa\lambda}^{abcd} \Biggl[} -(2\pi)^4 \delta^4(k_2) \delta^4(k-k_1-k_2) \chi_\lambda^d \chi_\kappa^c A_{v+}^b (k_1,t) \\
& \phantom{\frac{g^2}{(2\pi)^4} \int d^4k_1 d^4k_2 W_{\mu\nu\kappa\lambda}^{abcd} \Biggl[} +(2\pi)^8 \delta^4(k_1) \delta^4(k_2) \delta^4(k-k_1-k_2) \chi_\lambda^d \chi_v^b \chi_\kappa^c \Biggr] ={}& \\
& -g^2 W_{\mu\nu\kappa\lambda}^{abcd} [\chi_\lambda^d \chi_\kappa^c A_{\kappa+}^c (k,t) + \chi_\lambda^d \chi_\kappa^c A_{v+}^b (k,t)] \\
& +g^2 (2\pi)^4 W_{\mu\nu\kappa\lambda}^{abcd} \delta^4(k) \chi_\lambda^d \chi_v^b \chi_\kappa^c
\end{split}
\tag{6.1.26}
\end{equation}
$$

From Eq.(6.14)-Eq.(6.16) we obtain linear differential master equations
corresponding

to $A_{\mu-}^a(k,t)$ :

$$
\begin{equation}
\begin{split}
\frac{d}{dt} A_{\mu-}^a(k,t) = -\delta(t)\delta(k)\chi_{\mu}^a & - ((k^2 + m^2)\delta_{\mu\nu} - k_{\mu}k_{\nu})[A_{\nu-}^a(k,t) + (2\pi)^2\delta^4(k)m^2\chi_{\nu}^a] + \\
& + \eta_{\mu}^a(\omega, k, t) + \tilde{Y}_{\mu-}^{ab\{L\}a(1)}(k,t) + \tilde{Y}_{\mu-}^{ab\{L\}a(2)}(k,t),
\end{split}
\tag{6.1.27}
\end{equation}
$$

where

$$
\begin{align}
\tilde{Y}_{\mu-}^{a\{L\}a(1)}(k,t) &= \nonumber \\
&= gV_{\mu\kappa\lambda}^{abc}(k,0,-k)[\chi_{\kappa}^{b}A_{\lambda-}^{c}(k,t) + \chi_{\lambda}^{c}A_{\kappa-}^{b}(k,t)] + \nonumber \\
&\quad +(2\pi)^2 gV_{\mu\kappa\lambda}^{abc}(k,-k,0)\delta^4(k)\chi_{\kappa}^{b}\chi_{\lambda}^{c}. \tag{6.1.28} \\
\tilde{Y}_{\mu-}^{a\{L\}a(2)}(k,t) &= \nonumber \\
&= g^2 W_{\mu\nu\kappa\lambda}^{abcd}[\chi_{\nu}^{b}\chi_{\kappa}^{c}A_{\lambda-}^{d}(k,t) + \chi_{\lambda}^{d}\chi_{\kappa}^{c}A_{\nu-}^{b}(k,t)] + g^2(2\pi)^4 W_{\mu\nu\kappa\lambda}^{abcd}\delta(k)\chi_{\lambda}^{d}\chi_{\nu}^{b}\chi_{\kappa}^{c}. \nonumber \\
V_{\mu\kappa\lambda}^{abc}(k,0,-k) &= \left(-\frac{i}{2}\right)f^{abc} \times [k_{\lambda}\delta_{\mu\kappa} - k_{\mu}\delta_{\kappa\lambda}]. \nonumber
\end{align}
$$

From Eq.(6.27-6.28) we obtain linear integral master equation
$$
\begin{align}
A_{\mu-}^a(k,t) = & -\chi_\mu^a \int_0^t G_{\mu\nu}^{ab}(k;t,t',m) \delta(t') \delta^4(k) dt' + (2\pi)^2 m^2 \chi_v^a \int_0^t G_{\mu\nu}^{ab}(k;t,t',m) \delta(k) dt' + \nonumber \\
& + \int_0^t G_{\mu\nu}^{ab}(k;t,t',m) \tilde{Y}_{v'-}^{b\{\mathbf{L}\}\alpha(1)}(k,t') dt' + \int_0^t G_{\mu\nu}^{ab}(k;t,t',m) \tilde{Y}_{v'-}^{b\{\mathbf{L}\}\alpha(2)}(k,t') = \nonumber \\
& + \int_0^t G_{\mu\nu}^{ab}(k;t,t',m) \tilde{Y}_{v'-}^{b\{\mathbf{L}\}\alpha(1)}(k,t') dt' + \int_0^t G_{\mu\nu}^{ab}(k;t,t',m) \tilde{Y}_{v'-}^{b\{\mathbf{L}\}\alpha(2)}(k,t') dt', \tag{6.1.29} \\
& -\chi_\mu^a \delta^4(k) G_{\mu\nu}^{ab}(k;t,0,m) + (2\pi)^2 m^2 \chi_v^a \int_0^t G_{\mu\nu}^{ab}(k;t,t',m) \delta(k) dt' + \nonumber \\
& + \int_0^t G_{\mu\nu}^{ab}(k;t,t',m) \tilde{Y}_{v'-}^{b\{\mathbf{L}\}\alpha(1)}(k,t') dt' + \int_0^t G_{\mu\nu}^{ab}(k;t,t',m) \tilde{Y}_{v'-}^{b\{\mathbf{L}\}\alpha(2)}(k,t') dt', \nonumber \\
& \qquad \qquad \qquad \qquad \qquad \qquad \tilde{Y}_{v'-}^{b\{\mathbf{L}\}\alpha(1)}(k,t') = \tilde{Y}_{\mu-}^{b\{\mathbf{L}\}\alpha(1)}(k,t') \Big|_{\mu \triangleq v'} \nonumber \\
& \qquad \qquad \qquad \qquad \qquad \qquad \tilde{Y}_{v'-}^{b\{\mathbf{L}\}\alpha(2)}(k,t') = \tilde{Y}_{\mu-}^{b\{\mathbf{L}\}\alpha(2)}(k,t') \Big|_{\mu \triangleq v'} \nonumber
\end{align}
$$

where

$$
\begin{align}
G_{\mu\nu}^{ab}(k; t, t', m) &= \delta_{ab} G_{\mu\nu}(k; t, t', m), \tag{6.1.29'} \\
G_{\mu\nu}(k; t, t', m) &= \left\{ \left( \delta_{\mu\nu} - \frac{k_\mu k_\nu}{k^2 + m^2} \right) e^{-(k^2+m^2)(t-t')} \right\}, \nonumber
\end{align}
$$

where $G_{\mu\nu}^{ab}(k;t,t') = \delta_{ab}G_{\mu\nu}(k;t,t')$ is the zeroth order Green function, see Appendix 7.
From Eq.(6.24)-Eq.(6.26) we obtain the linear integral master equation
$$
\begin{align}
A_{\mu+}^a(k,t) &= \int_0^t G_{\mu\nu}^{ab}(k,t') \delta(t') \delta^4(k) dt' - (2\pi)^2 m^2 \chi_v^a \int_0^t G_{\mu\nu}^{ab}(k,t') \delta^4(k) dt' + \nonumber \\
&\quad + \int_0^t G_{\mu\nu}^{ab}(k,t') \eta_\mu^a (\omega, k, t) dt' + \nonumber \\
&\quad + \int_0^t G_{\mu\nu'}^{ab}(k,t') \tilde{Y}_{v+}^{b\{\mathbf{L}\}\alpha(1)}(k,t') dt' + \int_0^t G_{\mu\nu'}^{ab}(k,t') \tilde{Y}_{v+}^{b\{\mathbf{L}\}\alpha(2)}(k,t') = \nonumber \\
&= \chi_\mu^a \delta^4(k) G_{\mu\nu}^{ab}(k,t,0) - (2\pi)^2 m^2 \chi_v^a \int_0^t G_{\mu\nu}^{ab}(k,t') \delta^4(k) dt' + \tag{6.1.30} \\
&\quad + \int_0^t G_{\mu\nu'}^{ab}(k,t') \tilde{Y}_{v+}^{b\{\mathbf{L}\}\alpha(1)}(k,t') dt' + \int_0^t G_{\mu\nu'}^{ab}(k,t') \tilde{Y}_{v+}^{b\{\mathbf{L}\}\alpha(2)}(k,t'), \nonumber \\
&\qquad \tilde{Y}_{v+}^{b\{\mathbf{L}\}\alpha(1)}(k,t') = \tilde{Y}_{\mu+}^{b\{\mathbf{L}\}\alpha(1)}(k,t') \Big|_{\mu \triangleq v'} \nonumber \\
&\qquad \tilde{Y}_{v+}^{b\{\mathbf{L}\}\alpha(2)}(k,t') = \tilde{Y}_{\mu+}^{b\{\mathbf{L}\}\alpha(2)}(k,t') \Big|_{\mu \triangleq v'} \nonumber
\end{align}
$$

The linear equations (6.1.29) and (6.1.30) admit exact solutions denoted by

$$
A_{\mu-}^a(k, t; \{\eta_{\mu}^a\}, \{\chi_{\mu}^a\}), A_{\mu+}^a(k, t; \{\eta_{\mu}^a\}, \{\chi_{\mu}^a\}). \quad (6.1.31)
$$

Therefore the system of the transcendental master equations reads

$$
\langle [A_{\mu-}^a(x, t; \{\eta_\mu^a\}, \{\chi_\mu^a\})] A_{\mu+}^a(x, t; \{\eta_\mu^a\}, \{\chi_\mu^a\}) \rangle_\eta = 0. \quad (6.1.32)
$$

where

$$
\begin{align}
A_{\mu-}^a(x,t; \{\eta_\mu^a\}, \{\chi_\mu^a\}) &= \mathcal{F}^{-1}[A_{\mu-}^a(k,t; \{\eta_\mu^a\}, \{\chi_\mu^a\})], \tag{6.1.33} \\
A_{\mu+}^a(x,t; \{\eta_\mu^a\}, \{\chi_\mu^a\}) &= \mathcal{F}^{-1}[A_{\mu+}^a(k,t; \{\eta_\mu^a\}, \{\chi_\mu^a\})]. \nonumber
\end{align}
$$

For a sufficiently small $g < 1$, equations (6.1.29) and (6.1.30) can be solved iteratively

as an expansion in $g$, see Appendix 3, Theorem 3.2. We define now corresponding
countable sequences $\{A_{\mu-}^{a(n)}(k, t; \{\eta_{\mu}^{a}\}, \{\chi_{\mu}^{a}\})\}_{n \in \mathbb{N}}$ and $\{A_{\mu+}^{a(n)}(k, t; \{\eta_{\mu}^{a}\}, \{\chi_{\mu}^{a}\})\}_{n \in \mathbb{N}}$ of
the
iterations by recursions
$$
\begin{align}
A_{\mu-}^{a(n+1)}(k,t; \{\eta_{\mu}^a\}, \{\chi_{\mu}^a\}) &= \nonumber \\
&\quad -\chi_{\mu}^a \delta^4(k) G_{\mu\nu}^{ab}(k,t,0) + (2\pi)^2 m^2 \chi_{v'}^a \int_0^t G_{\mu\nu'}^{ab}(k;t,t') \delta(k) dt' + \nonumber \\
&\quad + \int_0^t G_{\mu\nu'}^{ab}(k;t,t') \eta_{v'}^a (\omega,k,t) dt' + \nonumber \\
&\quad + \int_0^t G_{\mu\nu'}^{ab}(k;t,t') \tilde{Y}_{v'-}^{b\{\mathbf{L}\}\alpha(1)}(k,t';n) dt' + \int_0^t G_{\mu\nu'}^{ab}(k;t,t') \tilde{Y}_{v'-}^{b\{\mathbf{L}\}\alpha(2)}(k,t';n), \tag{6.1.34} \\
\tilde{Y}_{v'-}^{b\{\mathbf{L}\}\alpha(1)}(k,t';n) &= \nonumber \\
&= g V_{v'\kappa\lambda}^{abc}(k,0,-k) [\chi_{\kappa}^b A_{\lambda-}^c(k,t;\{\eta_{\lambda}^c\}, \{\chi_{\lambda}^c\}) + \chi_{\lambda}^c A_{\kappa-}^b(k,t;\{\eta_{\kappa}^b\}, \{\chi_{\kappa}^b\})] + \nonumber \\
&\quad +(2\pi)^2 g V_{v'\kappa\lambda}^{abc}(k,-k,0) \delta(k) \chi_{\kappa}^b \chi_{\lambda}^c. \nonumber \\
\tilde{Y}_{v'-}^{\{\mathbf{L}\}a(2)}(k,t;n) &= \nonumber \\
&= g^2 W_{\mu\nu\kappa\lambda}^{abcd} [\chi_v^b \chi_{\kappa}^c A_{\lambda-}^d(k,t) + \chi_{\lambda}^d \chi_{\kappa}^c A_{v-}^b(k_1,t)] + g^2 (2\pi)^4 W_{\mu\nu\kappa\lambda}^{abcd} \delta^4(k) \chi_{\lambda}^d \chi_v^b \chi_{\kappa}^c. \nonumber \\
V_{\mu\kappa\lambda}^{abc}(k,0,-k) &= \left(-\frac{i}{2}\right) f^{abc} \times [k_{\lambda}\delta_{\mu\kappa} - k_{\mu}\delta_{\kappa\lambda}]. \nonumber
\end{align}
$$

We set now

$$
\begin{align}
A_{\mu-}^{a(1)}(k,t; \{\eta_{\mu}^{a}\}, \{\chi_{\mu}^{a}\}) &= \nonumber \\
&= -\chi_{\mu}^{a4} \delta(k) G_{\mu\nu}^{ab}(k; t, 0, m) + (2\pi)^2 m^2 \chi_{v'}^a \int_0^t G_{\mu\nu'}^{ab}(k; t, t', m) \delta(k) dt' + \nonumber \\
&\quad + \int_0^t G_{\mu\nu'}^{ab}(k; t, t', m) \eta_{v'}^a (\omega, k, t) dt'. \tag{6.1.35} \\
G_{\mu\nu}^{ab}(k; t, t', m) &= \nonumber \\
&\quad \delta_{ab} G_{\mu\nu}(k; t, t', m), \nonumber \\
G_{\mu\nu}(k; t, t', m) &= \left\{ \left( \delta_{\mu\nu} - \frac{k_{\mu}k_{\nu}}{k^2 + m^2} \right) e^{-(k^2+m^2)(t-t')} \right\}, \nonumber
\end{align}
$$

and
$$
\begin{align}
A_{\mu+}^{a(n+1)}(k,t; \{\eta_{\mu}^{a}\}, \{\chi_{\mu}^{a}\}) &= \nonumber \\
&= \chi_{\mu}^{a} \delta^{4}(k) G_{\mu\nu}^{ab}(k;t,0,m) - (2\pi)^{2} m^{2} \chi_{\nu'}^{a} \int_{0}^{t} G_{\mu\nu'}^{ab}(k;t,t',m) \delta^{4}(k) dt' + \nonumber \\
&\quad + \int_{0}^{t} G_{\mu\nu'}^{ab}(k;t,t',m) \tilde{Y}_{\nu'}^{b\{L\}a(1)}(k,t';n) dt' + \int_{0}^{t} G_{\mu\nu'}^{ab}(k;t,t',m) \tilde{Y}_{\nu'}^{b\{L\}a(2)}(k,t';n), \nonumber \\
&\qquad \tilde{Y}_{\nu'}^{b\{L\}a(1)}(k,t';n) = \nonumber \\
&\qquad g V_{\nu'\kappa\lambda}^{abc}(k,0,-k) [\chi_{\kappa}^{b} A_{\lambda+}^{c(n)}(k,t;\{\eta_{\lambda}^{c}\},\{\chi_{\lambda}^{c}\}) + \chi_{\lambda}^{c} A_{\kappa+}^{b(n)}(k,t;\{\eta_{\kappa}^{b}\},\{\chi_{\kappa}^{b}\})] + \nonumber \\
&\qquad +(2\pi)^{2} g V_{\nu'\kappa\lambda}^{abc}(k,-k,0) \delta^{4}(k) \chi_{\kappa}^{b} \chi_{\lambda}^{c}, \tag{6.1.36} \\
V_{\nu'\kappa\lambda}^{abc}(k,0,-k) &= \left(-\frac{i}{2}\right) f^{abc} \times [k_{\lambda} \delta_{\nu'\kappa} - k_{\nu'} \delta_{\kappa\lambda}]. \nonumber
\end{align}
$$

$$
\begin{align*}
\tilde{Y}_{\nu'}^{b\{L\}a(2)}(k,t;n) &= \\
&= -g^2 W_{\nu'\kappa\lambda}^{abcd} \left[ \chi_{\lambda}^{d} \chi_{\kappa}^{c} A_{\kappa+}^{c(n)}(k,t;\{\eta_{\kappa}^{c}\},\{\chi_{\kappa}^{c}\}) + \chi_{\lambda}^{d} \chi_{\kappa}^{c} A_{\nu+}^{b(n)}(k,t;\{\eta_{\nu}^{b}\},\{\chi_{\nu}^{b}\}) \right] + \\
&\quad +g^2 (2\pi)^4 W_{\nu'\kappa\lambda}^{abcd} \delta^4(k) \chi_{\lambda}^{d} \chi_{\nu}^{b} \chi_{\kappa}^{c} \\
W_{\nu'\kappa\lambda}^{abcd} &= -\frac{1}{6} \left[ f^{abe} f^{cde} (\delta_{\nu'\kappa} \delta_{\nu\lambda} - \delta_{\nu'\lambda} \delta_{\kappa\lambda}) + \right. \\
&\qquad \left. + f^{ace} f^{bde} (\delta_{\nu'\nu} \delta_{\kappa\lambda} - \delta_{\nu'\lambda} \delta_{\nu\kappa}) + \right. \\
&\qquad \left. + f^{ade} f^{bee} (\delta_{\nu'\kappa} \delta_{\nu\lambda} - \delta_{\nu'\nu} \delta_{\kappa\lambda}) \right].
\end{align*}
$$

We set now
$$
\begin{equation}
\begin{split}
A_{\mu+}^{a(1)}(k,t; \{\eta_{\mu}^a\}, \{\chi_{\mu}^a\}) ={}& \\
& (2\pi)^2 \chi_{\mu}^a \delta^4(k) G_{\mu\nu'}^{ab}(k;t,0,m) - (2\pi)^2 m^2 \chi_{\nu'}^a \int_0^t G_{\mu\nu'}^{ab}(k;t,t',m) \delta(k) dt' + \\
& + \int_0^t G_{\mu\nu'}^{ab}(k;t,t') \eta_{\nu'}^a (\omega,k,t) dt'.
\end{split}
\tag{6.1.37}
\end{equation}
$$

$$
\begin{align*}
G_{\mu\nu'}^{ab}(k; t, t', m) &= \delta_{ab} G_{\mu\nu'}(k; t, t', m), \\
G_{\mu\nu'}(k; t, t', m) &= \left( \delta_{\mu\nu'} - \frac{k_\mu k_{\nu'}}{k^2 + m^2} \right) e^{-(k^2+m^2)(t-t')}
\end{align*}
$$

From Eq.(6.1.29)-Eq.(6.1.29') we get

$$
\begin{equation}
\begin{aligned}
A_{\mu-}^{a(1)}(k,t; \{\eta_{\mu}^{a}\}, \{\chi_{\mu}^{a}\}) = &-(2\pi)^2 \chi_{\mu}^{a} \delta^{4}(k) \delta_{ab} \left(\delta_{\mu\nu'} - \frac{k_{\mu}k_{\nu'}}{k^2+m^2}\right) e^{-(k^2+m^2)t} + \\
&+(2\pi)^2 m^2 \delta^{4}(k) \delta_{ab} \chi_{\nu'}^{a} \int_{0}^{t} \left(\delta_{\mu\nu'} - \frac{k_{\mu}k_{\nu'}}{k^2+m^2}\right) e^{-(k^2+m^2)(t-t')} dt' + \\
&+ \int_{0}^{t} G_{\mu\nu'}^{ab}(k;t,t') \eta_{\nu'}^{a}(\omega,k,t) dt' = \\
&= -(2\pi)^2 \chi_{\mu}^{a} \delta^{4}(k) \left(1 - \frac{k_{\mu}^2}{k^2+m^2}\right) e^{-(k^2+m^2)t} + \\
&+(2\pi)^2 \chi_{\mu}^{a} \delta^{4}(k) \frac{m^2}{k^2+m^2} \left(1 - \frac{k_{\mu}^2}{k^2+m^2}\right) (1-e^{-(k^2+m^2)t}) + \\
&+ \int_{0}^{t} G_{\mu\nu'}^{ab}(k;t,t') \eta_{\nu'}^{a}(\omega,k,t) dt'.
\end{aligned}
\tag{6.1.38}
\end{equation}
$$

From Eq.(6.1.37) and Eq.(6.1.29') we get
$$
\begin{equation}
\begin{aligned}
A_{\mu+}^{a(1)}(k,t; \{\eta_{\mu}^{a}\}, \{\chi_{\mu}^{a}\}) = & \\
& (2\pi)^2 \chi_{\mu}^{a} \delta^4(k) \delta_{ab} \left( \delta_{\mu\nu'} - \frac{k_{\mu}k_{\nu'}}{k^2+m^2} \right) e^{-(k^2+m^2)t} - \\
& -(2\pi)^2 m^2 \chi_{\nu'}^{a} \int_0^t G_{\mu\nu'}^{ab}(k;t,t',m) \delta^4(k) dt' + \\
& + \int_0^t G_{\mu\nu'}^{ab}(k;t,t') \eta_{\nu'}^{a}(\omega,k,t) dt'.
\end{aligned}
\tag{6.1.39}
\end{equation}
$$

From Eq.(6.1.39)-Eq.(6.1.29') we get

$$
\begin{align}
A_{\mu+}^{a(1)}(k,t; \{\eta_{\mu}^{a}\}, \{\chi_{\mu}^{a}\}) = & -(2\pi)^2 \chi_{\mu}^{a} \delta^4(k) \delta_{ab} \left( \delta_{\mu\nu'} - \frac{k_{\mu}k_{\nu'}}{k^2+m^2} \right) e^{-(k^2+m^2)t} \nonumber \\
& - (2\pi)^2 m^2 \delta^4(k) \delta_{ab} \chi_{\nu'}^{a} \int_0^t \left( \delta_{\mu\nu'} - \frac{k_{\mu}k_{\nu'}}{k^2+m^2} \right) e^{-(k^2+m^2)(t-t')} dt' + \nonumber \\
& + \int_0^t G_{\mu\nu'}^{ab}(k;t,t') \eta_{\nu'}^{a}(\omega,k,t) dt' = \nonumber \\
& = -(2\pi)^2 \chi_{\mu}^{a} \delta^4(k) \left( 1 - \frac{k_{\mu}^2}{k^2+m^2} \right) e^{-(k^2+m^2)t} \nonumber \\
& - (2\pi)^2 \chi_{\mu}^{a} \delta^4(k) \frac{m^2}{k^2+m^2} \left( 1 - \frac{k_{\mu}^2}{k^2+m^2} \right) (1-e^{-(k^2+m^2)t}) + \nonumber \\
& + \int_0^t G_{\mu\nu'}^{ab}(k;t,t') \eta_{\nu'}^{a}(\omega,k,t) dt'. \tag{6.1.40}
\end{align}
$$

From Eq.(6.1.38) by using inverse Fourier transform we get
$$
\begin{align}
A_{\mu-}^{a(1)}(x,t; \{\eta_{\mu}^a\}, \{\chi_{\mu}^a\}) &= \mathcal{F}^{-1}[A_{\mu-}^{a(1)}(k,t; \{\eta_{\mu}^a\}, \{\chi_{\mu}^a\})](x,t) = \nonumber \\
&= -\chi_{\mu}^a \int e^{ikx} \delta(k) \left(1 - \frac{k_{\mu}^2}{k^2 + m^2}\right) e^{-(k^2+m^2)t} d^4k + \nonumber \\
&\quad +\chi_{\mu}^a \int e^{ikx} \delta(k) \frac{m^2}{k^2 + m^2} \left(1 - \frac{k_{\mu}^2}{k^2 + m^2}\right) (1 - e^{-(k^2+m^2)t}) d^4k + \nonumber \\
&\quad +\mathcal{F}^{-1}\left[\int_0^t G_{\mu\nu'}^{ab}(k;t,t')\eta_{\nu'}^a(\omega,k,t)dt'\right](x,t) = \nonumber \\
&= -\chi_{\mu}^a e^{-m^2t} + \chi_{\mu}^a(1 - e^{-m^2t}) + \mathcal{F}^{-1}\left[\int_0^t G_{\mu\nu'}^{ab}(k;t,t')\eta_{\nu'}^a(\omega,k,t)dt'\right](x,t). \tag{6.1.41}
\end{align}
$$

From Eq.(6.1.40) by using inverse Fourier transform we get

$$
\begin{align}
A_{\mu+}^{a(1)}(x,t; \{\eta_{\mu}^{a}\}, \{\chi_{\mu}^{a}\}) &= \mathcal{F}^{-1}[A_{\mu+}^{a(1)}(k,t; \{\eta_{\mu}^{a}\}, \{\chi_{\mu}^{a}\})](x,t) = \nonumber \\
&= \chi_{\mu}^{a} \int e^{ikx} \delta(k) \left(1 - \frac{k_{\mu}^{2}}{k^{2} + m^{2}}\right) e^{-(k^{2} + m^{2})t} d^{4}k - \nonumber \\
&\quad -\chi_{\mu}^{a} \int e^{ikx} \delta(k) \frac{m^{2}}{k^{2} + m^{2}} \left(1 - \frac{k_{\mu}^{2}}{k^{2} + m^{2}}\right) (1 - e^{-(k^{2} + m^{2})t}) d^{4}k + \nonumber \\
&\quad +\mathcal{F}^{-1}\left[\int_{0}^{t} G_{\mu\nu'}^{ab}(k;t,t') \eta_{\nu'}^{a}(\omega,k,t) dt'\right](x,t) = \nonumber \\
&= \chi_{\mu}^{a} e^{-m^{2}t} - \chi_{\mu}^{a}(1 - e^{-m^{2}t}) + \mathcal{F}^{-1}\left[\int_{0}^{t} G_{\mu\nu'}^{ab}(k;t,t') \eta_{\nu'}^{a}(\omega,k,t) dt'\right](x,t). \tag{6.1.42}
\end{align}
$$

From Eq.(6.1.41)-Eq.(6.1.42) in the limit $t \to \infty$ we get

$$
\begin{align}
\lim_{t \to \infty} \langle A_{\mu-}^{a(1)}(x,t; \{\eta_{\mu}^a\}, \{\chi_{\mu}^a\}) A_{\nu+}^{a(1)}(x',t; \{\eta_{\mu}^a\}, \{\chi_{\mu}^a\}) \rangle_{\eta} &= \tag{6.1.43} \\
&= -(\chi_{\mu}^a)^2 + \int d^4k e^{ik(x-x')} \frac{1}{k^2+m^2} \left( \delta_{\mu\nu} - \frac{k_{\mu}k_{\nu}}{k^2+m^2} \right). \nonumber
\end{align}
$$

The system of the transcendental master equations corresponding to correlations
$\langle [A_{\mu-}^a(x,t; \{\eta_\mu^a\})] A_{\nu+}^a(x,t; \{\eta_\nu^a\}) \rangle_\eta$ reads

$$
\langle [A_{\mu-}^{a}(x,t; \{\eta_{\mu}^{a}\}, \{\chi_{\mu}^{a}\})] A_{\nu+}^{a}(x,t; \{\eta_{\nu}^{a}\}, \{\chi_{\nu}^{a}\}) \rangle_{\eta} = 0. \quad (6.1.44)
$$

where
$$
\begin{align}
A_{\mu-}^a(x,t; \{\eta_\mu^a\}, \{\chi_\mu^a\}) &= \mathcal{F}^{-1}[A_{\mu-}^a(k,t; \{\eta_\mu^a\}, \{\chi_\mu^a\})], \nonumber \\
A_{\nu-}^a(x,t; \{\eta_\nu^a\}, \{\chi_\nu^a\}) &= \mathcal{F}^{-1}[A_{\nu-}^a(k,t; \{\eta_\nu^a\}, \{\chi_\nu^a\})], \nonumber \\
A_{\mu+}^a(x,t; \{\eta_\mu^a\}, \{\chi_\mu^a\}) &= \mathcal{F}^{-1}[A_{\mu+}^a(k,t; \{\eta_\mu^a\}, \{\chi_\mu^a\})], \nonumber \\
A_{\nu+}^a(x,t; \{\eta_\nu^a\}, \{\chi_\nu^a\}) &= \mathcal{F}^{-1}[A_{\nu+}^a(k,t; \{\eta_\nu^a\}, \{\chi_\nu^a\})]. \tag{6.1.45}
\end{align}
$$

For a sufficiently small $g < 1$, Eqs.(6.1.44)-(6.1.45) can be solved iteratively as an expansion in $g$, see Appendix 3, Theorem 3.2.

From Eq.(6.1.34) and Eq.(6.1.38) we get

$$
\begin{equation}
\begin{aligned}
A_{\mu-}^{a(n+1)}(k,t; \{\eta_{\mu}^{a}\}, \{\chi_{\mu}^{a}\}) = &-(2\pi)^2 \chi_{\mu}^{a} \delta^{4}(k) \delta_{ab} \left(\delta_{\mu\nu'} - \frac{k_{\mu}k_{\nu'}}{k^2+m^2}\right) e^{-(k^2+m^2)t} + \\
&+(2\pi)^2 \chi_{\mu}^{a} \delta^{4}(k) \frac{m^2}{k^2+m^2} \left(1 - \frac{k_{\mu}^2}{k^2+m^2}\right) \left(1 - e^{-(k^2+m^2)t}\right) + \\
&+\int_{0}^{t} G_{\mu\nu'}^{ab}(k;t,t') \eta_{\nu'}^{a}(\omega,k,t) dt' + \\
&+g \int_{0}^{t} dt' G_{\mu\nu'}^{ab}(k;t,t') V_{\nu'\kappa\lambda}^{abc}(k,0,-k) \times \\
&\qquad \times \left[\chi_{\kappa}^{b} A_{\lambda-}^{c(n)}(k,t; \{\eta_{\lambda}^{c}\}, \{\chi_{\lambda}^{c}\}) + \chi_{\lambda}^{c} A_{\kappa-}^{b(n)}(k,t; \{\eta_{\kappa}^{b}\}, \{\chi_{\kappa}^{b}\})\right] + \\
&+(2\pi)^2 g \int_{0}^{t} dt' G_{\mu\nu'}^{ab}(k;t,t') V_{\nu'\kappa\lambda}^{abc}(k,-k,0) \delta(k) \chi_{\kappa}^{b} \chi_{\lambda}^{c} + \\
&+g^2 \int_{0}^{t} dt' G_{\mu\nu'}^{ab}(k;t,t') W_{\mu\nu'\kappa\lambda}^{abcd} \left\{\left[\chi_{\nu'}^{b} \chi_{\kappa}^{c} A_{\lambda-}^{d(n)}(k,t) + \chi_{\lambda}^{d} \chi_{\kappa}^{c} A_{\nu'-}^{b(n)}(k,t)\right] + \right. \\
&\qquad \left.+(2\pi)^4 \delta^4(k) \chi_{\lambda}^{d} \chi_{\nu'}^{b} \chi_{\kappa}^{c}\right\}.
\end{aligned}
\tag{6.1.46}
\end{equation}
$$

From Eq.(6.1.36) and Eq.(6.1.38) we get
$$
\begin{equation}
\begin{aligned}
A_{\mu+}^{a(n+1)}(k,t; \{\eta_{\mu}^a\}, \{\chi_{\mu}^a\}) = {}& (2\pi)^2 \chi_{\mu}^a \delta^4(k) \delta_{ab} \left(\delta_{\mu\nu'} - \frac{k_{\mu}k_{\nu'}}{k^2+m^2}\right) e^{-(k^2+m^2)t} \\
& - (2\pi)^2 \chi_{\mu}^a \delta^4(k) \frac{m^2}{k^2+m^2} \left(1 - \frac{k_{\mu}^2}{k^2+m^2}\right) (1-e^{-(k^2+m^2)t}) + \\
& + \int_0^t G_{\mu\nu'}^{ab}(k;t,t') \eta_{\nu'}^a(\omega,k,t) dt' + \\
& + g \int_0^t dt' G_{\mu\nu'}^{ab}(k;t,t') V_{\nu'\kappa\lambda}^{abc}(k,0,-k) \times \\
& \qquad \times \left[ \chi_k^b A_{\lambda-}^{c(n)}(k,t; \{\eta_{\lambda}^c\}, \{\chi_{\lambda}^c\}) + \chi_{\lambda}^c A_{\kappa-}^{b(n)}(k,t; \{\eta_{\kappa}^b\}, \{\chi_{\kappa}^b\}) \right] + \\
& + (2\pi)^2 g \int_0^t dt' G_{\mu\nu'}^{ab}(k;t,t') V_{\nu'\kappa\lambda}^{abc}(k,-k,0) \delta(k) \chi_{\kappa}^b \chi_{\lambda}^c + \\
& + g^2 \int_0^t dt' G_{\mu\nu'}^{ab}(k;t,t') W_{\mu\nu'\kappa\lambda}^{abcd} \left\{ \left[ \chi_{\nu'}^b \chi_{\kappa}^c A_{\lambda-}^{d(n)}(k,t) + \chi_{\lambda}^d \chi_{\kappa}^c A_{\nu'-}^{b(n)}(k,1,t) \right] + \right. \\
& \qquad \left. + (2\pi)^4 \delta^4(k) \chi_{\lambda}^d \chi_{\nu'}^b \chi_{\kappa}^c \right\}.
\end{aligned}
\tag{6.1.47}
\end{equation}
$$

From Eq.(6.1.46) by using iverse Fourier transform we obtain
$$
\begin{align}
A_{\mu-}^{a(n+1)}(x,t; \{\eta_{\mu}^a\}, \{\chi_{\mu}^a\}) &= \mathcal{F}^{-1}[A_{\mu-}^{a(n+1)}(k,t; \{\eta_{\mu}^a\}, \{\chi_{\mu}^a\})] = \nonumber \\
&= -\chi_{\mu}^a \int d^4 k e^{ikx} \delta^4(k) \delta_{ab} \left(\delta_{\mu\nu'} - \frac{k_{\mu}k_{\nu'}}{k^2+m^2}\right) e^{-(k^2+m^2)t} + \nonumber \\
&\quad +\chi_{\mu}^a \int d^4 k e^{ikx} \delta^4(k) \frac{m^2}{k^2+m^2} \left(1-\frac{k_{\mu}^2}{k^2+m^2}\right) (1-e^{-(k^2+m^2)t}) + \nonumber \\
&\quad +\int d^4 k e^{ikx} \int_0^t G_{\mu\nu'}^{ab}(k;t,t') \eta_{\nu'}^a(\omega,k,t) dt' + \nonumber \\
&\quad +g \int d^4 k e^{ikx} \int_0^t dt' G_{\mu\nu'}^{ab}(k;t,t') V_{\nu'\kappa\lambda}^{abc}(k,0,-k) \times \tag{6.1.48} \\
&\quad \times \left[\chi_{\kappa}^b A_{\lambda-}^{c(n)}(k,t; \{\eta_{\lambda}^c\}, \{\chi_{\lambda}^c\}) + \chi_{\lambda}^c A_{\kappa-}^{b(n)}(k,t; \{\eta_{\kappa}^b\}, \{\chi_{\kappa}^b\})\right] + \nonumber \\
&\quad +g \int d^4 k e^{ikx} \int_0^t dt' G_{\mu\nu'}^{ab}(k;t,t') V_{\nu'\kappa\lambda}^{abc}(k,-k,0) \delta^4(k) \chi_{\kappa}^b \chi_{\lambda}^c + \nonumber \\
&\quad +g^2 \int d^4 k e^{ikx} \int_0^t dt' G_{\mu\nu'}^{ab}(k;t,t') W_{\mu\nu'\kappa\lambda}^{abcd} \left\{\left[\chi_{\nu'}^b \chi_{\kappa}^c A_{\lambda-}^{d(n)}(k,t) + \chi_{\lambda}^d \chi_{\kappa}^c A_{\nu-}^{b(n)}(k_1,t)\right] + \right. \nonumber \\
&\qquad \left. +(2\pi)^2 \int d^4 k e^{ikx} \delta^4(k) \chi_{\lambda}^d \chi_{\nu'}^b \chi_{\kappa}^c \right\}. \nonumber
\end{align}
$$

From Eq.(6.1.47) by using inverse Fourier transform we obtain
$$
\begin{align}
A_{\mu+}^{a(n+1)}(x', t; \{\eta_{\mu}^a\}, \{\chi_{\mu}^a\}) &= \mathcal{F}^{-1}[A_{\mu-}^{a(n+1)}(k', t; \{\eta_{\mu}^a\}, \{\chi_{\mu}^a\})] = \nonumber \\
&= \chi_{\mu}^a \int d^4 k' e^{ik'x'} \delta^4(k') \delta_{ab} \left(\delta_{\mu\nu'} - \frac{k'_{\mu} k'_{\nu'}}{k'^2 + m^2}\right) e^{-(k'^2+m^2)t} - \nonumber \\
&\quad -\chi_{\mu}^a \int d^4 k' e^{ik'x'} \delta^4(k') \frac{m^2}{k'^2 + m^2} \left(1 - \frac{k'_{\mu}^2}{k'^2 + m^2}\right) \left(1 - e^{-(k'^2+m^2)t}\right) + \nonumber \\
&\quad + \int d^4 k' e^{ik'x'} \int_0^t G_{\mu\nu'}^{ab}(k'; t, t') \eta_{\nu'}^a(\omega, k', t) dt' + \nonumber \\
&\quad +g \int d^4 k' e^{ik'x'} \int_0^t dt' G_{\mu\nu'}^{ab}(k'; t, t') V_{\nu'\kappa\lambda}^{abc}(k', 0, -k) \times \tag{6.1.49} \\
&\quad \times \left[\chi_{\kappa}^b A_{\lambda-}^{c(n)}(k', t; \{\eta_{\lambda}^c\}, \{\chi_{\lambda}^c\}) + \chi_{\lambda}^c A_{\kappa-}^{b(n)}(k', t; \{\eta_{\kappa}^b\}, \{\chi_{\kappa}^b\})\right] + \nonumber \\
&\quad +g \int d^4 k' e^{ik'x'} \int_0^t dt' G_{\mu\nu'}^{ab}(k'; t, t') V_{\nu'\kappa\lambda}^{abc}(k', -k', 0) \delta^4(k') \chi_{\kappa}^b \chi_{\lambda}^c + \nonumber \\
&\quad + g^2 \int d^4 k' e^{ik'x'} \int_0^t dt' G_{\mu\nu'}^{ab}(k'; t, t') W_{\mu\nu'\kappa\lambda}^{abcd} \left\{\left[\chi_{\nu'}^b \chi_{\kappa}^c A_{\lambda-}^{d(n)}(k', t) + \chi_{\lambda}^d \chi_{\kappa}^c A_{\nu'-}^{b(n)}(k'_1, t)\right] + \right. \nonumber \\
&\qquad \left. + (2\pi)^2 \int d^4 k' e^{ik'x'} \delta^4(k') \chi_{\lambda}^d \chi_{\nu'}^b \chi_{\kappa}^c \right\}. \nonumber
\end{align}
$$

From Eq.(6.1.48)-Eq.(6.1.49) we obtain
$$
\begin{aligned}
& \left\langle A_{\mu-}^{a(n+1)}(x, t; \{\eta_{\mu}^a\}, \{\chi_{\mu}^a\}) A_{\mu+}^{a(n+1)}(x', t; \{\eta_{\mu}^a\}, \{\chi_{\mu}^a\}) \right\rangle_{\eta} = O(e^{-m^2 t}) - (\chi_{\mu}^a)^2 + \\
& \qquad + \left\langle \left( \int d^4 k e^{ikx} \int_0^t G_{\mu\nu'}^{ab}(k; t, t') \eta_{\nu'}^a(\omega, k, t) dt' \right) \times \right. \\
& \qquad \left. \times \left( \int d^4 k' e^{ik'x'} \int_0^t G_{\mu\nu'}^{ab}(k'; t, t') \eta_{\nu'}^a(\omega, k', t) dt' \right) \right\rangle_{\eta} + \\
& \qquad g^2 \left\langle \left( \int d^4 k e^{ikx} \int_0^t dt' G_{\mu\nu'}^{ab}(k; t, t') V_{\nu'\kappa\lambda}^{abc}(k', 0, -k) \times \right. \right. \\
& \qquad \left. \left. \left[ \chi_{\kappa}^b A_{\lambda-}^{c(n)}(k, t; \{\eta_{\lambda}^c\}, \{\chi_{\lambda}^c\}) + \chi_{\lambda}^c A_{\kappa-}^{b(n)}(k, t; \{\eta_{\kappa}^b\}, \{\chi_{\kappa}^b\}) \right] \right) \times \right. \\
& \qquad \left. \left( \int d^4 k' e^{ik'x'} \int_0^t dt' G_{\mu\nu'}^{ab}(k'; t, t') V_{\nu'\kappa\lambda}^{abc}(k', 0, -k) \times \right. \right. \\
& \qquad \left. \left. \left[ \chi_{\kappa}^b A_{\lambda+}^{c(n)}(k', t; \{\eta_{\lambda}^c\}, \{\chi_{\lambda}^c\}) + \chi_{\lambda}^c A_{\kappa+}^{b(n)}(k', t; \{\eta_{\kappa}^b\}, \{\chi_{\kappa}^b\}) \right] \right) \right\rangle_{\eta} + O(g^4).
\end{aligned}
\quad (6.1.50)
$$

By using Eq.(6.1.43) we obtain

$$
\begin{aligned}
& \left\langle A_{\mu-}^{a(n+1)}(x, t; \{\eta_{\mu}^{a}\}, \{\chi_{\mu}^{a}\}) A_{\mu+}^{a(n+1)}(x', t; \{\eta_{\mu}^{a}\}, \{\chi_{\mu}^{a}\}) \right\rangle_{\eta} = \\
&= O(e^{-m^2 t}) - (\chi_{\mu}^{a})^2 + \int d^4 k e^{ik(x-x')} \frac{1}{k^2 + m^2} \left( \delta_{\mu\nu} - \frac{k_{\mu}k_{\nu}}{k^2 + m^2} \right) + \\
& g^2 \left\langle \left( \int d^4 k e^{ikx} \int_0^t dt' G_{\mu\nu'}^{ab}(k; t, t') V_{\nu'\kappa\lambda}^{abc}(k', 0, -k) \times \right. \right. \\
& \left. \left. \left[ \chi_{\kappa}^{b} A_{\lambda-}^{c(n)}(k, t; \{\eta_{\lambda}^{c}\}, \{\chi_{\lambda}^{c}\}) + \chi_{\lambda}^{c} A_{\kappa-}^{b(n)}(k, t; \{\eta_{\kappa}^{b}\}, \{\chi_{\kappa}^{b}\}) \right] \right) \times \right. \\
& \left. \left( \int d^4 k' e^{ik'x'} \int_0^t dt' G_{\mu\nu'}^{ab}(k'; t, t') V_{\nu'\kappa\lambda}^{abc}(k', 0, -k) \times \right. \right. \\
& \left. \left. \left[ \chi_{\kappa}^{b} A_{\lambda+}^{c(n)}(k', t; \{\eta_{\lambda}^{c}\}, \{\chi_{\lambda}^{c}\}) + \chi_{\lambda}^{c} A_{\kappa+}^{b(n)}(k', t; \{\eta_{\kappa}^{b}\}, \{\chi_{\kappa}^{b}\}) \right] \right) \right\rangle_{\eta} + O(g^4).
\end{aligned}
\quad (6.1.51)
$$

Therefore the system of the transcendental master equations (6.1.44) corresponding to correlations $\langle [A_{\mu-}^a(x, t; \{\eta_\mu^a\})] A_{\mu+}^a(x, t; \{\eta_\nu^a\}) \rangle_\eta$ in the limit $t \to \infty$ reads

$$
\langle [A_{\mu-}^{a}(x, t; \{\eta_{\mu}^{a}\})] A_{\mu+}^{a}(x, t; \{\eta_{\nu}^{a}\}) \rangle_{\eta} = 0. \quad (6.1.51')
$$

Thus we get
$$
-(\chi_{\mu}^{a})^{2} + \int d^{4}k e^{ik(x-x')} \frac{1}{k^{2} + m^{2}} \left( \delta_{\mu\nu} - \frac{k_{\mu}k_{\nu}}{k^{2} + m^{2}} \right) + \\
g^{2} \left\langle \left( \int d^{4}k e^{ikx} \int_{0}^{\infty} dt' G_{\mu\nu'}^{ab}(k; t, t') V_{\nu'\kappa\lambda}^{abc}(k', 0, -k) \times \right. \right. \\
\left. \left[ \chi_{\kappa}^{b} A_{\lambda-}^{c(n)}(k, t; \{\eta_{\lambda}^{c}\}, \{\chi_{\lambda}^{c}\}) + \chi_{\lambda}^{c} A_{\kappa-}^{b(n)}(k, t; \{\eta_{\kappa}^{b}\}, \{\chi_{\kappa}^{b}\}) \right] \right\rangle \times \\
\left( \int d^{4}k' e^{ik'x'} \int_{0}^{\infty} dt' G_{\mu\nu'}^{ab}(k'; t, t') V_{\nu'\kappa\lambda}^{abc}(k', 0, -k) \times \right. \\
\left. \left[ \chi_{\kappa}^{b} A_{\lambda+}^{c(n)}(k', t; \{\eta_{\lambda}^{c}\}, \{\chi_{\lambda}^{c}\}) + \chi_{\lambda}^{c} A_{\kappa+}^{b(n)}(k', t; \{\eta_{\kappa}^{b}\}, \{\chi_{\kappa}^{b}\}) \right] \right\rangle_{\eta} + O(g^{4}) \approx 0.
$$

For a sufficiently small $g < 1$, equations (6.1.52) can be solved iteratively
as an expansion in $g$, see Appendix 3, Theorem 3.2. We define now corresponding
countable sequences $\{\chi_{\mu}^{a(n)}\}_{n \in \mathbb{N} \cup \{0\}} \triangleq \{\chi_{\mu}^{a(n)}(g)\}_{n \in \mathbb{N} \cup \{0\}}$, $\chi_{\mu}^{a(0)} \triangleq \chi_{\mu}^{a}(g=0)$ of the
iterations by the recursions:

$$
\begin{align*}
(\chi_{\mu}^{a(n+1,n+1)}(g))^2 &= \\
&= \int d^4 k e^{ik(x-x')} \frac{1}{k^2 + m^2} \left( \delta_{\mu\nu} - \frac{k_\mu k_\nu}{k^2 + m^2} \right) + \\
&\quad + g^2 \left\langle \left( \int d^4 k e^{ikx} \int_0^\infty dt' G_{\mu\nu'}^{ab}(k; t, t') V_{\nu'\kappa\lambda}^{abc}(k', 0, -k) \times \right. \right. \\
&\qquad \left. \left[ \chi_{\kappa}^{b(n)} A_{\lambda-}^{c(n)}(k, t; \{\eta_\lambda^c\}, \{\chi_\lambda^{c(n)}\}) + \chi_\lambda^{c(n)} A_{\kappa-}^{b(n)}(k, t; \{\eta_\kappa^b\}, \{\chi_\kappa^{b(n)}\}) \right] \right\rangle \times \tag{6.1.53} \\
&\quad \left( \int d^4 k' e^{ik'x'} \int_0^\infty dt' G_{\mu\nu'}^{ab}(k'; t, t') V_{\nu'\kappa\lambda}^{abc}(k', 0, -k) \times \right. \\
&\qquad \left. \left[ \chi_{\kappa}^{b(n)} A_{\lambda+}^{c(n)}(k', t; \{\eta_\lambda^c\}, \{\chi_\lambda^{c(n)}\}) + \chi_\lambda^{c(n)} A_{\kappa+}^{b(n)}(k', t; \{\eta_\kappa^b\}, \{\chi_\kappa^{b(n)}\}) \right] \right\rangle_{\eta} + \\
&\qquad + O(g^4) \approx 0.
\end{align*}
$$

Where we set

$$
(\chi_{\mu}^{a(0)})^2 = (\chi_{\mu}^{a(0)}(0))^2 = \int d^4 k e^{ik(x-x')} \frac{1}{k^2 + m^2} \left( \delta_{\mu\nu} - \frac{k_{\mu}k_{\nu}}{k^2 + m^2} \right). \quad (6.1.54)
$$
$$
\begin{align}
A_{\mu-}^{a(0)}(x,t; \{\eta_{\mu}^a\}, \{\chi_{\mu}^{a(0)}\}) &= \mathcal{F}^{-1}[A_{\mu-}^{a(0)}(k,t; \{\eta_{\mu}^a\}, \{\chi_{\mu}^{a(0)}\})] \nonumber \\
A_{\mu-}^{a(0)}(k,t; \{\eta_{\mu}^a\}, \{\chi_{\mu}^{a(0)}\}) &= -(2\pi)^2 \chi_{\mu}^a \delta^4(k) \delta_{ab} \left(\delta_{\mu\nu'} - \frac{k_{\mu}k_{\nu'}}{k^2+m^2}\right) e^{-(k^2+m^2)t} + \nonumber \\
&\quad +(2\pi)^2 \chi_{\mu}^a \delta^4(k) \frac{m^2}{k^2+m^2} \left(1-\frac{k_{\mu}^2}{k^2+m^2}\right) \left(1-e^{-(k^2+m^2)t}\right) + \tag{6.1.55} \\
&\quad + \int_0^t G_{\mu\nu'}^{ab}(k;t,t') \eta_{\nu'}^a(k,k,t) dt' \nonumber
\end{align}
$$

$$
\begin{align}
A_{\mu+}^{a(0)}(x,t; \{\eta_{\mu}^{a}\}, \{\chi_{\mu}^{a(0)}\}) &= \mathcal{F}^{-1}[A_{\mu+}^{a(0)}(k,t; \{\eta_{\mu}^{a}\}, \{\chi_{\mu}^{a(0)}\})] \nonumber \\
A_{\mu+}^{a(0)}(k,t; \{\eta_{\mu}\}, \{\chi_{\mu}^{a(0)}\}) &= (2\pi)^2 \chi_{\mu}^{a} \delta^4(k) \delta_{ab} \left(\delta_{\mu\nu'} - \frac{k_{\mu}k_{\nu'}}{k^2+m^2}\right) e^{-(k^2+m^2)t} - \nonumber \\
&\quad -(2\pi)^2 \chi_{\mu}^{a} \delta^4(k) \frac{m^2}{k^2+m^2} \left(1-\frac{k_{\mu}^2}{k^2+m^2}\right) \left(1-e^{-(k^2+m^2)t}\right) + \tag{6.1.56} \\
&\quad + \int_0^t G_{\mu\nu'}^{ab}(k;t,t') \eta_{\nu'}^{a}(k,k,t) dt'. \nonumber
\end{align}
$$

Apendix1.Two point function of the free scalar fields.
Main properties.

$$
G_{(D)F}(x_1 - x_2; m) = (2\pi)^{-4} \int d^D k \frac{e^{ik(x_1 - x_2)}}{|k|^2 + m^2}, \quad (1)
$$

where $d^D k = dk_0 dk_1 \cdots dk_{D-1}$, $|k| = (\sum_{i=0}^{D-1} k_i^2)^{1/2}$.

In spherical coordinate system with $r = |k| = (\sum_{i=0}^{3} k_i^2)^{1/2}$, $\phi_1$- angle between vector $x_1 - x_2$ and vector $k$ we obtain
$$
\begin{align}
G_{(4)F}(x_1 - x_2; m) &= (2\pi)^{-4} \int d^4 k \frac{e^{ik(x_1 - x_2)}}{|k|^2 + m^2} = \nonumber \\
&\quad (2\pi)^{-4} \int_0^\infty \int_0^\infty \int_0^\infty \int_0^\infty \frac{e^{i|x_1-x_2||r\cos\varphi_1|}}{r^2 + m^2} \times r^3 dr \sin^2\varphi_1 \sin\varphi_2 d\varphi_1 d\varphi_2 d\varphi_3 = \nonumber \\
&= \frac{1}{\pi(2\pi)^{3/2}\Gamma(3/2)} \int_0^\infty \int_0^\infty \frac{e^{i|x_1-x_2||r\cos\varphi_1|}}{r^2 + m^2} \times r^3 dr \sin^2\varphi_1 d\varphi_1. \tag{2}
\end{align}
$$

Note that

$$
\int_0^\pi e^{i|x_1-x_2||r\cos\varphi_1|} \sin^2\varphi_1 d\varphi_1 = 2\sqrt{\pi} \Gamma(3/2) \frac{J_1(r|x_1-x_2|)}{r|x_1-x_2|}, \quad (3)
$$

where $J_1(z)$ is the Bessel function of the first kind, integer order 1, (see Definition 1)
and

$$
\int_0^\infty \frac{r^2}{r^2 + m^2} J_1(r|x_1 - x_2|) dr = m K_1(m|x_1 - x_2|), \quad (4)
$$

where $K_1(z)$ is the modified Bessel function of the second kind, integer order 1, (see
Definition 2) and where we used formula 6.566.2 of [12]. Thus finally we get

$$
\begin{align}
G_{(4)F}(x_1 - x_2) &= \lim_{\tau \to \infty} \langle \phi(x_1, \tau; \omega) \phi(x_2, \tau; \omega) \rangle = \nonumber \\
&\quad (2\pi)^{-2} \int d^4 k \frac{e^{ik(x_1-x_2)}}{k^2 + m^2} = (2\pi)^{-2} \left(\frac{m}{|x|}\right) K_1(m|x_1-x_2|). \tag{5}
\end{align}
$$

where $K_1$ is the modified Bessel functions of the second kind, integer order 1.

**Definition 1.** The Bessel function of the first kind $J_1(z)$, integer order 1 is defined by the power series:

$$
J_1(z) = \sum_{k=0}^{\infty} \frac{(-1)^k}{k!(k+1)!} \left(\frac{z}{2}\right)^{2k+1}, \quad (6)
$$

where $z \in \mathbb{C}$. For $z \to 0$ we have:

$$
J_1(z) \asymp z/2. \tag{7}
$$

**Definition 2.** The modified Bessel functions of the first kind $I_v(z)$ with $v \in \mathbb{R}$ and $z \in \mathbb{C}$ are defined by the power series

$$
I_v(z) = \sum_{k=0}^{\infty} \frac{1}{\Gamma(k+1)\Gamma(k+v+1)} \left(\frac{z}{2}\right)^{2k+v}. \quad (8)
$$

The modified Bessel functions of the second kind $K_v(z)$ are defined by
$$K_v(z) = \frac{\pi}{2} \frac{I_{-v}(z) - I_v(z)}{\sin v\pi}. \qquad (9)$$

**Remark 1.** Note that

$$ 
\begin{aligned}
1. & K_v(m|x|) \approx -\ln\left(\frac{m|x|}{2}\right) - \gamma \quad \text{as } m|x| \to 0, v = 0, \\
2. & K_v(m|x|) \approx \left(\frac{2}{m|x|}\right)^v \quad \text{as } m|x| \to 0, v > 0, \\
3. & K_1(m|x|) \approx \frac{1}{m|x|} \quad \text{as } m|x| \to 0,
\end{aligned} 
\qquad (10) $$

For $z \to \infty$ with $|\arg(z)|< \pi/2$ and for any $v$ we have :

$$ 
\begin{aligned}
K_v(z) &= \frac{1}{\sqrt{2\pi z}} e^{-z}, \\
K_1(m|x|) &\approx \frac{1}{\sqrt{2\pi m|x|}} e^{-m|x|}
\end{aligned} 
$$

where $\gamma$ is the Euler–Mascheroni constant. From Eq.(5) and Eq.(10) we get

$$ 
\begin{aligned}
G_{(4)F}(x_1 - x_2) &= \lim_{\tau \to \infty} \langle \phi(x_1, \tau; \omega) \phi(x_2, \tau; \omega) \rangle = \\
&= (2\pi)^{-2} \int d^4 k \frac{e^{ik(x_1-x_2)}}{k^2 + m^2} = (2\pi)^{-2} \left(\frac{m}{|x|}\right) K_1(m|x_1-x_2|) \approx (2\pi)^{-2} |x|^{-2}
\end{aligned} 
\qquad (11) $$

<figure><img src="image_10.png" /><figcaption>Figure 3.2.1. Plot of the modified Bessel functions of the second kind, integer order 0,1,2.</figcaption></figure>

**Definition3.**

$$ 
\begin{aligned}
G_{(D)F}(x_1 - x_2) &= \lim_{\tau \to \infty} \langle \phi(x_1, \tau; \omega) \phi(x_2, \tau; \omega) \rangle = \\
&= (2\pi)^{-D} \int d^D k \frac{e^{ik(x_1-x_2)}}{k^2 + m^2}.
\end{aligned} 
\qquad (12) $$

The integrand in Eq.(12) is not radial but one can choose a frame where
$k_{\mu}x^{\mu} = -kr\cos\theta$, $k \equiv |k_{\mu}k^{\mu}| = \sqrt{\sum_{\mu=1}^{D} k_{\mu}^{2}}$, $r = |x_{\mu}x^{\mu}| = \sqrt{\sum_{\mu=1}^{D} x_{\mu}^{2}}$, $\mu = 1, 2, \dots, D$ and the angular integral reads

$$ \int d\Omega_D e^{-ik \cdot x} = \Omega_{D-1} \int_0^\pi d\theta (\sin\theta)^{D-2} e^{ikr \cos\theta} \\ = \Omega_{D-1} \sqrt{\pi} \Gamma\left(\frac{D-1}{2}\right) \left(\frac{2}{kr}\right)^{\frac{D}{2}-1} J_{\frac{D}{2}-1}(kr), \qquad (13) $$

where $\Omega_D = 2\pi^{D/2}/\Gamma(D/2)$ is the volume of the unit D-ball. In the last line we used formula 3.915.5 of [12] Then,

$$ G_{(D)}(k; m) = \Gamma\left(\frac{D}{2}\right)\left(\frac{2}{k}\right)^{\frac{D}{2}-1} \int_{0}^{+\infty} dr r^{\frac{D}{2}} [\Omega_D G(r)] J_{\frac{D}{2}-1}(kr). \qquad (14) $$

Now we take the massive propagator in momentum space:

$$ G_{(D)}(k; m) = -\frac{\Gamma\left(\frac{D}{2}\right)}{\Gamma\left(\frac{D\alpha}{2}\right)} \left(\frac{2}{k}\right)^{\frac{D}{2}-1} \left(\frac{m}{2}\right)^{\frac{D}{2}-1} \times \\ \times \int_{0}^{+\infty} dr r K_{\frac{D}{2}-1}(mr) J_{\frac{D}{2}-1}(kr) = -m^{-2} F\left(\frac{D}{2}, 1; \frac{D}{2}; -\frac{k^2}{m^2}\right) = \\ = -\frac{1}{k^2 + m^2} \qquad (15) $$

where we used formula 6.576.3 and formula 9.121.1 of [12] with $D = 4$, and $F$ is the hypergeometric function

$$ F(a, b; c; z) = \sum_{n=0}^{\infty} \frac{\Gamma(a+n)\Gamma(b+n)}{\Gamma(a)\Gamma(b)} \frac{\Gamma(c)}{\Gamma(c+n)} \frac{z^n}{n!} \qquad (16) $$

**Remark 2.** Note that when $D = 4$, we exactly obtain massive propagator $G_{(4)}(k; m)$ in momentum space, see Eqs.(1-5).

**Remark 3.** If $D \ge 5$ Definition 3 no longer holds, since integral in RHS of Eq.(12) diverges as $|k'|^{D/2-1}$, $k' \to \infty$ since the estimate (17) holds

$$ \int d^D k \frac{e^{ik(x_1 - x_2)}}{k^2 + m^2} \sim \int_0^\infty |k|^{\frac{D}{2}-2} d|k|. \qquad (17) $$

**Remark 4.** In order to avoid these difficulties mentioned above, we replace Eq.(12) in def.3 by the weak limit taken in space $\mathcal{L}'(\mathbb{R}_x^D)$, $D \ge 5$

$$ G_{(D)F}(x_1 - x_2) = w\text{-lim}_{\eta \to 0^+} G_{(D)}(k; m, \eta) = w\text{-lim}_{\eta \to 0^+} \int d^D k \frac{e^{-\eta|k|+ik(x_1-x_2)}}{k^2 + m^2} \qquad (18) \\ \eta > 0 $$

Notice the definition by Eq.(18) is correct since in space $\mathcal{L}'(\mathbb{R}_k^D)$ :

$$ w\text{-lim}_{\eta \to 0^+} e^{-\eta|k|} (k^2 + m^2)^{-1} = (k^2 + m^2)^{-1}, \qquad (19) $$
and $\int (k^2 + m^2)^{-1} d^D k < \infty$.

## Appendix 2. Correlation functions for the $(\lambda\varphi^4)_d$ scalar theory in canonical Parisi and Wu stochastic quantization.

In this section we will analyze the canonical stochastic quantization for the $(\lambda\varphi^4)_d$ self-interaction scalar theory in comparison with double stochastic quantization for the $(\lambda\varphi^4)_d$ scalar theory. In this case the Langevin equation reads

$$ \frac{\partial}{\partial \tau} \varphi(\tau, x) = -(-\Delta + m_0^2)\varphi(\tau, x) - \frac{\lambda}{3!} \varphi^3(\tau, x) + \eta(\tau, x). \quad (2.1) $$

The two-point correlation function associated with the random field is given by the Einstein relation, while the other connected correlation functions vanish, i.e., The two-point correlation function associated with the random field is given by the Einstein relation, while the other connected correlation functions vanish, i.e.,

$$ \langle \eta(\tau_1, x_1) \eta(\tau_2, x_2) \dots \eta(\tau_{2k-1}, x_{2k-1}) \rangle_{\eta} = 0, \quad (2.2) $$

and also

$$ \langle \eta(\tau_1, x_1) \dots \eta(\tau_{2k}, x_{2k}) \rangle_{\eta} = \sum \langle \eta(\tau_1, x_1) \eta(\tau_2, x_2) \rangle_{\eta} \langle \eta(\tau_k, x_k) \eta(\tau_l, x_l) \rangle_{\eta} \dots, \quad (2.3) $$

where the sum is to be taken over all the different ways in which the $2k$ labels can be divided into $k$ parts, i.e., into $k$ pairs. Performing Gaussian averages over the white random noise, it is possible to prove that perturbatively [6]

$$ \lim_{\tau \to \infty} \langle \varphi(\tau_1, x_1) \varphi(\tau_2, x_2) \dots \varphi(\tau_n, x_n) \rangle_{\eta} = \frac{\int D[\varphi] \varphi(x_1) \varphi(x_2) \dots \varphi(x_n) e^{-S(\varphi)}}{\int D[\varphi] e^{-S(\varphi)}}, \quad (2.4) $$

where $S(\varphi) = S_0(\varphi) + S_I(\varphi)$ is the $d$-dimensional action. This result leads us to consider

the Euclidean path integral measure a stationary distribution of a stochastic process. Note that the solution of the Langevin equation needs a given initial condition. As for example

$$ \varphi(\tau, x)|_{\tau=0} = \varphi_0(x). \quad (2.5) $$

Let us use the Langevin equation to perturbatively solve the interacting field theory. One way to handle the Eq.(2.1) is with the method of Green's functions. We defined the retarded Green function for the diffusion problem in the Eq.(2.1). Let us assume that the coupling constant is a small quantity. Therefore to solve the Langevin equation

in the case of an interacting theory we use a perturbative series in $\lambda$. Therefore we can write
$$ \varphi(\tau, x) = \varphi^{(0)}(\tau, x) + \lambda\varphi^{(1)}(\tau, x) + \lambda^2\varphi^{(2)}(\tau, x) + \dots \quad (2.6) $$

Substituting the Eq.(2.6) in the Eq.(2.1), and if we equate terms of equal power in $\lambda$, the resulting equations are

$$ \left[ \frac{\partial}{\partial \tau} + (-\Delta_x + m_0^2) \right] \varphi^{(0)}(\tau, x) = \eta(\tau, x), \quad (2.7) $$

$$ \left[ \frac{\partial}{\partial \tau} + (-\Delta_x + m_0^2) \right] \varphi^{(1)}(\tau, x) = -\frac{1}{3!} (\varphi^{(0)}(\tau, x))^3, \quad (2.8) $$

$$ \left[ \frac{\partial}{\partial \tau} + (-\Delta_x + m_0^2) \right] \varphi^{(1)}(\tau, x) = -\frac{1}{3!} (\varphi^{(0)}(\tau, x))^3, \quad (2.9) $$

and so on. Using the retarded Green function and assuming that $\varphi^{(q)}(\tau, x)|_{\tau=0} = 0$, $\forall q$, the solution to the first equation given by Eq.(2.7) can be written formally as

$$ \varphi^{(0)}(\tau, x) = \int_0^\tau d\tau' \int_\Omega d^d x' G(\tau - \tau', x - x') \eta(\tau', x'). \quad (2.10) $$

The second equation given by Eq.(2.8) can also be solved using the above result. We obtain

$$ \varphi^{(1)}(\tau, x) = -\frac{1}{3!} \int_0^\tau d\tau_1 \int_\Omega d^d x_1 G(\tau - \tau_1, x - x_1) + \left( \int_0^{\tau_1} d\tau' \int_\Omega d^d x' G(\tau_1 - \tau', x_1 - x') \eta(\tau', x') \right)^3. \quad (2.11) $$

We have seen that we can generate all the tree diagrams with the noise field contributions. We can also consider the n-point correlation function. Substituting the above results in the n-point correlation

function, and taking the random averages over the white noise field using the Wick-decomposition property defined by Eq.(2.3) we generate the stochastic diagrams.

Each of these stochastic diagrams has the form of a Feynman diagram, apart from the

fact that we have to take into account that we are joining together two white random noise fields many times.

As simple examples let us show how to derive the two-point function in the zeroth order

$\langle \varphi(\tau_1, x_1)\varphi(\tau_2, x_2)\dots\varphi(\tau_n, x_n) \rangle_\eta$. and also the first order correction to the scalar two-point-function

given by $\langle \varphi(\tau_1, x_1)\varphi(\tau_2, x_2) \rangle_\eta^{(1)}$. Using the Eq.(2.10) and the Einstein relations we have

$$ \langle \varphi(\tau_1, x_1)\varphi(\tau_2, x_2) \rangle_\eta^{(0)} = 2 \int_0^{\min(\tau_1, \tau_2)} d\tau' \int_\Omega d^d x' G(\tau_1 - \tau', x_1 - x') G(\tau_2 - \tau', x_2 - x'), \quad (2.12) $$
which is just Eq.(2.13). For the first order correction we get:

$$ \langle \varphi(X_1)\varphi(X_2) \rangle_{\eta}^{(1)} = -\frac{\lambda}{3!} \langle \int dX_3 \int dX_4 (G(X_1-X_4)G(X_2-X_3) + G(X_1-X_3)G(X_2-X_4))\eta(X_3) (\int dX_5 G(X_4-X_5)\eta(X_5))^3 \rangle_{\eta}. \quad (2.13) $$

where, for simplicity, we have introduced a compact notation:

$$ \int_0^\tau d\tau \int_\Omega d^d x \equiv \int dX, \quad (2.14) $$

and also $\varphi(\tau, x) \equiv \varphi(X)$ and finally $\eta(\tau, x) \equiv \eta(X)$.

The process can be repeated and therefore the stochastic quantization can be used as an alternative approach to describe scalar quantum fields. We stress here that the

stochastic quantization is based in the fact that although one starts with the system out of equilibrium, the Markovian Langevin equation forces it into equilibrium.

Moreover, when the thermodynamic equilibrium is reached, the stochastic expectation

values will coincide with the Schwinger functions of the Euclidean field theory.

<figure><img src="image_11.png" /><figcaption>Figure1:Perturbative expansion for the scalar field where crosses denote noise fields.</figcaption></figure>

$$ \langle \varphi \varphi \rangle_{\eta} = \frac{\times}{\tau_1 \tau_2} + \frac{\times}{\tau_1 \tau_2} + \frac{\times}{\tau_1 \tau_2} \quad (a) \qquad (b) \qquad (c) $$

<caption>Figure2:The corrections up to one-loop to the two-point correlation function.</caption>
We can represent Eq.(6) graphically as figure (1) (the random noise field is represented by a cross). Using this diagrammatical expansion, it is possible to show that the two-point correlation function up to one-loop level is given by figure (2), where we represent the retarded Green function by a line and the free two-point function by a crossed line. The rules to obtain the algebraic values of the stochastic diagrams are similar to the usual Feynman rules. For instance the two-point function at one-loop level is given by

$$ (b) = -\frac{\lambda}{2} \delta^d(k_1 + k_2) \int d^d k \int_0^{\tau_1} d\tau G(k_1; \tau_1 - \tau) D(k; \tau, \tau) D(k_2; \tau_2, \tau). \quad (2.15) $$

$$ (c) = -\frac{\lambda}{2} \delta^d(k_1 + k_2) \int d^d k \int_0^{\tau_2} d\tau G(k_2; \tau_2 - \tau) D(k; \tau, \tau) D(k_1; \tau_1, \tau). \quad (2.16) $$

A simple computation shows that we recover the correct equilibrium result at equal asymptotic Markov parameters ($\tau_1 = \tau_2 \to \infty$):

$$ (b)|_{\tau_1=\tau_2 \to \infty} = -\frac{\lambda}{2} \delta^d(k_1+k_2) \frac{1}{(k_2^2+m_0^2)} \frac{1}{(k_1^2+k_2^2+2m_0^2)} \int \frac{d^d k}{(k^2+m_0^2)}. \quad (2.17) $$

Obtaining the Schwinger functions in the asymptotic limit does not guarantee that we

gain a finite physical theory. The next step is to implement a suitable regularization scheme. A crucial point to find a satisfactory regularization scheme is to use one that preserves the symmetries of the original model. In the stochastic regularization method the symmetries of the physical theory is maintained. There are in general two different ways to implement the stochastic regularization. The first one is to start from a Langevin equation with a memory kernel. It is known from the literature [9] that this method can at best only remove two degrees of divergence. Another possibility is to smear only the noise field in the probability functional [10]-[11]:

$$ \langle F[\varphi] \rangle_{\eta} = \frac{\int D[\eta] F[\varphi] \exp \left[ -\frac{1}{4} \int d^d x \int d\tau \int d\tau' \eta(\tau, x) K_{\Lambda}^{-1} \eta(\tau', x) \right]}{\int D[\eta] \exp \left[ -\frac{1}{4} \int d^d x \int d\tau \int d\tau' \eta(\tau, x) K_{\Lambda}^{-1} \eta(\tau', x) \right]}, \quad (2.18) $$

where $K_\Lambda$ is a memory kernel. In this case we change the Einstein relations of the noise

field to:

$$ \langle \eta(\tau, x) \eta(\tau', x') \rangle_{\eta} = 2 K_{\Lambda}(\tau, \tau'^d(x-x')). \quad (2.19) $$

The smearing function should be chosen such that, when $\Lambda \to \infty$:

$$ \lim_{\Lambda \to \infty} K_{\Lambda}(\tau - \tau') = \delta(\tau - \tau'), \quad (2.20) $$

recovering the usual theory.

Since the Langevin equation is unaffected by the stochastic regularization, the physical field is the same as in the regularized case. However, the zeroth-order two-point correlation function is given by:
$$ D(k; \tau, \tau') = 2\delta^d(k+k') \int_0^\tau ds \int_0^{\tau'} ds' G(k; \tau-s) G(k; \tau'-s') K_\Lambda(s-s') = \\ 2\delta^d(k+k') \int_0^\tau ds \int_0^{\tau'} ds' \exp[-(\tau+\tau'-s-s'^2+m_0^2)] K_\Lambda(s-s'). \qquad (2.21) $$

It is possible to prove that a necessary condition that the regularization function $K_\Lambda$ should satisfy in order to render the divergent loops finite is $K_\Lambda(\tau) |_{\tau=0} = 0$. The following series of kernels obeying this condition were proposed:

$$ K_{\Lambda}^{(n)}(\tau) = \frac{1}{2n!} \Lambda^2 (\Lambda^2 | \tau |)^n \exp[-\Lambda^2 | \tau |]. \qquad (2.22) $$

For the case $n = 0$ we obtain, for the free two-point correlation function:

$$ \lim_{\tau \to \infty} D(k; \tau, \tau) = \frac{\delta^d(k+k')}{(k^2+m_0^2)} \frac{\Lambda^2}{(\Lambda^2+k^2+m_0^2)}. \qquad (2.23) $$

Since the stochastic diagrams contains crossed lines in its loops, we have that the ultraviolet divergences can be regularized choosing an appropriate $n$. Note that it is possible to use a different regulator of the type $K_\sigma(\tau) = \frac{1}{2} \sigma \tau^{\sigma-1}$.

We now return to $\phi_4^4$ theory. To first order in $\lambda$, and in momentum space, the two point function in Euclidean space is

$$ G(p, -p) = \frac{1}{p^2 + m^2} + \frac{-\lambda}{2} \left( \int \frac{d^4 q}{(2\pi)} \frac{1}{p^2 + m^2} \right) \left( \frac{1}{p^2 + m^2} \right) + O(\lambda^2). \qquad (2.24) $$

Let us define now by $\mu_{\text{ren}}^2$ the effective or renormalized mass (squared) such that

$$ \frac{1}{p^2 + \mu_{\text{ren}}^2} = \frac{1}{p^2 + m^2} \left\{ 1 - \frac{\lambda}{2} \left( \int \frac{d^4 q}{(2\pi)} \frac{1}{p^2 + m^2} \right) \left( \frac{1}{p^2 + m^2} \right) + O(\lambda^2) \right\}. \qquad (2.25) $$

Again, to first order in $\lambda$, we can write the equivalent equation

$$ G(p, -p) = \frac{1}{p^2 + m^2 + \frac{\lambda}{2} \int \frac{d^4 q}{(2\pi)} \frac{1}{p^2 + m^2}} + O(\lambda^2). \qquad (2.26) $$

This equation leads us to define $\mu_{\text{ren}}^2$ by

$$ \mu_{\text{ren}}^2 = m^2 + \frac{\lambda}{2} \int \frac{d^4 q}{(2\pi)} \frac{1}{p^2 + m^2} = m^2 + \delta m^2, \qquad (2.27) $$

and the quantity $\mu_{\text{ren}}$ represents the physical (or renormalized) mass of the particle. Thus, the interaction changes the mass of the particle. However $\delta m^2 = \infty$ since that

$$ \int \frac{d^4 q}{(2\pi)} \frac{1}{p^2 + m^2} \approx \int_0^\infty \frac{q^3 dq}{q^2} = \int_0^\infty q dq = \infty. \qquad (2.28) $$

This is an example of a typical “ultraviolet” divergence in canonical perturbative expansion in quantum field theory.

## Appendix. 3.Generalized Banach fixed-point theorem.
**Theorem 3.1.** [15] Let $(X, d)$ be a complete metric space and $T: X \to X$ be a contraction

mapping with the Lipschitz constant $\alpha \in [0, 1)$. Then:

(1) $T$ has a unique fixed point $\bar{x}$ in $X$.

(2) For an arbitrary point $x_0 \in X$, the sequence $(x_n)_{n \in \mathbb{N}}$ generated by the Picard iteration

process (defined by $x_{n+1} = T(x_n), n \in \mathbb{N} \cup \{0\}$) converges to $\bar{x}$.

(3) $d(x_n, \bar{x}) \le \frac{\alpha^n}{1-\alpha} d(x_0, x_1)$ for all $n \in \mathbb{N}$.

**Definition 3.1.** [15] We say that $\bar{x} \in X$ is a contractive fixed point (abbr. CFP) of $T$ if $\bar{x} = T(\bar{x})$ and the Picard iterates $T_n(x)$ converge to $\bar{x}$ as $n \to \infty$ for all $x \in X$.

The operator $T$ from Definition 3.1 became known as a Picard operator (shortly PO),

**Definition 3.2.** We say that $\bar{x} \in X$ is a quasy contractive fixed point (abbr. QCFP) of $T$

if $\bar{x} = T(\bar{x})$ and there is an $x_0 \in X$ such that the Picard iterates $T_n(x_0)$ converge to $\bar{x}$ as $n \to \infty$.

**Definition 3.3.** The operator $T$ from Definition 3.2 became known as a quasy Picard operator (abbr. QPO)

**Theorem 3.2.** (Generalized Banach fixed-point theorem). Let $(X, d)$ be a complete metric

space and $T: X \to X$ is continuous. Assume that there is an $y_0 \in X$ such that the inequalities are satisfied

$$ 
\begin{align} 
d(y_1, y_0) &= d(T(y_0), y_0) \le \alpha \times \text{const}, \notag \\ 
d(y_2, y_1) &= d(T(y_1), T(y_0)) \le \alpha d(y_1, y_0), \notag \\ 
d(y_3, y_2) &= d(T(y_2), T(y_1)) \le \alpha d(y_2, y_1), \tag{3.1} \\ 
\multicolumn{4}{l}{\text{................................}} \notag \\ 
d(y_{n+1}, y_n) &= d(T(y_n), T(y_{n-1})) \le \alpha d(y_n, y_{n-1}), \notag \\ 
\multicolumn{4}{l}{\text{................................}} 
\end{align} 
$$

where $\alpha < 1$, $y_n = T(y_{n-1})$. Then there is a quasy contractive fixed point $\bar{x}$ of $T$, such that the Picard iterates $T_n(y_0)$ converge to $\bar{y} \in X$ as $n \to \infty$.

**Proof.** From inequalities (3.1) we obtain

$$ 
\begin{align} 
d(y_1, y_0) &= d(T(y_0), y_0) \le \alpha \times \text{const}, \notag \\ 
d(y_2, y_1) &= d(T(y_1), T(y_0)) \le \alpha d(y_1, y_0), \notag \\ 
d(y_3, y_2) &= d(T(y_2), T(y_1)) \le \alpha d(y_2, y_1) \le \alpha^2 d(y_1, y_0), \tag{3.2} \\ 
d(y_{n+1}, y_n) &= d(T(y_n), T(y_{n-1})) \le \alpha d(y_n, y_{n-1}) \le \alpha^n d(y_1, y_0), \notag \\ 
\multicolumn{4}{l}{\text{................................}} 
\end{align} 
$$
From inequalities (3.2) by using triangle inequality we obtain for some $\epsilon \ll 1$

$$
\begin{align}
d(y_n, y_{n+m}) &\le d(y_n, y_{n+1}) + d(y_{n+1}, y_{n+2}) + \dots + d(y_{n+m-1}, y_{n+m}) \le \nonumber \\
&\le (\alpha^n + \alpha^{n+1} + \dots + \alpha^{n+m-1}) d(y_1, y_0) = \alpha^n \frac{1 - \alpha^{n+m}}{1 - \alpha} d(y_1, y_0) < \tag{3.3} \\
&< \frac{\alpha^n}{1 - \alpha} d(y_1, y_0) < \epsilon. \nonumber
\end{align}
$$

Thus a sequence {$y_n$}$_{n=0}^\infty \subset X$ is a fundamental sequence in $X$ and there is exists $\lim_{n \to \infty} y_n = \bar{y} \in X$. We assume now that $T(\bar{y}) = \bar{\bar{y}}$, by using triangle inequality we obtain

$$
d(\bar{y}, \bar{\bar{y}}) \le d(\bar{y}, y_n) + d(y_n, y_{n+1}) + d(y_{n+1}, \bar{\bar{y}}). \quad (3.4)
$$

For any $\epsilon > 0$ we can choose $N = N(\epsilon)$ such that for any $n \ge N$

(i) $d(\bar{y}, y_n) < \epsilon/3$, since that is $\bar{y} = \lim_{n \to \infty} y_n$;

(ii) $d(y_n, y_{n+1}) < \epsilon/3$, since that {$y_n$}$_{n=0}^\infty \subset X$ is a fundamental sequence;

(iii) $d(y_{n+1}, \bar{\bar{y}}) = d(T(y_n), T(\bar{y})) \le \epsilon/3$, since that is $\lim_{n \to \infty} T(y_n) = T(\lim_{n \to \infty} y_n) = T(\bar{y})$.

Therefore for any $\epsilon$ such that $0 < \epsilon < \epsilon : d(\bar{y}, \bar{\bar{y}}) \le \epsilon$ it follows that $d(\bar{y}, \bar{\bar{y}}) = 0$ and consequently $\bar{y} = \bar{\bar{y}}$.

**Apendix. 4.1.Integration over random interval.**

**Definition 4.1.1.** Let $q_1(\omega)$ and $q_2(\omega)$ are $\mathbb{R}$- valued random variables defined on a probability space $\Sigma = (\Omega, \mathcal{F}, \mathbf{P})$, i.e., $q_{1,2}(\omega) : \Omega \to \mathbb{R}$. Assume that a.s.$-\infty < q_1(\omega) < q_2(\omega) < \infty$.

Let $\Xi(\Sigma, \mathbb{R})$ be a set of the all $\mathbb{R}$- valued random variables defined on a probability space

$$
\begin{equation}
\begin{aligned}
\Sigma &= (\Omega, \mathcal{F}, \mathbf{P}). \text{ Closed random interval } [q_1(\omega), q_2(\omega)] \text{ that is a subset} \\
       &\qquad [q_1(\omega), q_2(\omega)] \subset \Xi(\Sigma, \mathbb{R}) \text{ such that } -\infty < q_1(\omega) < q_2(\omega) < \infty \text{ and} \\
       &\qquad \forall q(\omega) \{q(\omega) \in [q_1(\omega), q_2(\omega)] \Leftrightarrow \{q(\omega) \in \Xi(\Sigma, \mathbb{R}) | \text{a.s. } (q_1(\omega) < q(\omega) < q_2(\omega))\}\}
\end{aligned}
\tag{4.1.1}
\end{equation}
$$

**Definition 4.1.2.** The lengths $l([q_1(\omega), q_2(\omega)])$ of the random interval $[q_1(\omega), q_2(\omega)]$ is defined by

$$
l([q_1(\omega), q_2(\omega)]) = \sup q_3(\omega), \quad (4.1.2)
$$

where $q_3(\omega) = q_2(\omega) - q_1(\omega)$.

**Notation 4.1.2.** Assume that a.s. $q_1(\omega) < q_2(\omega)$. Then we will write: $q_1(\omega) < q_2(\omega)$.
Assume that a.s. $q_1(\omega) \le q_2(\omega)$. Then we will write: $q_1(\omega) \leq q_2(\omega)$.

**Definition 4.1.3.** A partition $P$ of a closed random interval $[q_1(\omega), q_2(\omega)]$ is a finite system of random variables $x_0(\omega), \dots, x_n(\omega)$ such that

$$
q_1(\omega) = x_0(\omega) < x_1(\omega) < \dots < x_{n-1}(\omega) < x_n(\omega) = q_2(\omega).
$$

The closed random intervals $[x_{i-1}(\omega), x_i(\omega)]$, ($i = 1, \dots, n$) are called the intervals of the partition $P$.

**Definition 4.1.4.** The largest of the lengths of the intervals of the partition $P$, denoted $\lambda(P)$, is called the *mesh* of the partition.
**Definition 4.1.5.** We speak of a partition with distinguished points $(P, \xi)$ on the closed random interval $[q_1(\omega), q_2(\omega)]$ if we have a partition $P$ of $[q_1(\omega), q_2(\omega)]$ and a point $\xi_1(\omega) \in [x_{i-1}(\omega), x_i(\omega)]$ has been chosen in each of the intervals of the partition $[x_{i-1}(\omega), x_i(\omega)]$, $(i = 1, \dots, n)$.

We denote the set of points $(\xi_1(\omega), \dots, \xi_n(\omega))$ by the single letter $\xi(\omega)$.

**Definition 4.1.6.** In the set $\tilde{P}$ of partitions with distinguished points on a given random

interval $[q_1(\omega), q_2(\omega)]$, we consider the following base

$\mathbf{B} = \{B_d\}$. The element $B_d, d > 0$, of the base $\mathbf{B}$ consists of all partitions with distinguished points $(P, \xi)$ on $[q_1(\omega), q_2(\omega)]$ for which $\lambda(P) < d$.

**Proposition 4.1.1.** Let us verify that $\{B_d\}, d > 0$ is actually a base in $\tilde{P}$.

**Proof.** First $B_d \neq \emptyset$. In fact, for any number $d > 0$, it is obvious that there exists a partition $P$ of $[q_1(\omega), q_2(\omega)]$ with mesh $\lambda(P) < d$ (for example, a partition into $n$ congruent closed random intervals). But then there also exists a partition $(P, \xi(\omega))$ with

distinguished points for which $\lambda(P) < d$.

Second, if $d_1 > 0, d_2 > 0$, and $d = \min\{d_1, d_2\}$, it is obvious that $B_{d_1} \cap B_{d_2} = B_d \in \mathbf{B}$. Hence $\mathbf{B} = \{B_d\}$ is indeed a base in $\tilde{P}$.

**Definition 4.1.7. (Random Riemann Sums)** (i) If a function $f: \Xi(\Sigma, \mathbb{R}) \to \Xi(\Sigma, \mathbb{R})$ is defined on the closed random interval $[q_1(\omega), q_2(\omega)]$ and $(P, \xi(\omega))$ is a partition with distinguished points on this closed random interval, the sum

$$ \sigma[f, P, \xi(\omega)] = \sum_{i=1}^{n} f(\xi_i(\omega)) \Delta x_i(\omega), \quad (4.1.3) $$

where $\Delta x_i(\omega) = x_i(\omega) - x_{i-1}(\omega)$, is the random Riemann sum of the function $f: \Xi(\Sigma, \mathbb{R}) \to \Xi(\Sigma, \mathbb{R})$ corresponding to the partition $(P, \xi(\omega))$ with distinguished points on $[q_1(\omega), q_2(\omega)]$. Thus, when the function $f$ is fixed, the random Riemann sum $\sigma[f, P, \xi(\omega)]$ is a function $\Phi(p) = \sigma[f, P]$ on the set $\tilde{P}$ of all partitions $p = (P, \xi(\omega))$ with distinguished points on the closed interval $[q_1(\omega), q_2(\omega)]$.

Since there is a base $\mathbf{B}$ in $\tilde{P}$, one can ask about the limit of the function $\Phi(p(\omega))$ over that base.

(ii) If a function $f: \Omega \times \Xi(\Sigma, \mathbb{R}) \to \Xi(\Sigma, \mathbb{R})$ is defined on the closed random interval $[q_1(\omega), q_2(\omega)]$ and $(P, \xi(\omega))$ is a partition with distinguished points on this closed random interval, the sum

$$ \sigma[f, P, \xi(\omega)] = \sum_{i=1}^{n} f(\omega, \xi_i(\omega)) \Delta x_i(\omega), \quad (4.1.3') $$

where $\Delta x_i(\omega) = x_i(\omega) - x_{i-1}(\omega)$, is the random Riemann sum of the function $f: \Omega \times \Xi(\Sigma, \mathbb{R}) \to \Xi(\Sigma, \mathbb{R})$ corresponding to the partition $(P, \xi(\omega))$ with distinguished points on $[q_1(\omega), q_2(\omega)]$. Thus, when the function $f: \Omega \times [q_1(\omega), q_2(\omega)] \to \Xi(\Sigma, \mathbb{R})$ is fixed, the random Riemann sum $\sigma[f, P, \xi(\omega)]$ is a function $\Phi(p) = \sigma[f, P]$ on the set $\tilde{P}$ of

of
all partitions $p = (P, \xi(\omega))$ with distinguished points on the closed interval $[q_1(\omega), q_2(\omega)]$.

**Definition 4.1.8. (Riemann Integral on a random interval)**

(i) Let $f: \Xi(\Sigma, \mathbb{R}) \to \Xi(\Sigma, \mathbb{R})$ be a function restricted on a closed random interval $[q_1(\omega), q_2(\omega)]$.

The random variable $I(\omega)$ is the Riemann integral of the function

$f: \Xi(\Sigma, \mathbb{R}) \to \Xi(\Sigma, \mathbb{R})$ on the closed random interval $[q_1(\omega), q_2(\omega)]$ if for every $\varepsilon > 0$ there exists $\delta > 0$ such that

$$ a.s.: \left| I(\omega) - \sum_{i=1}^{n} f(\xi_i(\omega)) \Delta x_i(\omega) \right| < \varepsilon \quad (4.1.4) $$

for any partition $(P, \xi(\omega))$ with distinguished points on $[q_1(\omega), q_2(\omega)]$ whose mesh $\lambda(P)$

is less than $\delta$.

(ii) Let $f: \Omega \times \Xi(\Sigma, \mathbb{R}) \to \Xi(\Sigma, \mathbb{R})$ be a function restricted on a closed random interval $[q_1(\omega), q_2(\omega)]$ such that $f: \Omega \times [q_1(\omega), q_2(\omega)] \to \Xi(\Sigma, \mathbb{R})$.

The random variable $I(\omega)$ is the Riemann integral of the function

$f: \Omega \times \Xi(\Sigma, \mathbb{R}) \to \Xi(\Sigma, \mathbb{R})$ on the closed random interval $[q_1(\omega), q_2(\omega)]$ if for every $\varepsilon > 0$

there exists $\delta > 0$ such that

$$ a.s.: \left| I(\omega) - \sum_{i=1}^{n} f(\omega, \xi_i(\omega)) \Delta x_i(\omega) \right| < \varepsilon. \quad (4.1.4') $$

Since the partitions $p = (P, \xi(\omega))$ for which $\lambda(P) < \delta$ form the element $B_\delta$

of the base $\mathbf{B}$ introduced above in the set $\tilde{\mathbb{P}}$ of partitions with distinguished points, Definition 4.1.8 is equivalent to the statement

$$ a.s.: I(\omega) = \lim_{\mathbf{B}} \Phi(p(\omega)). \quad (4.1.5) $$

that is, the integral $I(\omega)$ is the limit over $\mathbf{B}$ of the Riemann sums of the function $f$ corresponding to partitions with distinguished points on $[q_1(\omega), q_2(\omega)]$.

It is natural to denote the base $\mathbf{B}$ by $\lambda(P) \to 0$, and then the definition of the Riemann

integral on a random interval can be rewritten as

$$ a.s.: I(\omega) = \lim_{\lambda(P) \to 0} \sum_{i=1}^{n} f(\xi_i(\omega)) \Delta x_i(\omega), \quad (4.1.6) $$

$$ a.s.: I(\omega) = \lim_{\lambda(P) \to 0} \sum_{i=1}^{n} f(\omega, \xi_i(\omega)) \Delta x_i(\omega). \quad (4.1.6) $$

**Notation 4.1.2.** The integral of $f(x(\omega)) (f(\omega, x(\omega)))$ over $[q_1(\omega), q_2(\omega)]$ is denoted
$$ 
\begin{aligned}
& \int_{q_1(\omega)}^{q_2(\omega)} f(x(\omega)) d[x(\omega)], \\
& \int_{q_1(\omega)}^{q_2(\omega)} f(\omega, x(\omega)) d[x(\omega)],
\end{aligned}
\qquad (4.1.7)
$$

in which the random variables $q_1(\omega)$ and $q_2(\omega)$ are called respectively the lower and upper limits of integration. The function $f$ is called the integrand, $f(x(\omega))d[x(\omega)]$ ($f(\omega, x(\omega))d[x(\omega)]$) is called the differential form, and $x(\omega)$ is the random variable of integration. Thus

$$ 
\begin{aligned}
a.s.: \quad & \int_{q_1(\omega)}^{q_2(\omega)} f(x(\omega)) d[x(\omega)] = \lim_{\lambda(P) \to 0} \sum_{i=1}^{n} f(\xi_i(\omega)) \Delta x_i(\omega), \\
a.s.: \quad & \int_{q_1(\omega)}^{q_2(\omega)} f(\omega, x(\omega)) d[x(\omega)] = \lim_{\lambda(P) \to 0} \sum_{i=1}^{n} f(\omega, \xi_i(\omega)) \Delta x_i(\omega).
\end{aligned}
\qquad (4.1.8)
$$

**Definition 4.1.9.** A function $f: \Omega \times \Xi(\Sigma, \mathbb{R}) \to \Xi(\Sigma, \mathbb{R})$ is Riemann integrable on the closed interval $[q_1(\omega), q_2(\omega)]$ if the limit of the Riemann sums in (3.1.8) exists a.s. as $\lambda(P) \to 0$ (that is, the Riemann integral of $f$ is defined).

**Notation 4.1.3.** The set of Riemann-integrable functions on a closed random interval $[q_1(\omega), q_2(\omega)]$ will be denoted $\mathfrak{R}[q_1(\omega), q_2(\omega)]$

By the definition of the integral (Definition 4.1.8) and its reformulation in the forms (4.1.5) and (4.1.8), an integral is the limit of a certain special function

$\Phi(p(\omega)) = \sigma[f, P, \xi(\omega)]$ the random Riemann sum, defined on the set $\tilde{P}$ of partitions $p(\omega) = (P, \xi(\omega))$ with distinguished points on $[q_1(\omega), q_2(\omega)]$. This limit is taken with respect to the base $\mathbf{B}$ in $\tilde{P}$ that we have denoted $\lambda(P) \to 0$.

Thus the integrability or nonintegrability of a function $f$ on $[q_1(\omega), q_2(\omega)]$ depends on the existence of this limit. By the Cauchy criterion, this limit exists a.s., if and only if for every $\varepsilon > 0$ there exists an element $B_\delta \in \mathbf{B}$ in the base such that

$$ a.s.: |\Phi(p'(\omega)) - \Phi(p''(\omega))| < \varepsilon \qquad (4.1.9) $$

for any two points $p'(\omega), p''(\omega)$ in $B_\delta$. In more detailed notation, what has just been said means that for any $\varepsilon > 0$ there exists $\delta > 0$ such that

$$ a.s.: |\sigma[f, P, \xi(\omega)] - \sigma[f, P, \xi(\omega)]| < \varepsilon \qquad (4.1.10) $$

or, what is the same,

$$ a.s.: \left| \sum_{i=1}^{n'} f(\omega, \xi_i'(\omega)) \Delta x_i'(\omega) - \sum_{i=1}^{n''} f(\omega, \xi_i''(\omega)) \Delta x_i''(\omega) \right| < \varepsilon \qquad (4.1.11) $$

for any partitions $(P', \xi'(\omega))$ and $(P'', \xi''(\omega))$ with distinguished points on the random
interval $[q_1(\omega), q_2(\omega)]$ with $\lambda(P') < \delta$ and $\lambda(P'') < \delta$.

**Proposition 4.1.2.** A necessary condition for a function $f(\omega, x(\omega))$ defined on $\Omega \times [q_1(\omega), q_2(\omega)]$ to be Riemann integrable on $[q_1(\omega), q_2(\omega)]$ is that $f$ be bounded a.s. on $\Omega \times [q_1(\omega), q_2(\omega)]$.

**Proof.** If $f$ is not bounded a.s. on $[q_1(\omega), q_2(\omega)]$, then for any partition $(P, \xi)$ of $[q_1(\omega), q_2(\omega)]$ the function $f$ is unbounded on at least one of the intervals $[x_{i-1}(\omega), x_i(\omega)]$ of $(P, \xi)$. This means that, by choosing the point

$$ \xi_i(\omega) \in [x_{i-1}(\omega), x_i(\omega)] $$

in different ways, we can make the quantity $|f(\omega, \xi_i(\omega))\Delta x_i(\omega)|$ a.s. as large as desired.

But then the Riemann sum $\sum_{i=1}^n f(\omega, \xi_i(\omega))\Delta x_i(\omega)$ can also be made as large as desired in absolute value by changing only the point $\xi_i(\omega)$ in this interval.

We agree that when a partition $P$

$$ q_1(\omega) = x_0(\omega) < x_1(\omega) < \dots < x_{n-1}(\omega) < x_n(\omega) = q_2(\omega). $$

is given on the closed random interval $[q_1(\omega), q_2(\omega)]$, we shall use the symbol $\Delta_i(\omega)$ to

denote the interval $[x_{i-1}(\omega), x_i(\omega)]$ along with $\Delta x_i(\omega)$ as a notation for the difference $x_i(\omega) - x_{i-1}(\omega)$.

If a partition $P^*$ of the closed random interval $[q_1(\omega), q_2(\omega)]$ is obtained from the partition $P$ by the adjunction of new points to $P$, we call $P^*$ a *refinement* of $P$.

When a refinement $P^*$ of a partition $P$ is constructed, some (perhaps all) of the closed random intervals $\Delta_i(\omega) = [x_{i-1}(\omega), x_i(\omega)]$ of the partition $P$ themselves undergo partitioning: $x_{i-1}(\omega) = x_{i_0}(\omega) < \dots < x_{i_{n_i}}(\omega) = x_i(\omega)$. In that connection, it will be

useful for us to label the points of $P^*$ by double indices. In the notation $x_{ij}(\omega)$ the first index means that $x_{ij}(\omega) \in \Delta_i(\omega)$, and the second index is the ordinal number of the point on the closed random interval $\Delta_i(\omega)$. It is now natural to set $\Delta x_{ij}(\omega)$

$$ = x_{i_j} - x_{i_{j-1}} $$

and $\Delta_{i_j}(\omega)$. Thus $\Delta x_i(\omega) = \Delta x_{i_1}(\omega) + \dots + \Delta x_{i_{n_i}}(\omega)$.

As an example of a partition that is a refinement of both the partition $P'$ and $P''$ one can take $P^* = P' \cup P''$, obtained as the union of the points of the two partitions $P'$ and $P''$.

We recall finally that $\Omega(f(\omega, x(\omega)), E, \omega')$ denotes the oscillation of the function $f(\omega, x)$ on

the random set $E(\omega)$, that is

$$ \Omega(f(\omega, x(\omega)), E(\omega)) = \sup_{\text{a.s.: } x_1(\omega)x_2(\omega) \in E(\omega)} |f(\omega, x_1(\omega)) - f(\omega, x_2(\omega))|. \quad (4.1.12) $$

In particular, $\Omega(f(\omega, x(\omega)), \Delta_i(\omega))$ is the oscillation of $f(\omega, x(\omega))$ on the closed random interval $[x_{i-1}(\omega), x_i(\omega)]$.
This oscillation is necessarily a.s. finite if $f(\omega, x)$ is a.s. bounded function of variable $x$.

**Proposition 4.1.3.** A sufficient condition for a.s. bounded function $f(\omega, x)$ to be integrable on a closed random interval $[q_1(\omega), q_2(\omega)]$ is that for every $\varepsilon > 0$ there exist a number $\delta > 0$ such that

$$ \sum_{i=1}^{n} \Omega(f(\omega, x(\omega)), \Delta_i(\omega)) \Delta x_i(\omega) < \varepsilon \quad (4.1.13) $$

for any partition $P$ of $[q_1(\omega), q_2(\omega)]$ with mesh $\lambda(P) < \delta$.

**Proof** Let $P$ be a partition of $[q_1(\omega), q_2(\omega)]$ and $P^*$ a refinement of $P$. Let us estimate

the difference between the random Riemann sums $\sigma(f, P^*, \xi^*) - \sigma(f, P, \xi)$. Using the notation introduced above, we can write

$$ \begin{aligned} & |\sigma(f, P^*, \xi^*)(\omega) - \sigma(f, P, \xi)(\omega)| = \\ & \left| \sum_{i=1}^{n} \sum_{j=1}^{n_i} f(\omega, \xi_{ij}(\omega)) \Delta x_{ij}(\omega) - \sum_{i=1}^{n} f(\omega, \xi_i(\omega)) \Delta x_i(\omega) \right| = \\ & \left| \sum_{i=1}^{n} \sum_{j=1}^{n_i} f(\omega, \xi_{ij}(\omega)) \Delta x_{ij}(\omega) - \sum_{i=1}^{n} f(\omega, \xi_i(\omega)) \Delta x_i(\omega) \right| = \\ & \left| \sum_{i=1}^{n} \sum_{j=1}^{n_i} [f(\omega, \xi_{ij}(\omega)) - f(\omega, \xi_i(\omega))] \Delta x_{ij}(\omega) \right| \le \quad (4.1.14) \\ & \sum_{i=1}^{n} \sum_{j=1}^{n_i} [f(\omega, \xi_{ij}(\omega)) - f(\omega, \xi_i(\omega))] |\Delta x_{ij}(\omega)| \le \\ & \sum_{i=1}^{n} \sum_{j=1}^{n_i} \Omega(f(\omega, x(\omega)), \Delta_i(\omega)) \Delta x_{ij}(\omega) = \\ & \sum_{i=1}^{n} \Omega(f(\omega, x(\omega)), \Delta_i(\omega)) \Delta x_i(\omega). \end{aligned} $$

In this computation we have used the relation $\Delta x_i(\omega) = \sum_{j=1}^{n_i} \Delta x_{ij}(\omega)$ and the inequality

$$ \text{a.s.: } |f(\omega, \xi_{ij}(\omega)) - f(\omega, \xi_i(\omega))| \le \Omega(f(\omega, x(\omega)), \Delta_i(\omega)), \quad (4.1.15) $$

which holds because a.s.: $\xi_{ij}(\omega) \in \Delta_{ij}(\omega) \subset \Delta_i(\omega)$ and a.s.: $\xi_i(\omega) \in \Delta_i(\omega)$.

It follows from the estimate for the difference of the random Riemann sums that if the function satisfies the sufficient condition given in the statement of Proposition 4.1.3, then for any $\varepsilon > 0$ we can find $\delta > 0$ such that

$$ \text{a.s.: } |\sigma(f, P^*, \xi^*)(\omega) - \sigma(f, P, \xi)(\omega)| < \varepsilon/2 \quad (4.1.16) $$
for any partition P of $[q_1(\omega), q_2(\omega)]$ with mesh $\lambda(P) < \delta$, any refinement $P^*$ of P,
and any choice of the sets of distinguished points $\xi(\omega)$ and $\xi^*(\omega)$.
Now if $(P', \xi')$ and $(P'', \xi'')$ are arbitrary partitions with distinguished
points on $[q_1(\omega), q_2(\omega)]$ whose meshes satisfy $\lambda(P') < \delta$ and $\lambda(P'') < \delta$, then, by
what

has just been proved, the partition $P^* = P' \cup P''$, which is a refinement of both of
them,
must satisfy

$$
\begin{align}
a.s.: & |\sigma(f, P^*, \xi^*(\omega)) - \sigma(f, P', \xi'(\omega))| < \varepsilon/2, \tag{4.1.17} \\
a.s.: & |\sigma(f, P^*, \xi^*(\omega)) - \sigma(f, P'', \xi''(\omega))| < \varepsilon/2. \nonumber
\end{align}
$$

It follows that

$$
|\sigma(f, P', \xi'(\omega)) - \sigma(f, P'', \xi''(\omega))| < \varepsilon, \quad (4.1.18)
$$

provided $\lambda(P') < \delta$ and $\lambda(P'') < \delta$. Therefore, by the Cauchy criterion, the
limit of the random Riemann sums exists a.s.:

$$
a.s.: \exists \lim_{\lambda(P) \to 0} \sum_{i=1}^{n} f(\omega, \xi_i(\omega)) \Delta x_i(\omega), \quad (4.1.19)
$$

that is $f \in \Re[q_1(\omega), q_2(\omega)]$.

**Definition 4.1.10.** (i) A function $f(\omega, x(\omega)) : \Omega \times \Xi(\Sigma, \mathbb{R}) \to \Xi(\Sigma, \mathbb{R})$ is a.s. continuous at point $x_0(\omega) \in [q_1(\omega), q_2(\omega)]$ if

1. $f(\omega, x_0(\omega))$ is defined, so that $x_0(\omega)$ is in the domain of $f(\omega, x(\omega))$.

2. $\lim_{x(\omega) \to x_0(\omega)} f(\omega, x(\omega))$ a.s. exists for $x(\omega)$ in the domain of $f(\omega, x(\omega))$.

3. a.s.: $\lim_{x(\omega) \to x_0(\omega)} f(\omega, x(\omega)) = f(\omega, x_0(\omega))$.

(ii) A function $f(\omega, x(\omega)) : \Omega \times \Xi(\Sigma, \mathbb{R}) \to \Xi(\Sigma, \mathbb{R})$ is a.s. continuous on the closed random interval $[q_1(\omega), q_2(\omega)]$ if $f(\omega, x(\omega))$ is a.s. continuous at any point $x(\omega) \in [q_1(\omega), q_2(\omega)]$

**Notation 4.1.4.** The set of a.s. continuous functions on a closed random interval $[q_1(\omega), q_2(\omega)]$ will be denoted $C_{a.s.}[q_1(\omega), q_2(\omega)]$

**Definition 4.1.11.**

**Corollary 4.1.1.** $f(\omega, x(\omega)) \in C_{a.s.}[q_1(\omega), q_2(\omega)] \Rightarrow f(\omega, x(\omega)) \in \Re[q_1(\omega), q_2(\omega)]$, that is,

every a.s. continuous function $f(\omega, x(\omega))$ on a
closed random interval $[q_1(\omega), q_2(\omega)]$ is integrable on that closed random interval.

**Proof.** If a function is continuous on a closed random interval, it is a.s. bounded there,

so that the necessary condition for integrability is satisfied in this case. But
a.s. continuous function on a closed random interval $[q_1(\omega), q_2(\omega)]$ is uniformly a.s.
continuous on that interval. Therefore, for every $\varepsilon > 0$ there exists $\delta > 0$ such that
a.s.: $\Omega(f(\omega, x(\omega)), \Delta(\omega)) < \frac{\varepsilon}{q_2(\omega) - q_1(\omega)}$ on any closed interval

$$ \Delta(\omega) \subset [q_1(\omega), q_2(\omega)] $$

of length less than $\delta$. Then for any partition $P$ with mesh $\lambda(P) < \delta$ we have that

$$ \begin{aligned} \text{a.s.: } \sum_{i=1}^{n} \Omega(f(\omega, x(\omega)), \Delta_i(\omega)) < \frac{\varepsilon}{q_2(\omega) - q_1(\omega)} \sum_{i=1}^{n} \Delta_i(\omega) &= \\ &= \frac{\varepsilon[q_2(\omega) - q_1(\omega)]}{q_2(\omega) - q_1(\omega)} = \varepsilon. \end{aligned} \quad (4.1.20) $$

By Proposition 3.1.3, we can now conclude that $f \in \Re[q_1(\omega), q_2(\omega)]$.

**Corollary 4.1.2.** If a.s. bounded function $f$ on a closed random interval $[q_1(\omega), q_2(\omega)]$ is a.s. continuous everywhere except at a finite set of random points $x_i(\omega), i = 1, \dots, k$ then $f \in \Re[q_1(\omega), q_2(\omega)]$.

**Proof.** Let a.s.: $\Omega(f(\omega, x(\omega)), [q_1(\omega), q_2(\omega)]) < C < oo$, and suppose $f$ has $k$ points of discontinuity on $[q_1(\omega), q_2(\omega)]$. We shall verify that the sufficient condition for integrability of the function $f$ is satisfied.

For a given $\varepsilon > 0$ we choose the number $\delta_1 = \frac{\varepsilon}{8C \times k}$ and construct the

$\delta_1$-neighborhood of each of the $k$ points of a.s. discontinuity of $f$ on $[q_1(\omega), q_2(\omega)]$.

The

complement of the union of these neighborhoods in $[q_1(\omega), q_2(\omega)]$ consists of a finite number of closed random intervals, on each of which $f$ is a.s. continuous and hence a.s. uniformly continuous. Since the number of these intervals is finite, given $\varepsilon > 0$ there exists $\delta_2 > 0$ such that on each interval $\Delta_i(\omega) = [x_i(\omega), x_{i-1}(\omega)], i = 2, \dots, k$

whose

length $l[\Delta_i(\omega)] = x_i(\omega) - x_{i-1}(\omega)$ a.s. is less than $\delta_2$ and which is entirely contained in one of the closed random intervals just mentioned, on which $f$ is a.s. continuous, we have

$$ \text{a.s.: } \Omega(f(\omega, x(\omega)), \Delta_i(\omega)) < \frac{\varepsilon}{2[q_2(\omega) - q_1(\omega)]} $$

We now choose $\delta = \min\{\delta_1, \delta_2\}$. Let $P$ be an arbitrary partition of $[q_1(\omega), q_2(\omega)]$ for which $\lambda(P) < \delta$. We break the sum $\sum_{i=1}^n \Omega(f(\omega, x(\omega)), \Delta_i(\omega))$ corresponding to the partition $P$ into two parts:

$$ \sum_{i=1}^{n} \Omega(f(\omega, x(\omega)), \Delta_i(\omega)) \Delta x_i(\omega) = \sum_{i=1}^{n} \Omega(f(\omega, x(\omega)), \Delta_i(\omega)) \Delta x_i(\omega) + \sum_{i=n}^{n-1} \Omega(f(\omega, x(\omega)), \Delta_i(\omega)) \Delta x_i(\omega). \quad (4.1.21) $$

The sum $\sum'$ contains the terms corresponding to random intervals $\Delta_i(\omega)$ of the partition having no points in common with any of the $\delta_1$-neighborhoods of the points of discontinuity. For these random intervals $\Delta_i(\omega)$ we have

$$ \text{a.s.: } \Omega(f(\omega, x(\omega)), \Delta_i(\omega)) < \frac{\varepsilon}{2(q_2(\omega) - q_1(\omega))} $$
and so

$$ a.s.: \sum' \Omega(f(\omega, x(\omega)), \Delta_i(\omega)) \Delta x_i(\omega) < \frac{\varepsilon}{2(q_2(\omega) - q_1(\omega))} \sum' \Delta x_i(\omega) \le \frac{\varepsilon}{2(q_2(\omega) - q_1(\omega))(q_2(\omega) - q_1(\omega))} = \frac{\varepsilon}{2}. \quad (4.1.22) $$

The sum of the lengths of the remaining intervals of the partition $P$, as one can easily see, is at most $(\delta + 2\delta_1 + \delta) \le \frac{4\varepsilon}{8C \times k}k = \varepsilon/2C$, and therefore

$$ a.s.: \sum'' \Omega(f(\omega, x(\omega)), \Delta_i(\omega)) \Delta x_i(\omega) \le C \frac{\varepsilon}{2C} = \frac{\varepsilon}{2}. \quad (4.1.23) $$

Thus we find that for $\lambda(P) < \delta$:

$$ a.s.: \sum_{i=1}^{n} \Omega(f(\omega, x(\omega)), \Delta_i(\omega)) \Delta x_i(\omega) < \varepsilon. \quad (4.1.24) $$

that is, the sufficient condition for integrability holds, and so $f \in \Re[q_1(\omega), q_2(\omega)]$.

**Corollary 4.1.3.** Any a.s. monotonic function on a closed random interval $[q_1(\omega), q_2(\omega)]$ is integrable on that random interval.

**Proof.** It follows from the a.s. monotonicity of $f(\omega, x(\omega))$ on $[q_1(\omega), q_2(\omega)]$ that $a.s.: \sum_{i=1}^n \Omega(f(\omega, x(\omega)), [q_1(\omega), q_2(\omega)]) = |f(\omega, q_2(\omega)) - f(\omega, q_1(\omega))|$. Suppose $\varepsilon > 0$ is given. We set $\delta(\omega) = \frac{\varepsilon}{|f(\omega, q_2(\omega)) - f(\omega, q_1(\omega))|}$. We assume

that $a.s.: f(\omega, q_2(\omega)) - f(\omega, q_1(\omega)) \ne 0$, since otherwise a.s. is constant, and there is no doubt as to its integrability. Let $P$ be an arbitrary partition of $[q_1(\omega), q_2(\omega)]$ with mesh $a.s.: \lambda(P) < \delta(\omega)$. Then, taking account of the a.s.monotonicity of $f$, we have
(3.1.25)

Thus / satisfies the sufficient condition for integrability, and therefore $f \in \Re[q_1(\omega), q_2(\omega)]$.

**Remark 4.1.1.** A monotonic function may have a (countably) infinite set of discontinuities on a closed random $[q_1(\omega), q_2(\omega)]$ interval. For example, the function defined by the relations

# Appendix 4.2.Integration over generalized random interval.

**Definition 4.2.1.** Let for any $\omega \in \Omega: q_1(\omega, \omega')$ and $q_2(\omega, \omega')$ are $\Omega' = \Xi(\Sigma, \mathbb{R})$-valued random variables defined on a generalized probability space $\Sigma = (\Omega, \mathcal{F}, \tilde{\mathbf{P}})$, i.e., $\forall \omega[\omega \in \Omega \Rightarrow q_{1,2}(\omega, \omega') : \Omega \to \Omega' = \Xi(\Sigma, \mathbb{R})]$.

Let $\tilde{\Xi}(\Omega, \Sigma, \mathbb{R})$ be a set of the all $\Xi(\Sigma, \mathbb{R})$-valued random variables defined on a probability space $\Sigma = (\Omega, \mathcal{F}, \tilde{\mathbf{P}})$, thus $q_{1,2}(\omega, \omega') \in \tilde{\Xi}(\Omega, \Sigma, \mathbb{R})$.

**Definition 4.2.2.** Let $q_1(\omega, \omega')$ and $q_2(\omega, \omega')$ are $\Xi(\Sigma, \mathbb{R})$-valued random variables
defined on a generalized probability space $\Sigma = (\Omega, \mathcal{F}, \tilde{\mathbf{P}})$, i.e.,

$q_{1,2}(\omega, \omega') : \Omega \to \Xi(\Sigma, \mathbb{R})$. Assume that a.s.: $-\infty < q_1(\omega, \omega') < q_2(\omega, \omega') < \infty$.

Let $\tilde{\Xi}(\Omega, \Sigma, \mathbb{R})$ be a set of the all $\Xi(\Sigma, \mathbb{R})$-valued random variables defined on a generalized probability space $\Sigma = (\Omega, \mathcal{F}, \tilde{\mathbf{P}})$. Closed generalized random interval $[q_1(\omega, \omega'), q_2(\omega, \omega')]$ that is a subset $[q_1(\omega, \omega'), q_2(\omega, \omega')] \subset \tilde{\Xi}(\Omega, \Sigma, \mathbb{R})$ such that a.s.: $-\infty < q_1(\omega, \omega') < q_2(\omega, \omega') < \infty$ and

$$ \begin{array}{l} \forall q(\omega, \omega')\{q(\omega, \omega') \in [q_1(\omega, \omega'), q_2(\omega, \omega')] \Leftrightarrow \\ \qquad \{q(\omega, \omega') \in \Xi(\Sigma, \mathbb{R}) | \text{a.s. } (q_1(\omega, \omega') < q(\omega, \omega') < q_2(\omega, \omega')) \}\}. \end{array} \quad (4.2.1) $$

**Notation 4.2.1.** Assume that a.s.: $q_1(\omega, \omega') < q_2(\omega, \omega')$. Then we will write:

$$ q_1(\omega, \omega') < q_2(\omega, \omega'). $$

Assume that a.s.: $q_1(\omega, \omega') \le q_2(\omega, \omega')$. Then we will write:

$$ q_1(\omega, \omega') \le q_2(\omega, \omega'). $$

**Definition 4.2.3.** Let $E$ be a separable complete metric space and let $\Sigma$ be its Borel $\sigma$-algebra. *Generalized random measure* it is a function $\mu: \Sigma \to \Xi(\Sigma, \mathbb{R})$, that satisfies:

(1) If $E_1 \subset E_2$, then a.s.: $\mu(E_1) < \mu(E_2)$.

(2) If $E_n \in S$, $n = 1, 2, \dots$ and $E_i \cap E_j = \emptyset$ ($i \ne j$), then a.s.: $\mu(\cup_n E_n) = \sum_n \mu(E_n)$.

If we further impose a condition $\mu(\Omega) = 1$, then the generalized random measure so defined is called the generalized random probability measure and is usually denoted by $\tilde{\mathbf{P}}$.

**Definition 4.2.4.** The tuple $(\Omega, S, \tilde{\mathbf{P}})$ is then called the generalized probability space.

Suppose $(\Omega, S)$ and $(\Omega', S')$ are two measurable spaces. The function $X: \Omega \to \Omega'$ is called a *generalized random element*, if for every $A \in S'$, $X^{-1}(A) \in S$. If the first measurable space is equipped with a generalized random probability measure $\tilde{\mathbf{P}}$, then

the generalized random element $X: \Omega \to \Omega'$ induces a generalized random probability

measure on the second space $(\Omega', S')$ and given by $P = \tilde{\mathbf{P}} \circ X^{-1}$, called the generalized random distribution of $X$.

**Definition 4.2.5.** If the second measurable space $(\Omega', S')$ is taken to be $(\Xi(\Sigma, \mathbb{R}), \mathfrak{B}_{\Xi(\Sigma, \mathbb{R})})$, where $\mathfrak{B}_{\Xi(\Sigma, \mathbb{R})}$ is the Borel $\sigma$-algebra on $\Xi(\Sigma, \mathbb{R})$ the function $X(\omega): \Omega \to \Omega' = (\Xi(\Sigma, \mathbb{R}), \mathfrak{B}_{\Xi(\Sigma, \mathbb{R})})$ is called a generalized random variable. For this case, the generalized random distribution of $X$ is completely determined by the random distribution function defined as

$$ F_{X(\omega)}(x(\omega, \omega')) = \tilde{\mathbf{P}}(X(\omega) \le x(\omega, \omega')) \quad (4.2.2) $$
**Definition 4.2.6.** The lengths $l([q_1(\omega), q_2(\omega)])$ of the random interval $[q_1(\omega), q_2(\omega)]$ is defined by

$$l([q_1(\omega, \omega'), q_2(\omega, \omega')]) = \operatorname{ess} \sup q_3(\omega, \omega'), \quad (4.2.2)$$

where $q_3(\omega, \omega') = q_2(\omega, \omega') - q_1(\omega, \omega')$.

**Notation 4.2.2.** Assume that a.s. $q_1(\omega, \omega') < q_2(\omega, \omega')$. Then we will write:

$$q_1(\omega, \omega') < q_2(\omega, \omega').$$

Assume that a.s. $q_1(\omega, \omega') \le q_2(\omega, \omega')$. Then we will write:

$$q_1(\omega, \omega') \le q_2(\omega, \omega').$$

**Definition 4.2.7.** A partition $P$ of a closed random interval $[q_1(\omega, \omega'), q_2(\omega, \omega')]$ is a finite system of random variables $x_0(\omega, \omega'), \dots, x_n(\omega, \omega')$ such that

$$q_1(\omega, \omega') = x_0(\omega, \omega') < x_1(\omega, \omega') < \dots < x_{n-1}(\omega, \omega') < x_n(\omega, \omega') = q_2(\omega, \omega').$$

The closed random intervals $[x_{i-1}(\omega), x_i(\omega)]$, ($i = 1, \dots, n$) are called the intervals of the partition $P$.

**Definition 4.2.8.** The largest of the lengths of the intervals of the partition $P$, denoted $\lambda(P)$, is called the *mesh* of the partition.

**Definition 4.2.9.** We speak of a partition with distinguished points $(P, \xi)$ on the closed random interval $[q_1(\omega, \omega'), q_2(\omega, \omega')]$ if we have a partition $P$ of $[q_1(\omega, \omega'), q_2(\omega, \omega')]$ and a point $\xi_1(\omega, \omega') \in [x_{i-1}(\omega, \omega'), x_i(\omega, \omega')]$ has been chosen in each of the intervals of the partition $[x_{i-1}(\omega, \omega'), x_i(\omega, \omega')]$, ($i = 1, \dots, n$).

We denote the set of points $(\xi_1(\omega, \omega'), \dots, \xi_n(\omega, \omega'))$ by the single letter $\xi(\omega, \omega')$.

**Definition 4.2.10.** In the set $\tilde{P}$ of partitions with distinguished points on a given random interval $[q_1(\omega, \omega'), q_2(\omega, \omega')]$, we consider the following base

$\mathbf{B} = \{B_d\}$. The element $B_d$, $d > 0$, of the base $\mathbf{B}$ consists of all partitions

with distinguished points $(P, \xi)$ on $[q_1(\omega, \omega'), q_2(\omega, \omega')]$ for which $\lambda(P) < d$.

**Proposition 4.2.1.** Let us verify that $\{B_d\}$, $d > 0$ is actually a base in $\tilde{P}$.

**Proof.** First $B_d \ne \emptyset$. In fact, for any number $d > 0$, it is obvious that there exists a partition $P$ of $[q_1(\omega, \omega'), q_2(\omega, \omega')]$ with mesh $\lambda(P) < d$ (for example, a partition into $n$

congruent closed random intervals). But then there also exists a partition $(P, \xi(\omega, \omega'))$ with distinguished points for which $\lambda(P) < d$.

Second, if $d_1 > 0, d_2 > 0$, and $d = \min\{d_1, d_2\}$, it is obvious that $B_{d_1} \cap B_{d_2} = B_d \in \mathbf{B}$. Hence $B = \{B_d\}$ is indeed a base in $\tilde{P}$.

**Definition 4.2.11. (Random Riemann Sums)** (i) If a function $f: \tilde{\Xi}(\Sigma, \mathbb{R}) \to \tilde{\Xi}(\Sigma, \mathbb{R})$ is defined on the closed random interval $[q_1(\omega, \omega'), q_2(\omega, \omega')]$ and $(P, \xi(\omega))$ is a partition with distinguished points on this closed random interval, the sum

$$\sigma[f, P, \xi(\omega, \omega')] = \sum_{i=1}^{n} f(\xi_i(\omega, \omega')) \Delta x_i(\omega, \omega'), \quad (4.2.3)$$
where $\Delta x_i(\omega, \omega') = x_i(\omega, \omega') - x_{i-1}(\omega, \omega')$, is the random Riemann sum of the function

$f: \tilde{\Xi}(\Sigma, \mathbb{R}) \to \tilde{\Xi}(\Sigma, \mathbb{R})$ corresponding to the partition $(P, \xi(\omega, \omega'))$ with distinguished points on $[q_1(\omega, \omega'), q_2(\omega, \omega')]$. Thus, when the function $f$ is fixed, the random

Riemann

sum $\sigma[f, P, \xi(\omega, \omega')]$ is a function $\Phi(p) = \sigma[f, P]$ on the set $\tilde{P}$ of all partitions

$p = (P, \xi(\omega, \omega'))$ with distinguished points on the closed interval $[q_1(\omega, \omega'), q_2(\omega, \omega')]$.

Since there is a base $\mathbf{B}$ in $\tilde{P}$, one can ask about the limit of the function $\Phi(p(\omega, \omega'))$ over that base.

(ii) If a function $f: \Omega \times \Xi(\Sigma, \mathbb{R}) \to \tilde{\Xi}(\Sigma, \mathbb{R})$ is defined on the closed random interval $[q_1(\omega, \omega'), q_2(\omega, \omega')]$ and $(P, \xi(\omega, \omega'))$ is a partition with distinguished points on this closed random interval, the sum

$$ \sigma[f, P, \xi(\omega, \omega')] = \sum_{i=1}^{n} f(\omega, \xi_i(\omega, \omega')) \Delta x_i(\omega, \omega'), \quad (4.2.3') $$

where $\Delta x_i(\omega, \omega') = x_i(\omega, \omega') - x_{i-1}(\omega, \omega')$, is the random Riemann sum of the function

$f: \Omega \times \Xi(\Sigma, \mathbb{R}) \to \Xi(\Sigma, \mathbb{R})$ corresponding to the partition $(P, \xi(\omega, \omega'))$ with

distinguished

points on $[q_1(\omega, \omega'), q_2(\omega, \omega')]$. Thus, when the function

$$ f: \Omega \times [q_1(\omega, \omega'), q_2(\omega, \omega')] \to \tilde{\Xi}(\Omega, \Sigma, \mathbb{R}) $$

is fixed, the random Riemann sum $\sigma[f, P, \xi(\omega, \omega')]$ is a function $\Phi(p) = \sigma[f, P]$ on the set $\tilde{P}$ of all partitions $p = (P, \xi(\omega, \omega'))$ with distinguished points on the closed interval $[q_1(\omega, \omega'), q_2(\omega, \omega')]$.

**Definition 4.2.12. (Riemann Integral on a generalized random interval)**

(i) Let $f: \Xi(\Sigma, \mathbb{R}) \to \Xi(\Sigma, \mathbb{R})$ be a function restricted on a closed random interval $[q_1(\omega, \omega'), q_2(\omega, \omega')]$.

The random variable $I(\omega, \omega')$ is the Riemann integral of the function

$f: \Xi(\Sigma, \mathbb{R}) \to \Xi(\Sigma, \mathbb{R})$ on the closed random interval $[q_1(\omega, \omega'), q_2(\omega, \omega')]$ if for every $\varepsilon > 0$ there exists $\delta > 0$ such that

$$ a.s.: \left| I(\omega, \omega') - \sum_{i=1}^{n} f(\xi_i(\omega, \omega')) \Delta x_i(\omega, \omega') \right| < \varepsilon \quad (4.2.4) $$

for any partition $(P, \xi(\omega, \omega'))$ with distinguished points on $[q_1(\omega, \omega'), q_2(\omega, \omega')]$ whose mesh $\lambda(P)$ is less than $\delta$.

(ii) Let $f: \Omega \times \Xi(\Sigma, \mathbb{R}) \to \Xi(\Sigma, \mathbb{R})$ be a function restricted on a closed random interval $[q_1(\omega), q_2(\omega)]$ such that $f: \Omega \times [q_1(\omega, \omega'), q_2(\omega, \omega')] \to \Xi(\Sigma, \mathbb{R})$.

The random variable $I(\omega, \omega')$ is the Riemann integral of the function

$f: \Omega \times \Xi(\Sigma, \mathbb{R}) \to \Xi(\Sigma, \mathbb{R})$ on the closed random interval $[q_1(\omega, \omega'), q_2(\omega, \omega')]$ if for every

$\varepsilon > 0$ there exists $\delta > 0$ such that

$$ a.s.: \left| I(\omega, \omega') - \sum_{i=1}^{n} f(\omega, \xi_i(\omega, \omega')) \Delta x_i(\omega, \omega') \right| < \varepsilon. \quad (4.2.4') $$

Since the partitions $p = (P, \xi(\omega, \omega'))$ for which $\lambda(P) < \delta$ form the element $B_\delta$
of the base **B** introduced above in the set $\tilde{\mathcal{P}}$ of partitions with distinguished points, Definition 3.2.12 is equivalent to the statement

$$ a.s.: I(\omega, \omega') = \lim_{\mathbf{B}} \Phi(p(\omega, \omega')). \qquad (4.2.5) $$

that is, the integral $I(\omega, \omega')$ is the limit over **B** of the Riemann sums of the function $f$ corresponding to partitions with distinguished points on $[q_1(\omega, \omega'), q_2(\omega, \omega')]$. It is natural to denote the base **B** by $\lambda(P) \to 0$, and then the definition of the Riemann

integral on a random interval can be rewritten as

$$ \begin{aligned} a.s.: I(\omega, \omega') &= \lim_{\lambda(P) \to 0} \sum_{i=1}^{n} f(\xi_i(\omega, \omega')) \Delta x_i(\omega, \omega'), \\ a.s.: I(\omega, \omega') &= \lim_{\lambda(P) \to 0} \sum_{i=1}^{n} f(\omega, \xi_i(\omega, \omega')) \Delta x_i(\omega, \omega'). \end{aligned} \qquad (4.2.6) $$

**Notation 4.2.2.** The integral of $f(x(\omega, \omega'))$ over $[q_1(\omega, \omega'), q_2(\omega, \omega')]$ is denoted

$$ \begin{aligned} & \int_{q_1(\omega)}^{q_2(\omega)} f(x(\omega, \omega')) d[x(\omega, \omega')], \\ & \int_{q_1(\omega)}^{q_2(\omega)} f(\omega, x(\omega, \omega')) d[x(\omega, \omega')], \end{aligned} \qquad (4.2.7) $$

in which the random variables $q_1(\omega, \omega')$ and $q_2(\omega, \omega')$ are called respectively the lower and upper limits of integration. The function $f$ is called the integrand, $f(x(\omega, \omega'))d[x(\omega, \omega')]$ is called the differential form, and $x(\omega, \omega')$ is the random variable

of integration. Thus

$$ \begin{aligned} a.s.: \quad & \int_{q_1(\omega, \omega')}^{q_2(\omega, \omega')} f(x(\omega, \omega')) d[x(\omega, \omega')] = \lim_{\lambda(P) \to 0} \sum_{i=1}^{n} f(\xi_i(\omega, \omega')) \Delta x_i(\omega, \omega'), \\ a.s.: \quad & \int_{q_1(\omega, \omega')}^{q_2(\omega, \omega')} f(\omega, x(\omega, \omega')) d[x(\omega, \omega')] = \lim_{\lambda(P) \to 0} \sum_{i=1}^{n} f(\omega, \xi_i(\omega, \omega')) \Delta x_i(\omega, \omega'). \end{aligned} \qquad (4.2.8) $$

**Definition 4.2.13.** A function $f: \Omega \times \Xi(\Sigma, \mathbb{R}) \to \Xi(\Sigma, \mathbb{R})$ is Riemann integrable on the closed interval $[q_1(\omega, \omega'), q_2(\omega, \omega')]$ if the limit of the Riemann sums in (4.2.8) exists a.s. as $\lambda(P) \to 0$ (that is, the Riemann integral of $f$ is defined).

**Notation 4.2.3.** The set of Riemann-integrable functions on a closed random interval $[q_1(\omega, \omega'), q_2(\omega, \omega')]$ will be denoted $\mathfrak{R}[q_1(\omega, \omega'), q_2(\omega, \omega')]$

By the definition of the integral (Definition 4.2.8) and its reformulation in the forms
(4.2.5) and (4.2.8), an integral is the limit of a certain special function $\Phi(p(\omega, \omega')) = \sigma[f, P, \xi(\omega, \omega')]$ the random Riemann sum, defined on the set $\tilde{P}$ of partitions $p(\omega, \omega') = (P, \xi(\omega, \omega'))$ with distinguished points on $[q_1(\omega, \omega'), q_2(\omega, \omega')]$. This limit is taken with respect to the base $\mathbf{B}$ in $\tilde{P}$ that we have denoted $\lambda(P) \to 0$. Thus the integrability or nonintegrability of a function $f$ on $[q_1(\omega, \omega'), q_2(\omega, \omega')]$ depends

on the existence of this limit. By the Cauchy criterion, this limit exists a.s., if and only if for every $\varepsilon > 0$ there exists an element $B_\delta \in \mathbf{B}$ in the base such that

$$ a.s.: |\Phi(p'(\omega, \omega')) - \Phi(p''(\omega, \omega'))| < \varepsilon \quad (4.2.9) $$

for any two points $p'(\omega, \omega'), p''(\omega, \omega')$ in $B_\delta$. In more detailed notation, what has just been said means that for any $\varepsilon > 0$ there exists $\delta > 0$ such that

$$ a.s.: |\sigma[f, P, \xi(\omega, \omega')] - \sigma[f, P, \xi(\omega, \omega')]| < \varepsilon \quad (4.2.10) $$

or, what is the same,

$$ a.s.: \left| \sum_{i=1}^{n'} f(\omega, \xi_i'(\omega, \omega')) \Delta x_i'(\omega, \omega') - \sum_{i=1}^{n''} f(\omega, \xi_i''(\omega, \omega')) \Delta x_i''(\omega, \omega') \right| < \varepsilon \quad (4.2.11) $$

for any partitions $(P', \xi'(\omega, \omega'))$ and $(P'', \xi''(\omega, \omega'))$ with distinguished points on the random interval $[q_1(\omega, \omega'), q_2(\omega, \omega')]$ with $\lambda(P') < \delta$ and $\lambda(P'') < \delta$.

**Proposition 4.2.2.** A necessary condition for a function $f(\omega, x(\omega, \omega'))$ defined on $\Omega \times [q_1(\omega, \omega'), q_2(\omega, \omega')]$ to be Riemann integrable on $[q_1(\omega, \omega'), q_2(\omega, \omega')]$ is that $f$ be bounded a.s. on $\Omega \times [q_1(\omega, \omega'), q_2(\omega, \omega')]$.

**Proof.** If $f$ is not bounded a.s. on $[q_1(\omega, \omega'), q_2(\omega, \omega')]$, then for any partition $(P, \xi)$ of $[q_1(\omega, \omega'), q_2(\omega, \omega')]$ the function $f$ is unbounded on at least one of the intervals $[x_{i-1}(\omega, \omega'), x_i(\omega, \omega')]$ of $(P, \xi)$. This means that, by choosing the point $\xi_i(\omega, \omega') \in [x_{i-1}(\omega, \omega'), x_i(\omega, \omega')]$

in different ways, we can make the quantity $|f(\omega, \xi_i(\omega, \omega')) \Delta x_i(\omega, \omega')|$ a.s. as large as desired. But then the Riemann sum $\sum_{i=1}^n f(\omega, \xi_i(\omega, \omega')) \Delta x_i(\omega, \omega')$ can also be made

as large as desired in absolute value by changing only the point $\xi_i(\omega, \omega')$ in this interval. We agree that when a partition $P$

$$ q_1(\omega, \omega') = x_0(\omega, \omega') < x_1(\omega, \omega') < \dots < x_{n-1}(\omega, \omega') < x_n(\omega, \omega') = q_2(\omega, \omega'). $$

is given on the closed random interval $[q_1(\omega, \omega'), q_2(\omega, \omega')]$, we shall use the symbol $\Delta_i(\omega, \omega')$ to denote the interval $[x_{i-1}(\omega, \omega'), x_i(\omega, \omega')]$ along with $\Delta x_i(\omega, \omega')$ as a notation

for the difference $x_i(\omega, \omega') - x_{i-1}(\omega, \omega')$.

If a partition $P^*$ of the closed random interval $[q_1(\omega, \omega'), q_2(\omega, \omega')]$ is obtained from the

partition $P$ by the adjunction of new points to $P$, we call $P^*$ a *refinement* of $P$.

When a refinement $P^*$ of a partition $P$ is constructed, some (perhaps all)
of the closed random intervals $\Delta_i(\omega, \omega') = [x_{i-1}(\omega, \omega'), x_i(\omega, \omega')]$ of the partition $P$
themselves undergo partitioning: $x_{i-1}(\omega, \omega') = x_{i_0}(\omega, \omega') < \dots < x_{i_{n_i}}(\omega, \omega') = x_i(\omega, \omega')$.

In that connection, it will be useful for us to label the points of $P^*$ by double indices.

In the notation $x_{ij}(\omega, \omega')$ the first index means that $x_{ij}(\omega, \omega') \in \Delta_i(\omega, \omega')$, and the
second

index is the ordinal number of the point on the closed random interval $\Delta_i(\omega)$. It is
now

natural to set $\Delta x_{ij}(\omega, \omega') = x_{ij}(\omega, \omega') - x_{i_{j-1}}(\omega, \omega')$ and $\Delta \Delta_{ij}(\omega, \omega')$.

$$
\Delta x_i(\omega, \omega') = \Delta x_{i_1}(\omega, \omega') + \dots + \Delta x_{i_{n_i}}(\omega, \omega').
$$

As an example of a partition that is a refinement of both the partition $P'$ and $P''$ one
can

take $P^* = P' \cup P''$, obtained as the union of the points of the two partitions $P'$ and $P''$.

We recall finally that $\Omega(f(\omega, x(\omega, \omega')), E, \omega')$ denotes the oscillation of the function $f(\omega, x)$ on the random set $E(\omega, \omega')$, that is

$$
\begin{equation} \tag{4.2.12}
\begin{aligned}
\Omega(f(\omega, x(\omega, \omega')), E(\omega, \omega')) &= \\
&\quad \sup_{\text{a.s.: } x_1(\omega, \omega'), x_2(\omega, \omega') \in E(\omega, \omega')} \left| f(\omega, x_1(\omega, \omega')) - f(\omega, x_2(\omega, \omega')) \right|.
\end{aligned}
\end{equation}
$$

In particular, $\Omega(f(\omega, x(\omega, \omega'))$, $\Delta_i(\omega, \omega')$ is the oscillation of $f(\omega, x(\omega, \omega'))$ on the closed random interval $[x_{i-1}(\omega, \omega'), x_i(\omega, \omega')]$.

This oscillation is necessarily a.s. finite if $f(\omega, x)$ is a.s. bounded function of variable $x$.

**Proposition 4.1.3.** A sufficient condition for a.s. bounded function $f(\omega, x)$ to be integrable on a closed random interval $[q_1(\omega, \omega'), q_2(\omega, \omega')]$ is that for every $\varepsilon > 0$ there

exist a number $\delta > 0$ such that

$$
\sum_{i=1}^{n} \Omega(f(\omega, x(\omega, \omega'))), \Delta_i(\omega, \omega')) \Delta x_i(\omega, \omega') < \varepsilon \quad (4.2.13)
$$

for any partition $P$ of $[q_1(\omega, \omega'), q_2(\omega, \omega')]$ with mesh $\lambda(P) < \delta$.

**Proof** Let $P$ be a partition of $[q_1(\omega, \omega'), q_2(\omega, \omega')]$ and $P^*$ a refinement of $P$. Let us estimate the difference between the random Riemann sums $\sigma(f, P^*, \xi^*) - \sigma(f, P, \xi)$. Using the notation introduced above, we can write
$$|\sigma(f, P^*, \xi^*(\omega, \omega')) - \sigma(f, P, \xi(\omega, \omega'))| =$$

$$\left| \sum_{i=1}^{n} \sum_{j=1}^{n_i} f(\omega, \xi_{ij}(\omega, \omega')) \Delta x_{ij}(\omega, \omega') - \sum_{i=1}^{n} f(\omega, \xi_i(\omega, \omega')) \Delta x_i(\omega, \omega') \right| =$$

$$\left| \sum_{i=1}^{n} \sum_{j=1}^{n_i} f(\omega, \xi_{ij}(\omega, \omega')) \Delta x_{ij}(\omega, \omega') - \sum_{i=1}^{n} f(\omega, \xi_i(\omega, \omega')) \Delta x_i(\omega, \omega') \right| =$$

$$\left| \sum_{i=1}^{n} \sum_{j=1}^{n_i} [f(\omega, \xi_{ij}(\omega, \omega')) - f(\omega, \xi_i(\omega, \omega'))] \Delta x_{ij}(\omega, \omega') \right| \leq \qquad (4.2.14)$$

$$\sum_{i=1}^{n} \sum_{j=1}^{n_i} |[f(\omega, \xi_{ij}(\omega)) - f(\omega, \xi_i(\omega))]| \Delta x_{ij}(\omega, \omega') \leq$$

$$\sum_{i=1}^{n} \sum_{j=1}^{n_i} \Omega(f(\omega, x(\omega, \omega')), \Delta_i(\omega, \omega')) \Delta x_{ij}(\omega, \omega') =$$

$$\sum_{i=1}^{n} \Omega(f(\omega, x(\omega, \omega')), \Delta_i(\omega, \omega')) \Delta x_i(\omega, \omega').$$

In this computation we have used the relation $\Delta x_i(\omega, \omega') = \sum_{j=1}^{n_i} \Delta x_{ij}(\omega, \omega')$ and the inequality

$$a.s.: |f(\omega, \xi_{ij}(\omega, \omega')) - f(\omega, \xi_i(\omega, \omega'))| \leq \Omega(f(\omega, x(\omega, \omega')), \Delta_i(\omega, \omega')), \quad (4.2.15)$$

which holds because $a.s.: \xi_{ij}(\omega) \in \Delta_{ij}(\omega) \subset \Delta_i(\omega)$ and $a.s.: \xi_i(\omega) \in \Delta_i(\omega)$.

It follows from the estimate for the difference of the random Riemann sums that if the function satisfies the sufficient condition given in the statement of Proposition 4.2.3, then for any $\epsilon > 0$ we can find $\delta > 0$ such that

$$a.s.: |\sigma(f, P^*, \xi^*(\omega, \omega')) - \sigma(f, P, \xi(\omega, \omega'))| < \epsilon/2 \qquad (4.2.16)$$

for any partition $P$ of $[q_1(\omega), q_2(\omega)]$ with mesh $\lambda(P) < \delta$, any refinement $P^*$ of $P$, and any choice of the sets of distinguished points $\xi(\omega, \omega')$ and $\xi^*(\omega, \omega')$.

Now if $(P', \xi')$ and $(P'', \xi'')$ are arbitrary partitions with distinguished points on $[q_1(\omega, \omega'), q_2(\omega, \omega')]$ whose meshes satisfy $\lambda(P') < \delta$ and $\lambda(P'') < \delta$, then,

by

what has just been proved, the partition $P^* = P' \cup P''$, which is a refinement of both of them, must satisfy

$$a.s.: |\sigma(f, P^*, \xi^*(\omega, \omega')) - \sigma(f, P', \xi'(\omega, \omega'))| < \epsilon/2, \qquad (4.2.17)$$

$$a.s.: |\sigma(f, P^*, \xi^*(\omega, \omega')) - \sigma(f, P'', \xi''(\omega, \omega'))| < \epsilon/2.$$

It follows that
$$|\sigma(f, P', \xi'(\omega, \omega')) - \sigma(f, P'', \xi''(\omega, \omega'))| < \varepsilon, \quad (4.2.18)$$

provided $\lambda(P') < \delta$ and $\lambda(P'') < \delta$. Therefore, by the Cauchy criterion, the limit of the random Riemann sums exists a.s.:

$$\text{a.s.: } \exists \lim_{\lambda(P) \to 0} \sum_{i=1}^{n} f(\omega, \xi_i(\omega, \omega')) \Delta x_i(\omega, \omega'), \quad (4.2.19)$$

that is $f \in \Re[q_1(\omega, \omega'), q_2(\omega, \omega')]$.

**Definition 4.2.10.** (i) A function $f(\omega, x(\omega, \omega')): \Omega \times \tilde{\Xi}(\Omega, \Sigma, \mathbb{R}) \to \tilde{\Xi}(\Omega, \Sigma, \mathbb{R})$ is a.s. continuous at point $x_0(\omega, \omega') \in [q_1(\omega, \omega'), q_2(\omega, \omega')]$ if

1. $f(\omega, x_0(\omega, \omega'))$ is defined, so that $x_0(\omega, \omega')$ is in the domain of $f(\omega, x(\omega, \omega'))$.

2. $\lim_{x(\omega) \to x_0(\omega)} f(\omega, x(\omega))$ a.s. exists for $x(\omega)$ in the domain of $f(\omega, x(\omega))$.

3. a.s.: $\lim_{x(\omega) \to x_0(\omega)} f(\omega, x(\omega)) = f(\omega, x_0(\omega))$.

(ii) A function $f(\omega, x(\omega, \omega')): \Omega \times \tilde{\Xi}(\Omega, \Sigma, \mathbb{R}) \to \tilde{\Xi}(\Omega, \Sigma, \mathbb{R})$ is a.s. continuous on the closed random interval $[q_1(\omega), q_2(\omega)]$ if $f(\omega, x(\omega))$ is a.s. continuous at any point $x(\omega, \omega') \in [q_1(\omega, \omega'), q_2(\omega, \omega')]$

**Notation 4.2.4.** The set of a.s. continuous functions on a closed random interval $[q_1(\omega), q_2(\omega)]$ will be denoted $C_{\text{as.}}[q_1(\omega), q_2(\omega)]$

**Definition 4.2.11.**

**Corollary 4.2.1.** $f(\omega, x(\omega, \omega')) \in C_{\text{as.}}[q_1(\omega, \omega'), q_2(\omega, \omega')] \Rightarrow f(\omega, x(\omega, \omega')) \in \Re[q_1(\omega, \omega'), q_2(\omega, \omega')],$ that is, every a.s. continuous function $f(\omega, x(\omega))$

on a closed random interval $[q_1(\omega, \omega'), q_2(\omega, \omega')]$ is integrable on that closed random interval.

**Proof.** If a function is continuous on a closed random interval, it is a.s. bounded there,

so that the necessary condition for integrability is satisfied in this case. But a.s. continuous function on a closed random interval $[q_1(\omega, \omega'), q_2(\omega, \omega')]$ is uniformly a.s. continuous on that interval. Therefore, for every $\varepsilon > 0$ there exists $\delta > 0$ such that

a.s.: $\Omega(f(\omega, x(\omega, \omega'))), \Delta(\omega, \omega')) < \frac{\varepsilon}{q_2(\omega, \omega') - q_1(\omega, \omega')}$ on any closed interval

$\Delta(\omega, \omega') \subset [q_1(\omega, \omega'), q_2(\omega, \omega')]$ of length less than $\delta$. Then for any partition $P$ with mesh $\lambda(P) < \delta$ we have that

$$\begin{align}
a.s.: \sum_{i=1}^{n} \Omega(f(\omega, x(\omega, \omega'))), \Delta_i(\omega, \omega')) &< \frac{\varepsilon}{q_2(\omega, \omega') - q_1(\omega, \omega')} \sum_{i=1}^{n} \Delta_i(\omega, \omega') = \nonumber \\
&= \frac{\varepsilon[q_2(\omega, \omega') - q_1(\omega, \omega')]}{q_2(\omega, \omega') - q_1(\omega, \omega')} = \varepsilon. \tag{4.2.20}
\end{align}$$

By Proposition 4.2.3, we can now conclude that $f \in \Re[q_1(\omega, \omega'), q_2(\omega, \omega')]$.
**Corollary 4.2.2.** If a.s. bounded function $f$ on a closed random interval $[q_1(\omega, \omega'), q_2(\omega, \omega')]$ is a.s. continuous everywhere except at a finite set of random points $x_i(\omega, \omega')$, $i = 1, \dots, k$ then $f \in \Re[q_1(\omega, \omega'), q_2(\omega, \omega')]$.

**Proof.** Let a.s.: $\Omega(f(\omega, x(\omega, \omega'))), [q_1(\omega, \omega'), q_2(\omega, \omega')] < C < \infty$, and suppose $f$ has $k$ points of discontinuity on $[q_1(\omega, \omega'), q_2(\omega, \omega')]$. We shall verify that the sufficient condition for integrability of the function $f$ is satisfied.

For a given $\varepsilon > 0$ we choose the number $\delta_1 = \frac{\varepsilon}{8C \times k}$ and construct the $\delta_1$-neighborhood of each of the $k$ points of a.s. discontinuity of $f$ on $[q_1(\omega, \omega'), q_2(\omega, \omega')]$.

The complement of the union of these neighborhoods in $[q_1(\omega, \omega'), q_2(\omega, \omega')]$ consists of a finite number of closed random intervals, on each of which $f$ is a.s. continuous and hence a.s. uniformly continuous. Since the number of these intervals is finite, given $\varepsilon > 0$ there exists $\delta_2 > 0$ such that on each interval $\Delta_i(\omega, \omega') = [x_i(\omega, \omega'), x_{i-1}(\omega)]$, $i = 2, \dots, k$ whose length $l[\Delta_i(\omega, \omega')] = x_i(\omega, \omega') - x_{i-1}(\omega, \omega')$ a.s. is less than $\delta_2$ and which is entirely contained in one of the closed random intervals just mentioned, on which $f$ is a.s. continuous, we have

$$ \text{a.s.: } \Omega(f(\omega, x(\omega, \omega'))), \Delta_i(\omega, \omega')) < \frac{\varepsilon}{2[q_2(\omega, \omega') - q_1(\omega, \omega')]}. $$

We now choose $\delta = \min\{\delta_1, \delta_2\}$. Let $P$ be an arbitrary partition of $[q_1(\omega), q_2(\omega)]$ for which $\lambda(P) < \delta$. We break the sum $\sum_{i=1}^n \Omega(f(\omega, x(\omega, \omega'))), \Delta_i(\omega))$ corresponding to the partition $P$ into two parts:

$$ \begin{aligned} \sum_{i=1}^{n} \Omega(f(\omega, x(\omega, \omega'))), \Delta_i(\omega, \omega')) \Delta x_i(\omega, \omega') &= \\ & \sum' \Omega(f(\omega, x(\omega, \omega'))), \Delta_i(\omega, \omega')) \Delta x_i(\omega, \omega') + \\ & \sum'' \Omega(f(\omega, x(\omega, \omega'))), \Delta_i(\omega, \omega')) \Delta x_i(\omega, \omega'). \end{aligned} \quad (4.2.21) $$

The sum $\sum'$ contains the terms corresponding to random intervals $\Delta_i(\omega)$ of the partition having no points in common with any of the $\delta_1$-neighborhoods of the points of discontinuity. For these random intervals $\Delta_i(\omega, \omega')$ we have

$$ \text{a.s.: } \Omega(f(\omega, x(\omega, \omega'))), \Delta_i(\omega)) < \frac{\varepsilon}{2(q_2(\omega, \omega') - q_1(\omega, \omega'))} $$

and so
$$
\begin{equation} \tag{4.2.22}
\begin{aligned}
& a.s.: \sum' \Omega(f(\omega, x(\omega, \omega'))), \Delta_i(\omega, \omega')) \Delta x_i(\omega, \omega') < \\
& \phantom{a.s.: \sum' \Omega(f(\omega, x(\omega, \omega'))), \Delta_i(\omega, \omega')) \Delta} \frac{\varepsilon}{2(q_2(\omega, \omega') - q_1(\omega, \omega'))} \sum' \Delta x_i(\omega, \omega') \le \\
& \phantom{a.s.: \sum' \Omega(f(\omega, x(\omega, \omega'))), \Delta_i(\omega, \omega')) \Delta x_i(\omega, \omega')} \frac{\varepsilon}{2(q_2(\omega, \omega') - q_1(\omega, \omega'))} (q_2(\omega, \omega') - q_1(\omega, \omega')) = \frac{\varepsilon}{2}.
\end{aligned}
\end{equation}
$$

The sum of the lengths of the remaining intervals of the partition $P$, as
one can easily see, is at most $(\delta + 2\delta_1 + \delta) \le \frac{4\varepsilon}{8C \times k} = \varepsilon/2C$, and therefore

$$
a.s.: \sum'' \Omega(f(\omega, x(\omega, \omega'))), \Delta_i(\omega, \omega')) \Delta x_i(\omega, \omega') \leq C \frac{\varepsilon}{2C} = \frac{\varepsilon}{2}. \quad (4.2.23)
$$

Thus we find that for $\lambda(P) < \delta$ :

$$
a.s.: \sum_{i=1}^{n} \Omega(f(\omega, x(\omega, \omega'))), \Delta_i(\omega, \omega')) \Delta x_i(\omega, \omega') < \varepsilon. \quad (4.2.24)
$$

that is, the sufficient condition for integrability holds, and so
$f \in \Re[q_1(\omega, \omega'), q_2(\omega, \omega')]$.

**Corollary 4.2.3.** Any a.s. monotonic function on a closed random interval $[q_1(\omega, \omega'), q_2(\omega, \omega')]$ is integrable on that random interval.

**Proof.** It follows from the a.s. monotonicity of $f(\omega, x(\omega, \omega'))$ on $[q_1(\omega, \omega'), q_2(\omega, \omega')]$ that

$$
a.s.: \sum_{i=1}^{n} \Omega(f(\omega, x(\omega, \omega'))), [q_1(\omega, \omega'), q_2(\omega, \omega')]) = |f(\omega, q_2(\omega, \omega')) - f(\omega, q_1(\omega, \omega'))|.
$$

Suppose $\varepsilon > 0$ is given. We set

$$
\delta(\omega, \omega') = \frac{\varepsilon}{|f(\omega, q_2(\omega, \omega')) - f(\omega, q_1(\omega, \omega'))|}
$$

We assume

that $a.s.: f(\omega, q_2(\omega, \omega')) - f(\omega, q_1(\omega, \omega')) \neq 0$, since otherwise $f$ a.s. is constant, and
there is no doubt as to its integrability. Let $P$ be an arbitrary partition of
$[q_1(\omega, \omega'), q_2(\omega, \omega')]$ with mesh $a.s.: \lambda(P) < \delta(\omega, \omega')$. Then, taking account of the
a.s.monotonicity of $f$, we have

$$
(4.2.25)
$$

Thus $f$ satisfies the sufficient condition for integrability, and therefore
$f \in \Re[q_1(\omega, \omega'), q_2(\omega, \omega')]$.

**Remark 4.1.1.** A monotonic function may have a (countably) infinite set of discontinuities on a closed random $[q_1(\omega, \omega'), q_2(\omega, \omega')]$interval. For example, the function

defined by the relations

Apendix 5.Generalized random variables and double
stochastic processes.

**Definition 5.1.** Let $\Xi(\Sigma, \mathbb{R})$ be a set of the all $\mathbb{R}$- valued random variables
defined on a generalized probability space $\Sigma = (\Omega, \mathcal{F}, \tilde{\mathbf{P}})$, see def.5.4.

**Definition 5.2.** Let $q_1(\omega, \omega')$ and $q_2(\omega, \omega')$ are $\Xi(\Sigma, \mathbb{R})$-valued random variables defined on a generalized probability space $\Sigma = (\Omega, \mathcal{F}, \tilde{\mathbf{P}})$, i.e.,

$q_{1,2}(\omega, \omega') : \Omega \to \Omega' = \Xi(\Sigma, \mathbb{R})$. Assume that a.s.: $-\infty < q_1(\omega, \omega') < q_2(\omega, \omega') < \infty$.

Let $\tilde{\Xi}(\Omega, \Sigma, \mathbb{R})$ be a set of the all $\Xi(\Sigma, \mathbb{R})$-valued random variables defined on a generalized probability space $\Sigma = (\Omega, \mathcal{F}, \tilde{\mathbf{P}})$. Closed random interval

$$[q_1(\omega, \omega'), q_2(\omega, \omega')]$$

that is a subset $[q_1(\omega, \omega'), q_2(\omega, \omega')] \subset \tilde{\Xi}(\Omega', \Sigma, \mathbb{R})$ such that

a.s.: $-\infty < q_1(\omega, \omega') < q_2(\omega, \omega') < \infty$ and

$$\begin{array}{l} \forall \omega \{q(\omega, \omega') \in [q_1(\omega, \omega'), q_2(\omega, \omega')] \Leftrightarrow \\ \{q(\omega, \omega') \in \tilde{\Xi}(\Omega', \Sigma, \mathbb{R}) \wedge \text{a.s.: } (q_1(\omega, \omega') < q(\omega, \omega') < q_2(\omega, \omega'))\}. \end{array} \quad (5.1)$$

**Notation 5.1.** Assume that a.s.: $q_1(\omega, \omega') < q_2(\omega, \omega')$. Then we will write:

$$q_1(\omega, \omega') < q_2(\omega, \omega').$$

Assume that a.s. $q_1(\omega) \le q_2(\omega)$. Then we will write:

$$q_1(\omega, \omega') \le q_2(\omega, \omega').$$

**Definition 5.3.** Let $E$ be a separable complete metric space and let $\Sigma$ be its Borel $\sigma$-algebra. *Generalized random measure* it is a function $\mu: \Sigma \to \Xi(\Sigma, \mathbb{R})$, that satisfies:

(1) If $E_1 \subset E_2$, then a.s.: $\mu(E_1) < \mu(E_2)$.

(2) If $E_n \in S$, $n = 1, 2, \dots$ and $E_i \cap E_j = \emptyset$ $(i \ne j)$, then a.s.: $\mu(\cup_n E_n) = \sum_n \mu(E_n)$.

If we further impose a condition $\mu(\Omega) = 1$, then the generalized random measure so defined is called the generalized random probability measure and is usually denoted by $\tilde{\mathbf{P}}$.

**Definition 5.4.** The tuple $(\Omega, S, \tilde{\mathbf{P}})$ is then called the generalized probability space.

Suppose $(\Omega, S)$ and $(\Omega', S')$ are two measurable spaces. The function $X: \Omega \to \Omega'$ is called a *generalized random element*, if for every $A \in S'$, $X^{-1}(A) \in S$. If the first measurable space is equipped with a generalized random probability measure $\tilde{\mathbf{P}}$, then

the generalized random element $X: \Omega \to \Omega'$ induces a generalized random probability

measure on the second space $(\Omega', S')$ and given by $P \equiv \tilde{\mathbf{P}} \circ X^{-1}$, called the generalized random distribution of $X$.

**Definition 5.5.** If the second measurable space $(\Omega', S')$ is taken to be

$$(\Xi(\Sigma, \mathbb{R}), \mathfrak{B}_{\Xi(\Sigma, \mathbb{R})}),$$

where $\mathfrak{B}_{\Xi(\Sigma, \mathbb{R})}$ is the Borel $\sigma$-algebra on $\Xi(\Sigma, \mathbb{R})$ the function $X(\omega): \Omega \to \Omega'$ is called a
generalized random variable. For this case, the generalized random distribution of X is

completely determined by the random distribution function defined as

$$F_{X(\omega)}(x(\omega, \omega')) = \tilde{\mathbf{P}}(X(\omega) \le x(\omega, \omega')) \quad (5.2)$$

where $X \le x$ is defined as the set $\{X(\omega) \le x(\omega, \omega')\} = \{\omega \in \Omega | X(\omega) \le x(\omega, \omega')\}$.

The distribution function is continuously non-decreasing and gives the generalized random measure associated with $(x(\omega, \omega'), x(\omega') + d[x(\omega, \omega')]) \in \mathfrak{B}$ as $d[F_{X(\omega)}(x(\omega, \omega'))]$.

**Definition 5.6.** A special case is when the distribution function can be written in the form

$$F(x(\omega, \omega')) = \int_{-\infty(\omega, \omega')}^{x(\omega, \omega')} p(x(\omega, \omega'))d[x(\omega, \omega')] \quad (5.3)$$

In this case, the function $p(x(\omega, \omega')): \tilde{\Xi}(\Omega, \Sigma, \mathbb{R}) \to \tilde{\Xi}(\Omega, \Sigma, \mathbb{R})$ is called the random probability density function of generalized random variable $X(\omega, \omega')$, and the generalized

random measure associated with $d[x(\omega, \omega')]$ is

$$d[F_{X(\omega)}(x(\omega, \omega'))] = p(x(\omega, \omega'))d[x(\omega, \omega')].$$

**Definition 5.7.** Gaussian random distribution is a type of continuous random probability distribution for a real-valued generalized random variable. The general form

of its random probability density function is

$$p(x(\omega, \omega')) = \frac{1}{\sqrt{2\pi(\omega, \omega')\sigma^2}} \exp\left[-\frac{(x(\omega, \omega') - \mu)^2}{2\sigma^2}\right], \quad (5.3)$$

where a.s.: $\pi(\omega, \omega') = \pi$

With the knowledge of the distribution function, one can define the expectation value of

any function $h(X)$ as

$$\mathbb{E}[h(X(\omega, \omega'))] = \int_{-\infty(\omega, \omega')}^{\infty(\omega, \omega')} h(x(\omega, \omega'))d[F_{X(\omega)}(x(\omega, \omega'))]. \quad (5.4)$$

**Definition 5.8.** A double stochastic process is an indexed set of generalized random variables $X_t(\omega), t \in T$, where $T$ is called the index set and is usually taken (in the continuous case) to be $T = [0, \infty)$. In a physical setting, the index is time and a stochastic process can be interpreted as, for each $t \in T$, $X_t(\omega)$ picks an event $E \in \mathcal{S}$ with random probability $\tilde{\mathbf{P}}(E)$, and returns $X_t(E)$.

Consider generalized probability space $(\Omega, \mathcal{F}, \tilde{\mathbf{P}})$ and an 1-dimensional time-dependent

generalized random variable $X_t(\omega) : \Omega \subset \mathbb{R} \to \Xi(\Sigma, \mathbb{R})$, i.e. a $\Xi(\Sigma, \mathbb{R})$-valued function that
maps elements of the sample space to $\Xi(\Sigma, \mathbb{R})$.

Assume now that the random variable is time-dependent: $X(t, \omega) : \Omega \subset \mathbb{R} \rightarrow \Omega' = \Xi(\Sigma, \mathbb{R})$,

then, given a sequence of timesteps $t_1, t_2, \dots, t_N$ with $t_1 < t_2 < \dots < t_N$, we can write

$$ \{X(t_1, \omega) = x_1(\omega, \omega'), X(t_2, \omega) = x_2(\omega, \omega'), \dots, X(t_N, \omega) = x_N(\omega, \omega')\}, \quad (5.5) $$

where $x_1(\omega, \omega'), x_2(\omega, \omega'), \dots, x_N(\omega, \omega') \in \tilde{\Xi}(\Omega, \Sigma, \mathbb{R})$.

The sequence defined in Eq.(5.5) is a double stochastic process if the joint random probability density $p_\omega(x_1(\omega, \omega'), t_1, \dots, x_2(\omega, \omega'), t_2, \dots, x_N(\omega, \omega'), t_N)$, fully describes the system.

Depending on how the joint random probability density is defined, we can classify the double stochastic processes. Here, we consider two cases.

1. Purely double stochastic process or separable stochastic process. If successive values of $X(t)$ are statistically independent, then we the joint random probability density is written as

$$ a.s.: p(x_1(\omega, \omega'), t_1, \dots, x_2(\omega, \omega'), t_2, \dots, x_N(\omega, \omega'), t_N) = \prod_{i=1}^{N} p(x_i(\omega, \omega'), t_i). \quad (5.6) $$

The underlying idea is that the random probability of an event $x_i(\omega, \omega')$ occurring at a time $t_i$ does not depend on the past and in no way determines the future. In terms of conditional random probabilities, we can write

$$ \begin{aligned} a.s.: & p(x_N(\omega, \omega'), t_N | x_1(\omega, \omega'), t_1, \dots, x_2(\omega, \omega'), t_2, \dots, x_{N-1}(\omega, \omega'), t_{N-1}) \\ & = p(x_N(\omega, \omega'), t_N). \end{aligned} \quad (5.7) $$

2. A second example of a stochastic process is the Markov process, whose joint probability density is written as

$$ \begin{aligned} & p(x_1(\omega, \omega'), t_1, \dots, x_2(\omega, \omega'), t_2, \dots, x_N(\omega, \omega'), t_N) = \\ & = \prod_{i=2}^{N} p(x_i(\omega, \omega'), t_i | x_{i-1}(\omega, \omega'), t_{i-1}) p(x_1(\omega, \omega'), t_1), \end{aligned} \quad (5.8) $$

or in terms of conditional probabilities as

$$ \begin{aligned} & p(x_N(\omega, \omega'), t_N | x_1(\omega, \omega'), t_1, \dots, x_2(\omega, \omega'), t_2, \dots, x_{N-1}(\omega, \omega'), t_{N-1}) = \\ & \qquad p(x_N(\omega, \omega'), t_N | x_{N-1}(\omega, \omega'), t_{N-1}), \end{aligned} \quad (5.9) $$

i.e., double stochastic Markovian process is a process without memory, whose temporal evolution depends only on the present state, not on the past.

## Appendix 6. Double stochastic space-time white noise.
**Definition 6.1.** A distribution valued double stochastic Gaussian process with almost surely mean zero {$\dot{W}(t,x;\omega,\omega') : t \in [0,T], x \in \mathbb{R}^d$} is a double stochastic space-time white noise if

$$ \mathbb{E}(\dot{W}(t,x;\omega,\omega')\dot{W}(s,y;\omega,\omega')) = \delta(t-s)\delta(x-y). \quad (6.1) $$

More precisely :

**Definition 6.2.** We denote by $\mathcal{D}((0,T) \times \mathbb{R}^d; \mathbb{R}^d)$ the space of the infinitely differentiable functions with compact support in $(0,T) \times \mathbb{R}^d$ and values in $\mathbb{R}^d$.

1. For any $\xi \in \mathcal{D}((0,T) \times \mathbb{R}^d)$ the random variable $\dot{W}(\xi)$ is double stochastic Gaussian variable with almost surely mean zero

$$ \mathbb{E}(\dot{W}(\xi) = 0 \qquad (2.6.10) $$

For any $\xi_1(t,x), \xi_2(t,x) \in \mathcal{D}((0,T) \times \mathbb{R}^d)$ the double stochastic random variables $\dot{W}(\xi_1), \dot{W}(\xi_2)$ almost surely have covariance

$$ \bullet \quad \mathbb{E}(\dot{W}(\xi_1)\dot{W}(\xi_2)) = \int_{[0,T]\times\mathbb{R}^d} \xi_1(t,x) \cdot \xi_2(t,x)dxdt. \quad (2.6.11) $$

It is not difficult to construct a space-time white noise. In fact, let $\{f_j : j \in \mathbb{N}\}$ be a complete orthonormal basis of $L^2([0,T] \times \mathbb{R}^d)$ and $\{Z_j(\omega, \omega') : j \in \mathbb{N}\}$ be a family of independent Gaussian generalized random variables with mean zero and variance one. Then

$$ \dot{W}(t,x;\omega,\omega') := \sum_{j=1}^{\infty} f_j(t,x)Z_j(\omega,\omega') \quad (2.6.12) $$

is a space-time white noise, where the action is defined as

$$ (\dot{W}, \xi)_{D'} = \sum_{j=1}^{\infty} (\xi, f_j) Z_j(\omega, \omega'). \quad (2.6.13) $$

**Remark 2.6.1.** It is well known that the last action can be extended to

$\xi \in L^2([0, T] \times \mathbb{R}^d)$

using Itô isometry.

Let $\rho : \mathbb{R}^d \to [0, \infty)$ be an infinitely differentiable symmetric function with compact support such that $\int_{\mathbb{R}^d} \rho(x) dx = 1$. We will consider the mollifiers $\rho_n(x) = n^d \rho(nx)$, with $n \in \mathbb{N}$.

**Definition 2.6.3.** The regularization by $\rho$ of the space-time white noise $\dot{W}(t,x;\omega,\omega')$, denoted by $\dot{W}_{\rho_n}$, are defined to be

$$ \dot{W}_{\rho_n}(t,x;\omega,\omega') := \rho_n * \dot{W}(t,x;\omega,\omega'). \quad (2.6.14) $$

**Remark 2.6.2.** Note that $\dot{W}_{\rho_n}$ is white in time and colored in space, in fact we have that $\dot{W}_{\rho_n}(t,x)$ is a distribution valued Gaussian process with mean zero and covariance,

$$ \mathbb{E}(\dot{W}_{\rho_n}(t,x;\omega,\omega')\dot{W}_{\rho_n}(s,y;\omega,\omega')) = \delta(t-s)h_n(x-y) \quad (2.6.15) $$

where $h_n : \mathbb{R}^d \to \mathbb{R}$ is given by

$$ h_n(z) = \int_{\mathbb{R}^d} \rho_n(u)\rho_n(u+z)du. \quad (2.6.16) $$
In terms of the expansion (2.6.13) we have that

$$ \dot{W}_{\rho_n}(t, x; \omega, \omega') := \sum_{j=1}^{\infty} \rho_n * f_j(t, x) Z_j(\omega, \omega'). \qquad (2.6.17) $$

The mollified cylindrical Wiener process $W_{\rho_n,t}(x)$ associated with the space-time white noise $\dot{W}(t,x)$ is defined by

$$ W_{\rho_n,t}(x; \omega, \omega') := W_t(\rho_n(x - \cdot)). \qquad (2.6.18) $$

The distributional time derivative of $W_{\rho_n,t}(x; \omega, \omega')$ is $\dot{W}_{\rho_n}(t, x; \omega, \omega')$. We have that $W_{\rho_n}(x; \omega, \omega')$ is a double stochastic Brownian motion with quadratic variation,

$$ \langle W_{\rho_n}(x; \omega, \omega') \rangle_t = \|\rho\|^2 n^d \cdot t. \qquad (2.6.19) $$

**Proposition 2.6.1.** ($W_{\rho_n}(x)$) is a good weak approximation to the cylindrical double stochastic Wiener process $W$.

# Appendix 6. The Dirac equation.

The Dirac equation in the form originally proposed by Dirac reads

$$ (\beta mc^2 + c \sum_{n=1}^{3} \alpha_n p_n) \psi(x, t) = i\hbar \frac{\partial \psi(x, t)}{\partial t} \qquad (6.1) $$

where $\psi(x,t)$ is the wave function for an electron of rest mass $m$ with spacetime coordinates $(x,t), x = (x_1, x_2, x_3)$,

$$ p_1 = -i\hbar \frac{\partial}{\partial x_1}, p_2 = -i\hbar \frac{\partial}{\partial x_2}, p_3 = -i\hbar \frac{\partial}{\partial x_3} \qquad (6.2) $$

are the components of the momentum, understood to be the momentum operator in

the Schrödinger equation. $c$ is the speed of light, and $\hbar$ is the reduced Planck constant;

these fundamental physical constants reflect special relativity and quantum mechanics,

respectively. $\alpha_n$ and $\beta$ are 4 × 4 gamma matrices.

# Appendix 7. Double stochastic quantization of massive free Abelian gauge field

Free Abelian gauge field in the 4-dimensional Euclidean space is characterized by the

action integral
$$S[A] = \frac{1}{2} \int d^4 x [A_{\mu}(x)(\Box \delta_{\mu\nu} + \delta_{\mu\nu}m^2 + \partial_{\mu}\partial_{\nu})A_{\nu}(x)] \quad (7.1.1)$$

From (7.1.1) in terms of Fourier transforms $A_\mu(k, t)$ and $\eta(\omega, k, t)$, $\tilde{\eta}(\omega, k, t)$ we have the double stochastic Langevin equation

$$\frac{\partial}{\partial t} A_{\mu}^{a}(k, t) = -(k^{2} + m^{2}) \left( \delta_{\mu\nu} - \frac{k_{\mu}k_{\nu}}{k^{2} + m^{2}} \right) A_{\mu}^{a}(k, t) + \eta_{\mu}^{a}(\omega, k, t) + \tilde{\eta}_{\mu}^{a}(\omega, k, t) \quad (7.1.2)$$

where

$$\begin{align}
\langle \eta_{\mu}^{a}(\omega, k, t) \rangle_{\eta} &= \langle \tilde{\eta}_{\mu}^{a}(\omega, k, t) \rangle_{\tilde{\eta}} = 0, \nonumber \\
\langle \eta_{\mu}^{a}(\omega, k, t) \eta_{\nu}^{b}(\omega, k, t) \rangle_{\eta} &= 2\delta_{ab}\delta_{\mu\nu}\delta^{4}(k + k')\delta(t - t'), \tag{7.1.3} \\
\langle \eta_{\mu}^{a}(\omega, k, t) \eta_{\nu}^{b}(\omega, k, t) \rangle_{\tilde{\eta}} &= 2\delta_{ab}\delta_{\mu\nu}\delta^{4}(k + k')\delta(t - t'). \nonumber
\end{align}$$

Differential master equation corresponding to double stochastic Langevin equation (7.1.2) reads

$$\frac{\partial}{\partial t} A_{\mu}^{a}(k, t) = -(k^{2} + m^{2}) \left( \delta_{\mu\nu} - \frac{k_{\mu}k_{\nu}}{k^{2} + m^{2}} \right) A_{\mu}^{a}(k, t) + \eta_{\mu}^{a}(\omega, k, t). \quad (7.1.4)$$

Solving (7.1.4) under a special initial condition $A_\mu(k, 0) = 0$, one obtains

$$A_{\mu}^{a}(k, t) = \int_{0}^{\infty} G_{\mu\nu}^{ab}(k, t - t') \eta_{\nu}^{b}(\omega, k, t') dt', \quad (7.1.5)$$

where $G_{\mu\nu}^{ab}(k, t - t')$ is the Green function given by

$$\begin{align}
G_{\mu\nu}^{ab}(k, t - t') &= \delta_{ab} G_{\mu\nu}(k, t - t'), \nonumber \\
G_{\mu\nu}(k, t - t') &= \left\{ \left( \delta_{\mu\nu} - \frac{k_{\mu}k_{\nu}}{k^2+m^2} \right) e^{-(k^2+m^2)(t-t')} + \frac{k_{\mu}k_{\nu}}{k^2+m^2} \right\} \theta(t-t'), \tag{7.1.6}
\end{align}$$

where $G_{\mu\nu}(k, t - t')$ is the solution of the equation

$$\begin{align}
\left[ \delta_{\mu\kappa} \frac{\partial}{\partial t} + ((k^2 + m^2)\delta_{\mu\kappa} - k_{\mu}k_{\nu}) \right] G_{\mu\nu}(k, t - t') &= \delta_{\mu\nu} \delta(t - t'), \tag{7.1.7} \\
G_{\mu\nu}(k, 0_{-}) &= G_{\mu\nu}(k, 0_{+}) = 0. \nonumber
\end{align}$$

Thus correlation functions reads

$$\langle A_{\mu}^{a}(k, t) A_{\nu}^{b}(k', t') \rangle_{\eta} = (2\pi)^{4} \delta_{ab} \delta^{4}(k + k') D_{\mu\nu}^{(0)}(k, t; k', t'), \quad (7.1.8)$$

where

$$D_{\mu\nu}^{(0)}(k, t; k', t') = \left( \delta_{\mu\nu} - \frac{k_{\mu}k_{\nu}}{k^2+m^2} \right) e^{-(k^2+m^2)(t-t')} \frac{e^{2(k^2+m^2)\min(t,t')}}{k^2+m^2} + \frac{2k_{\mu}k_{\nu}}{k^2+m^2} \min(t,t'). \quad (7.1.9)$$
Thus

$$ \langle A_{\mu}^{a}(k,t) A_{\nu}^{b}(k',t') \rangle_{\eta} \rightarrow_{t=t' \rightarrow \infty} (2\pi)^{4} \delta_{ab} \delta^{4}(k+k') \times \\ \times \left\{ \frac{1}{k^{2} + m^{2}} \left( \delta_{\mu\nu} - \frac{k_{\mu}k_{\nu}}{k^{2} + m^{2}} \right) + 2t \frac{k_{\mu}k_{\nu}}{k^{2} + m^{2}} \right\}. \qquad (7.1.10) $$

## References

[1] G. Parisi and Wu-yong-shi, Sci. Sin. 24, 484 (1981).

[2] D. Zwanziger, Nucl. Phys. B192, 259 (1981); B193, 163 (1981);

[3] G. Marchesini, ibid. B191, 214 (1981).

[4] G. Parisi, Nucl. Phys. B180, 378 (1981); B205, 337 (1982).

[5] M. Masujima, Path Integral Quantization and Stochastic Quantization 2009. https://link.springer.com/book/10.1007/978-3-540-87851-3

[6] E. Gozzi, Functional-integral approach to Parisi-Wu stochastic quantization: Scalar theory. Phys. Rev. D 28, 1922 – Published 15 October 1983 DOI: https://doi.org/10.1103/PhysRevD.28.1922

[7] G. Menezes, N. F. Svaiter, Stochastic Quantization of Scalar Theory: Generalized Langevin Equation with Memory Kernel. Physica A374:617-630, 2007 DOI: https://doi.org/10.1016/j.physa.2006.07.038 https://arxiv.org/abs/hep-th/0511224v1

[8] J. Foukzon, Double Stochastic Quantization: Non perturbative approach to Stochastic Quantization. https://www.researchgate.net/publication/392621837_Double_Stochastic_QuantizationNon

[9] Z. Bern, Nucl. Phys. B251, 633 (1985).

[10] J. D. Breit, S. Gupta and A. Zaks, Nucl. Phys. B233, 61 (1984).

[11] J. Alfaro, Nucl. Phys. B253, 464 (1985).

[12] I.S. Gradshteyn and I.M. Ryzhik, Table of integrals, series, and products, Academic Press, London U.K. (2007).

[13] Г. Н. Ватсон, Теории бесседевых функций. Перев. с англ., ч.1, М., ИЛ 1949. https://ikfia.ysn.ru/wp-content/uploads/2018/01/Vatson_ch1_1949ru.pdf

[14] Y. Ageeva, P. Petrov, and M. Shaposhnikov, Nonrenormalizable theories and finite formulation of QFT, Phys. Rev. D 111, 025020 – Published 24 January, 2025 DOI: https://doi.org/10.1103/PhysRevD.111.025020

[15] J. Jachymski, I. Jó'zwik M. Terepeta, Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat. (2024) 118:140 https://doi.org/10.1007/s13398-024-01636-6

[16] P.H. DAMGAARD, STOCHASTIC QUANTIZATION WITH FERMIONS, Nuclear Physics B235[FS11] (1984) 75-92 North-Holland Pubhstung Company
[17] H. DAMGAARD and H. HUFFEL, STOCHASTIC QUANTIZATION, PHYSICS REPORTS (Review Section of Physics Letters) 152, Nos. 5 & 6 (1987) 227-398.
https:// homepage.univie.ac.at/helmuth.hueffel/PhysRep.pdf

[18] E. ABDALLA and R. L. VIANA, ANALYTIC STOCHASTIC REGULARIZATION IN QCD AND ITS SUPERSYMMETRIC EXTENSION, Modem Physics Letters A Vol. 4, No. 5 (1989) 491-499

[19] M.B. Gavela and H. Huffel, CERN Sep, 1985 13 pages, Published in: Nucl.Phys.B
275 (1986) 721-733 Published: 1986 DOI: 10.1016/0550-3213(86)90581-X
Report number: CERN-TH-4276/85

[20] T. Mogami, On quantization of massive non-Abelian gauge fields, https://arxiv.org/abs/0707.0025v2

[21] Mikio NAMIKI, Ichiro OHBA, Keisuke OKANO and Yoshiya YAMANAKA, Stochastic Quantization of Non-Abelian Gauge Field, Progress of Theoretical Physics, Vol. 69, No.5, May 1983

[22] M. Reed, B. Simon, Fourier Analysis, Self-Adjointness (Methods of Modern Mathematical Physics, Vol. 2) ISBN-10:0125850506, ISBN-13:978-0125850506