# Feynman-Kac formula for the heat equation driven by time-homogeneous white
noise potential.

Ramiro Scorolli Dipartimento di Scienze Statistiche Paolo Fortunati,
Università di Bologna, Bologna, Italy. e-mail<EMAIL_ADDRESS>
###### Abstract

We present a Feynman-Kac formula for the $1$-dimensional stochastic heat
equation (SHE) driven by a time-homogeneous Gaussian white noise potential,
where the noise is interpreted in the Wick-Itô-Skorokhod sense. Our approach
consist in constructing a Wong-Zakai-type approximation for the SHE from which
we are able to obtain an “approximating Feynman-Kac” representation via the
reduction of the approximated SHE to a deterministic partial differential
equation (PDE). Then we will show that those “approximating Feynman-Kac”
converge to a well defined object we will call “formal Feynman-Kac”
representation which happens to coincide with the unique solution of SHE.

Key words and phrases: Stochastic heat equation, Feynman-Kac, Parabolic
Anderson Model, Wick product, Wiener chaos expansion.

AMS 2000 classification: 60H10; 60H30; 60H05.

##  1 Introduction.

In this work we will deal with the $1$-dimensional stochastic heat equation

$\displaystyle\begin{cases}\partial_{t}u(t,x)=\frac{1}{2}\partial_{xx}^{2}u(t,x)+u(t,x)\diamond\frac{d}{dx}W_{x},\;(t,x)\in[0,T]\times\mathbb{R}\\\
u(0,x)=u_{0}(x),\end{cases}$ (1.1)

driven by the (distributional) derivative of the Brownian motion
$\\{W_{x}\\}_{x\in\mathbb{R}}$. From now on, in a slight abuse of notation we
will denote $\dot{W}_{x}=\frac{d}{dx}W_{x}$.

The initial condition $u_{0}:\mathbb{R}\to\mathbb{R}$ is assumed to be a
bounded, deterministic Borel-measurable function. In (1.1) the symbol
“$\diamond$” denotes the Wick product (e.g. [7] or [5] ) and implies that the
noise is interpreted in the Itô-Skorohod sense.

In the case of space-time white noise potential equation (1.1) has been
extensively studied (e.g. [26],[11],[12],[1] and references therein). On the
other hand if the noise is assumed to be space-homogeneous white noise the
equation (1.1) is just a particular case of the well known Zakai equation from
non-linear filtering theory (see for instance the original paper [29] or the
review [6] and references therein).

Nevertheless the case of space-only white noise hasn’t received the same
attention. In [28] the author has showed that in this one-dimensional setting,
equation (1.1) admits a unique weak-solution which is square integrable for
any $(t,x)\in[0,T]\times\mathbb{R}$; such a solution is constructed employing
the Wiener Chaos expansion (see also Theorem $3.9$ of [10] ).

In [9][8] the author treated the SHE with time-homogeneous noise in the
$d$-dimensional case, showing existence, and providing numerous estimations of
the Lyapunov exponentials of the solutions. The former treats the case of
fractional Brownian motion, while in the latter the solution is showed to
exist in a flat Hilbert space similar to those introduced by Kondratiev (see
for instance [7]).

In [10] the authors studied, among other things, the existence and regularity
of the multidimensional version of (1.1) when the covariance structure of
$\\{W_{x}\\}_{x\in\mathbb{R}^{d}}$ satisfies certain conditions. They also
propose some formal Feynman-Kac representations for the solutions of the SHE
with space-time, and time-homogeneous Gaussian noise both in the Skorohod and
Stratonovich sense. Nevertheless no representation was proposed for the
solution of (1.1).

In [16] and [17] the authors study the SHE with time-homogeneous white noise
potential in a bounded interval using the concept of Wiener chaos solution and
propagator introduced in [23] (see also [22]). They obtain estimation of the
regularity of the solutions and well as existence and uniqueness results.

The aim of this article is to construct a Feynman-Kac representation for the
unique solution of (1.1), this hasn’t been achieved in [10] due to the great
generality under which the authors analyze the problem. In this simpler
framework we were able to use some of the techniques proposed by them to
construct such a representation.

Our analysis relies on the use of a “Wong-Zakai-type” approximation of
equation (1.1) where we replace the singular white noise $\\{\dot{W}_{x}\\}$
with a truncated Karuhnen-Loève-like series.

As in [20] this “approximating equation” is reduced to a deterministic partial
differential equation for which we are able to derive a Feynman-Kac
representation. Then we show that as the number of terms in the truncated
series goes to infinity this representation converges to a well-defined random
variable (for fixed $t$ and $x$) and that this limit-object is the unique
solution of (1.1) present in the literature. It’s worth noticing that due to
the structure of the approximated noise the sequence of “approximated
solutions” converges not only in the $\mathbb{L}^{p}$ norm for any
$p\in[1,\infty)$, but also almost surely.

##  2 Preliminaries

Let $(\Omega,\mathfrak{B},\mathbb{P}^{W})$ be a complete probability space
which carries a one dimensional Brownian motion $\\{W_{x}\\}_{x\in\mathbb{R}}$
indexed by the real line. Then consider the following Gaussian Hilbert space

$\displaystyle\mathfrak{H}(W):=\bigg{\\{}\int_{\mathbb{R}}h(x)dW_{x};h\in\mathbb{\mathbb{missing}}L^{2}(\mathbb{R})\bigg{\\}},$

where the stochastic integral over the real line is defined in [15] (Chapter
7, section 2). From now on we will let $\mathfrak{B}=\sigma(\mathfrak{H}(W))$,
i.e. the sigma algebra generated by the Gaussian Hilbert space
$\mathfrak{H}(W)$ then we have that the family of “stochastic exponentials”
(also known as Wick exponentials)

$\displaystyle\bigg{\\{}\mathcal{E}^{f}:=\exp\left(\int_{\mathbb{R}}f(x)dW_{x}-\frac{1}{2}\int_{\mathbb{R}}|f(x)|^{2}dx\right),f\in\mathbb{L}^{2}(\mathbb{R})\bigg{\\}},$

is total in $\mathbb{L}^{2}(\Omega,\mathcal{B},\mathbb{P}^{W})$
($\mathbb{L}^{2}(P)$ for short). According to the Wiener chaos decomposition,
any random variable $X\in\mathbb{L}^{2}(P)$ can be represented as:

$\displaystyle X=\sum_{n=0}^{\infty}I_{n}(h_{n}),\;\text{convergence in
}\mathbb{L}^{2}(\mathbb{P}^{W})$

where $I_{n}(\bullet)$ denotes the $n$-th multiple stochastic integral (e.g.
[15] Theorem 7.26), and the kernels
$h_{n}\in\mathbb{\mathbb{missing}}L^{2}(\mathbb{R}^{n})$ are symmetric
deterministic functions.

If $A:\mathbb{L}^{2}(\mathbb{P}^{W})\to\mathbb{L}^{2}(\mathbb{P}^{W})$ is a
bounded linear operator and we assume that $A$ is a contraction , then its
“second quantization operator”
$\Gamma(A):\mathbb{L}^{2}(\mathbb{P}^{W})\to\mathbb{L}^{2}(\mathbb{P}^{W})$ is
given by the action:

$\displaystyle\Gamma(A)X=\sum_{n=0}^{\infty}I_{n}(A^{\otimes n}h_{n}),$

notice that $A$ being a contraction is a sufficient condition for $\Gamma(A)$
to map $\mathbb{L}^{2}(\mathbb{P}^{W})$ into itself.

In the following we will also need an complete orthonormal system (CONS for
short) of the Hilbert space $\mathbb{L}^{2}(\mathbb{R})$. In particular we
will use the family of Hermite functions, $\\{e_{j}\\}_{j\in\mathbb{N}}$ that
are defined by:

$\displaystyle
e_{j}(x)=(-1)^{j-1}\left(\sqrt{\pi}2^{j-1}(j-1)!\right)^{-1/2}e^{-x^{2}/2}\left(\frac{d}{dx}\right)^{j-1}e^{-x^{2}},j\in\mathbb{N}$
(2.1)

where we used a different indexing in order to avoid having a $0$-order
element.

It’s straightforward to see that taking tensor products we obtain a CONS for
the space $\mathbb{\mathbb{missing}}L^{2}(\mathbb{R}^{n})$, i.e.
$\\{e_{i_{1}}\otimes\cdots\otimes
e_{i_{n}}\\}_{(i_{1},...,i_{n})\in\mathbb{N}^{n}}$.

One could show that the family $\\{e_{j}\\}_{j\in\mathbb{N}}$ belongs to the
Schwartz space of rapidly decreasing functions $S(\mathbb{R})$, and this is
particularly useful since this implies that we can grab any element of its
dual space $S^{\prime}(\mathbb{R})$ (space of tempered distributions) and
expand it into series of Hermite functions (e.g. [25]). In particular we will
work with the Dirac’s delta function $\delta_{x}\in
S^{\prime}(\mathbb{R}),x\in\mathbb{R}$, that can be written as:

$\displaystyle\delta_{x}(\bullet)=\sum_{n=0}^{\infty}e_{j}(x)e_{j}(\bullet),$
(2.2)

where the series clearly diverges in the classical sense.

We will now briefly discuss a particular type of product between random
variables; let $X,Y\in\mathbb{L}^{2}(\mathbb{P}^{W})$ with Wiener chaos
expansions given by

$\displaystyle
X=\sum_{n=0}^{\infty}I_{n}(h_{n}),\;\;\;Y=\sum_{n=0}^{\infty}I_{n}(g_{n})$

the new element

$\displaystyle X\diamond
Y=\sum_{n=0}^{\infty}I_{n}(k_{n}),\;\;k_{n}:=\sum_{j=1}^{n}h_{j}\odot
g_{n-j},$

where $``\odot"$ denotes the symmetric tensor product is called the Wick
product of $X$ and $Y$. It’s worth noticing that
$\mathbb{L}^{2}(\mathbb{P}^{W})$ is not closed under Wick multiplication. This
product has great relevance in several areas of stochastic analysis and
quantum physics, the interested reader is referred to [7] and [18] from a
complete account on the Wick product.

Finally all throughout this article we will denote with $p_{t-s}(x-y)$ the
heat kernel

$\displaystyle
p_{t-s}(x-y):=\frac{1}{\sqrt{2\pi(t-s)}}e^{-\frac{(x-y)^{2}}{2(t-s)}},$

and $(P_{t}f)(x)$ will denote the action of the heat semigroup on the function
$f$, i.e.

$\displaystyle(P_{t}f)(x)=\int_{\mathbb{R}}p_{t}(x-y)f(y)dy.$

##  3 Construction of the approximating equation.

From now on we will work on the probability space
$(\Omega,\mathfrak{B},\mathbb{P}^{W})$ introduced in the previous section. In
this section we will propose an approximation of equation (1.1), and thus the
first thing to do is to construct an opportune smooth approximation of the
singular White noise process $\\{\dot{W}_{x}\\}_{x\in\mathbb{R}}$. It’s known
that the singular white noise at $x\in\mathbb{R}$ could be formally seen as

$\displaystyle\dot{W}_{x}=\int_{\mathbb{R}}\delta_{x}(y)dW_{y},$

i.e. the stochastic integral of a Dirac delta function with mass at
$x\in\mathbb{R}$ (see [18]).

One possible approximation can be obtained by truncating the series in (2.2)
up to a certain finite value $K$ yielding:

$\displaystyle\dot{W}_{x}^{K}:=\sum_{j=1}^{K}e_{j}(x)\int_{\mathbb{R}}e_{j}(y)dW_{y},$
(3.1)

(notice that the latter is nothing more than the derivative of a truncated
Karuhnen-Loève expansion of the Brownian motion $W$) clearly $\dot{W}^{K}$
converges to $\dot{W}$ in some space of generalized random variables (see
formula $2.3.33$ of [7]).

If we substitute the singular white noise in (1.1) with (3.1) we obtain the
following “approximating equation”:

$\displaystyle\begin{cases}\partial_{t}u_{t,x}^{K}=\frac{1}{2}\partial_{xx}^{2}u_{t,x}^{K}+u_{t,x}^{K}\diamond\dot{W}_{x}^{K},\;(t,x)\in\mathbb{[}0,T]\times\mathbb{R}\\\
u(0,x)=u_{0}(x).\end{cases}$ (3.2)

Since the equation above involves non-trivial operations, such as taking the
Wick product between the solution and a random potential we should state what
a “solution” of the latter actually is. Following [12] we have:

######  Definition 3.1.

Let $K$ be any arbitrary positive integer, then random field
$u^{K}:[0,T]\times\mathbb{R}\times\Omega\to\mathbb{R}$ is said to be a weak
solution of (3.2) if for any fixed $(t,x)\in[0,T]\times\mathbb{R}$ we have
that $u^{K}_{t,x}\in\mathbb{L}^{2}(\mathbb{P}^{W})$ and for any random
variable $F\in\mathbb{D}^{1,2}$ it holds that:

$\displaystyle\mathbb{E}\left[u^{K}_{t,x}F\right]$
$\displaystyle=(P_{t}u_{0})(x)\mathbb{E}[F]$
$\displaystyle+\mathbb{E}\left[\bigg{\langle}\int_{0}^{t}\int_{\mathbb{R}}p_{t-s}(x-y)\left(\sum_{j=1}^{K}e_{j}(y)e_{j}(\bullet)\right)u_{s,y}^{K}dy\;ds,D_{(\bullet)}F\bigg{\rangle}_{\mathbb{\mathbb{missing}}L^{2}(\mathbb{R})}\right],$
(3.3)

where $D$ denotes the Malliavin derivative and $\mathbb{D}^{1,2}$ the Sobolev-
Malliavin space [24].

If in (3.2) we consider a space-independent potential $\dot{W}_{t}^{K}$ the
change of variables
$v^{K}(t,x)=u^{K}(t,x)\diamond\exp^{\diamond}\left(\int_{0}^{t}W_{s}^{K}ds\right)$
(see section $3.6$ of [7] for a detailed explanation) reduces (3.2) to a
standard homogeneous heat equation. Unfortunately, since the potential in our
case is time-homogeneous (space-dependent) this approach is not applicable.
For this reason in order to deal with equation (3.2) we will employ the
approach proposed [20] (see also [19] ). The following remark offers a brief
explanation of the latter.

######  Remark 3.2.

From now on we will let
$Z_{j}(\omega):=\int_{\mathbb{R}}e_{j}(y)dW_{y}(\omega),j\in\\{1,...,K\\},\;\omega\in\Omega$,
then we can rewrite equation (3.2) as

$\displaystyle\begin{cases}\partial_{t}u_{t,x}^{K}=\frac{1}{2}\partial_{xx}^{2}u_{t,x}^{K}+\sum_{j=1}^{K}e_{j}(x)u_{t,x}^{K}\diamond
Z_{j},\;(t,x)\in\mathbb{[}0,T]\times\mathbb{R}\\\ u(0,x)=u_{0}(x).\end{cases}$
(3.4)

It’s known (e.g. formula $(2.5)$ of [13] or Theorem $9.20$ of [18] for an
alternative but equivalent formulation) that the Wick product between a random
variable $X\in\mathbb{D}^{1,2}$ and a random variable
$I(g),\,g\in\mathbb{L}^{2}(\mathbb{R})$ in first Wiener Chaos is given by:

$\displaystyle X\diamond I(g)=X\cdot I(g)-\langle
DX,g\rangle_{\mathbb{\mathbb{missing}}L^{2}(\mathbb{R})}.$

Then proceeding formally we can write:

$\displaystyle u_{t,x}^{K}\diamond Z_{j}=u_{t,x}^{K}\cdot Z_{j}-\langle
Du_{t,x}^{K},e_{j}\rangle_{\mathbb{\mathbb{missing}}L^{2}(\mathbb{R})},$

and thus we are left to consider the equation

$\displaystyle\begin{cases}\partial_{t}u_{t,x}^{K}=\frac{1}{2}\partial_{xx}^{2}u_{t,x}^{K}+u_{t,x}^{K}\cdot
Z_{j}-\langle
Du_{t,x}^{K},e_{j}\rangle_{\mathbb{\mathbb{missing}}L^{2}(\mathbb{R})},\;(t,x)\in\mathbb{[}0,T]\times\mathbb{R}\\\
u(0,x)=u_{0}(x).\end{cases}$

If we assume that the solution of the equation above is of the form

$\displaystyle
u_{t,x}^{K}(\omega)=\mathfrak{u}^{K}(t,x,Z_{1}(\omega),...,Z_{K}(\omega))$

for some continuous function
$\mathfrak{u}:[0,T]\times\mathbb{R}\times\mathbb{R}^{K}\to\mathbb{R}$ that
must be determined we can use the chain rule for Malliavin derivatives and
obtain the following partial differential equation (PDE for short):

$\displaystyle\begin{cases}\partial_{t}\mathfrak{u}_{t,x,z}^{K}=\frac{1}{2}\partial_{xx}^{2}\mathfrak{u}_{t,x,z}^{K}+\mathfrak{u}_{t,x,z}^{K}\left(\sum_{j=1}^{K}e_{j}(x)z_{j}\right)-\sum_{j=1}^{K}e_{j}(x)\partial_{z_{j}}\mathfrak{u}_{t,x,z}^{K},\;(t,x,z)\in[0,T]\times\mathbb{R}\times\mathbb{R}^{K}\\\
\mathfrak{u}^{K}(0,x)=u_{0}(x).\end{cases}$

Upon multiplying both sides of the equation by
$\exp\left(-\frac{1}{2}\sum_{j=1}^{K}z_{j}^{2}\right)$ and defining
$\mathfrak{v}_{t,x,z}^{K}:=\mathfrak{u}_{t,x,z}^{K}\exp\left(-\frac{1}{2}\sum_{j=1}^{K}z_{j}^{2}\right)$,
we are able to get rid of the zero-order term above and obtain the following:

$\displaystyle\begin{cases}\partial_{t}\mathfrak{v}^{K}=\frac{1}{2}\partial_{xx}^{2}\mathfrak{v}^{K}-\sum_{j=1}^{K}e_{j}(x)\partial_{z_{j}}\mathfrak{v}^{K},\\\
\mathfrak{v}^{K}(0,x)=u_{0}(x)\times\exp\left(-\frac{1}{2}\sum_{j=1}^{K}z_{j}^{2}\right).\end{cases}$
(3.5)

Formally applying the classical Feynman-Kac formula we obtain:

$\displaystyle\mathfrak{v}^{K}(t,x,z)=\mathbb{E}^{B}\left[u_{0}(B_{t}^{x})\exp\bigg{\\{}-\frac{1}{2}\sum_{j=1}^{K}\left(z_{j}-\int_{0}^{t}e_{j}(B_{s}^{x})ds\right)^{2}\bigg{\\}}\right],$
(3.6)

where $\\{B_{t}\\}_{t\in[0,T]}$ is a $1$-dimensional Brownian motion defined
on the auxiliary filtered probability space
$(\tilde{\Omega},\mathcal{G},\\{\mathcal{G}^{B}_{t}\\}_{t\in[0,T]},\mathbb{P}^{B})$
and $\mathbb{E}^{B}$ denotes the expectation in this space.

The expression given by (3.6) is sometimes referred to as a “generalized
solution” of (3.5). It is worth mentioning that the latter becomes a classical
solution if suitable regularity assumptions on the coefficients of (3.5) are
in force. For more details see [4] (page 122). By definition the latter
implies that

$\displaystyle\mathfrak{u}^{K}(t,x,z)=\mathbb{E}^{B}\left[u_{0}(B_{t}^{x})\exp\bigg{\\{}\sum_{j=1}^{K}z_{j}\left(\int_{0}^{t}e_{j}(B_{s}^{x})ds\right)-\frac{1}{2}\sum_{j=1}^{K}\left(\int_{0}^{t}e_{j}(B_{s}^{x})ds\right)^{2}\bigg{\\}}\right].$

Letting
$u^{K}_{t,x}(\omega):=\mathfrak{u}^{K}(t,x,Z_{1}(\omega),...,Z_{K}(\omega))$
we obtain the formula in theorem 4.1.

##  4 Statements of the theorems.

######  Theorem 4.1.

The random field $u^{K}:[0,T]\times\mathbb{R}\times\Omega\to\mathbb{R}$
defined by:

$\displaystyle
u_{t,x}^{K}=\mathbb{E}^{B}\left[u_{0}(B_{t}^{x})\times\exp\bigg{\\{}\int_{\mathbb{R}}\sum_{j=1}^{K}\left(\int_{0}^{t}e_{j}(B_{s}^{x})ds\right)e_{j}(y)dW(y)-\frac{1}{2}\sum_{j=1}^{K}\left(\int_{0}^{t}e_{j}(B_{s}^{x})ds\right)^{2}\bigg{\\}}\right],$
(4.1)

is a weak solution of (3.2) (in the sense of definition 3.1 )

######  Theorem 4.2.

The family of random variables $\\{\Psi_{t,x}^{K};\;K\in\mathbb{N}\\}$ given
by:

$\displaystyle\Psi_{t,x}^{K}:=\int_{\mathbb{R}}\sum_{j=1}^{K}\left(\int_{0}^{t}e_{j}(B_{s}^{x})ds\right)e_{j}(y)dW(y)-\frac{1}{2}\sum_{j=1}^{K}\left(\int_{0}^{t}e_{j}(B_{s}^{x})ds\right)^{2},$

converges in $\mathbb{L}^{2}(\mathbb{P}^{W}\otimes\mathbb{P}^{B})$ to a well
defined random variable denoted by

$\displaystyle\Psi_{t,x}=\int_{0}^{t}\int_{\mathbb{R}}\delta(B_{s}^{x}-y)\;dW(y)\;ds-\frac{1}{2}\int_{\mathbb{R}}L_{a}(t)^{2}da.$

Furthermore conditional on $\mathcal{G}_{T}^{B}$ it holds that $\Psi_{t,x}\sim
N\left(-\frac{1}{2}\int_{\mathbb{R}}|L_{a}(t)|^{2}da,\int_{\mathbb{R}}|L_{a}(t)|^{2}da\right)$,
where $\\{L_{a}(t);(t,a)\in[0,T]\times\mathbb{R}\\}$ is the local time of
$\\{B_{t}\\}_{t\in[0,T]}$.

######  Theorem 4.3.

For fixed $(x,t)\in[0,T]\times\mathbb{R}$, $p\in[1,\infty)$, let
$u_{t,x}^{K}:=\mathbb{E}^{B}\left[u_{0}(B_{t}^{x})\exp\\{\Psi_{t,x}^{K}\\}\right]$
and denote
$u_{t,x}:=\mathbb{E}^{B}\left[u_{0}(B_{t}^{x})\exp\\{\Psi_{t,x}\\}\right]$,
then it holds that:

$\displaystyle\lim_{K\to\infty}\|u_{t,x}^{K}-u_{t,x}\|_{\mathbb{L}^{p}(\mathbb{P}^{W})}=0,$
(4.2)

and

$\displaystyle\lim_{K\to\infty}u_{t,x}^{K}=u_{t,x},\;\mathbb{P}^{W}-a.s.$
(4.3)

Furthermore we have that

$\displaystyle[0,T]\times\mathbb{R}\times\Omega\ni(t,x,\omega)\mapsto
u_{t,x}(\omega)$

is the unique solution for (1.1) (in the sense of theorem $3.1$ of [28]).

Using our Feynman-Kac representation we are able to derive the following
formulae for the moments of the solution (this formula has also been obtained
in [10] but no proof is provided).

######  Theorem 4.4.

Let $q\geq 2$ then the $q$-th moment of the unique solution of (1.1) is given
by

$\displaystyle\mathbb{E}^{W}\left[\left(u_{t,x}\right)^{q}\right]=\mathbb{E}^{B}\left[\left(\prod_{i=1}^{q}u_{0}(B_{t}^{(i)}+x)\right)\exp\bigg{\\{}\sum_{i<j}^{q}\int_{0}^{t}\int_{0}^{t}\delta_{0}(B_{s}^{(i)}-B_{r}^{(j)})dsdr\bigg{\\}}\right],$
(4.4)

$(B^{(1)},...,B^{(q)})$ are $q$ independent $1$-dimensional Brownian motions
and $\int_{0}^{t}\int_{0}^{t}\delta_{0}(B_{s}-B_{r}^{\prime})dsdr$ denotes the
intersection local time of the Brownian motions $B$ and $B^{\prime}$(e.g.
[21])

##  5 Proof of theorem 4.1

Letting $F=\mathcal{E}^{\xi},\;\xi\in\mathbb{L}^{2}(\mathbb{R})$ we have that
$\mathbb{E}\left[u_{t,x}^{K}F\right]$ coincides with the $S$-transform
(e.g.[18]) of $u_{t,x}^{K}$; i.e.

$\displaystyle\mathbb{E}^{W}\left[u_{t,x}^{K}F\right]=\mathcal{S}(u_{t,x}^{K})(\xi)$
$\displaystyle=\mathbb{E}^{W}\left[\mathbb{E}^{B}\left[u_{0}(B_{t}^{x})\times\exp\\{\Psi_{t,x}^{K}\\}\right]\times\mathcal{E}^{\xi}\right]$
$\displaystyle=\mathbb{E}^{B}\left[u_{0}(B_{t}^{x})\mathbb{E}^{W}\left[\exp\\{\Psi_{t,x}^{K}\\}\times\mathcal{E}^{\xi}\right]\right]$
$\displaystyle=\mathbb{E}^{B}\left[u_{0}(B_{t}^{x})\exp\bigg{\\{}\int_{0}^{t}\sum_{j=1}^{K}e_{j}(B_{s}^{x})\int_{\mathbb{R}}e_{j}(y)\xi(y)dy\;ds\bigg{\\}}\right],$

hence by the classical Feynman-Kac formula we can see that the latter is the
solution of

$\displaystyle\begin{cases}\partial_{t}\mathcal{S}_{t,x}(\xi)=\frac{1}{2}\partial_{xx}^{2}\mathcal{S}_{t,x}(\xi)+\mathcal{S}_{t,x}(\xi)\cdot\left(\sum_{j=1}^{K}e_{j}(x)\int_{\mathbb{R}}e_{j}(y)\xi(y)dy\right),\;(t,x)\in[0,T]\times\mathbb{R}\\\
\mathcal{S}_{0,x}(\xi)=u_{0}(x),\end{cases}$

for any $\xi\in\mathbb{L}^{2}(\mathbb{R})$.

The solution of this equation can be written in mild form as

$\displaystyle\mathcal{S}_{t,x}(\xi)$
$\displaystyle=(P_{t}u_{0})(x)+\int_{0}^{t}\int_{\mathbb{R}}p_{t-s}(x-y)\mathcal{S}_{s,y}(\xi)\sum_{j=1}^{K}e_{j}(y)\int_{\mathbb{R}}e_{j}(z)\xi(z)dz\;dy,$
$\displaystyle=(P_{t}u_{0})(x)+\int_{0}^{t}\int_{\mathbb{R}}p_{t-s}(x-y)\mathbb{E}^{W}\left[u_{t,x}^{K}\mathcal{E}^{\xi}\right]\sum_{j=1}^{K}e_{j}(y)\int_{\mathbb{R}}e_{j}(z)\xi(z)dz\;dy,$

or which is equivalent,

$\displaystyle\mathbb{E}^{W}\left[u_{t,x}^{K}\mathcal{E}^{\xi}\right]=(P_{t}u_{0})(x)+\mathbb{E}^{W}\left[\bigg{\langle}\int_{0}^{t}\int_{\mathbb{R}}p_{t-s}(x-y)\left(\sum_{j=1}^{K}e_{j}(z)e_{j}(\bullet)\right)u_{s,y}^{K}dyds,\xi(\bullet)\mathcal{E}^{\xi}\bigg{\rangle}_{\mathbb{\mathbb{missing}}L^{2}(\mathbb{R})}\right],$

which implies (3.1) since $\mathbb{E}^{W}[\mathcal{E}^{\xi}]=1$ and
$D_{(\bullet)}\mathcal{E}^{\xi}=\xi(\bullet)\mathcal{E}^{\xi}$ and the fact
that the stochastic exponentials are a dense family in $\mathbb{D}^{1,2}$.

##  6 Proof of theorem 4.2

We start by showing that $\\{\Psi_{t,x}^{K}\\}_{K\in\mathbb{N}}$ is a Cauchy
sequence in $\mathbb{L}^{2}(\mathbb{P}^{W}\otimes\mathbb{P}^{B})$, and we let
$\vvvert\bullet\vvvert_{p}$ be the norm on the Hilbert space
$\mathbb{L}^{p}(\mathbb{P}^{W}\otimes\mathbb{P}^{B})$.

Without lost of generality assume that $N\geq M$

$\displaystyle\vvvert\Psi_{t,x}^{N}-\Psi_{t,x}^{M}\vvvert_{2}^{2}$
$\displaystyle=\left\vvvert\sum_{j=M+1}^{N}z_{j}\times\left(\int_{0}^{t}e_{j}(B_{s}^{x})ds\right)-\frac{1}{2}\sum_{j=1}^{K}\left(\int_{0}^{t}e_{j}(B_{s}^{x})ds\right)^{2}\right\vvvert_{2}^{2}$
$\displaystyle=\mathbb{E}^{B}\mathbb{E}^{W}\left[\bigg{|}\sum_{j=M+1}^{N}z_{j}\times\left(\int_{0}^{t}e_{j}(B_{s}^{x})ds\right)-\frac{1}{2}\sum_{j=M+1}^{N}\left(\int_{0}^{t}e_{j}(B_{s}^{x})ds\right)^{2}\bigg{|}^{2}\right]$
$\displaystyle=\mathbb{E}^{B}\mathbb{E}^{W}\Bigg{[}\left(\sum_{j=M+1}^{N}z_{j}\times\left(\int_{0}^{t}e_{j}(B_{s}^{x})ds\right)\right)^{2}$
$\displaystyle-\left(\sum_{j=M+1}^{N}z_{j}\times\left(\int_{0}^{t}e_{j}(B_{s}^{x})ds\right)\right)\left(\sum_{j=M+1}^{N}\left(\int_{0}^{t}e_{j}(B_{s}^{x})ds\right)^{2}\right)$
$\displaystyle+\frac{1}{4}\left(\sum_{j=M+1}^{N}\left(\int_{0}^{t}e_{j}(B_{s}^{x})ds\right)^{2}\right)^{2}\Bigg{]}$
$\displaystyle=\mathbb{E}^{B}\left[\sum_{j=M+1}^{N}\left(\int_{0}^{t}e_{j}(B_{s}^{x})ds\right)^{2}+\frac{1}{4}\left(\sum_{j=M+1}^{N}\left(\int_{0}^{t}e_{j}(B_{s}^{x})ds\right)^{2}\right)^{2}\right].$

Then if would suffices to show that

$\displaystyle\lim_{N,M\to\infty}\mathbb{E}^{B}\left[\sum_{j=M+1}^{N}\left(\int_{0}^{t}e_{j}(B_{s}^{x})ds\right)^{2}+\frac{1}{4}\left(\sum_{j=M+1}^{N}\left(\int_{0}^{t}e_{j}(B_{s}^{x})ds\right)^{2}\right)^{2}\right]=0.$

Using (9.1) we have that

$\displaystyle\sum_{j=M+1}^{N}\left(\int_{0}^{t}e_{j}(B_{s}^{x})ds\right)^{2}\leq\int_{\mathbb{R}}|L_{a}(t)|^{2}da=\alpha_{t},$

for any positive integers $N\geq M$. This together with the fact that the
random variable $\alpha_{t}$ is exponentially integrable (e.g. [21] p. $178$)
allows us to use the Dominated Convergence theorem to bring the limit inside
the expectation.

Finally from (9.1) we know that the sequence

$\displaystyle
S_{n}=\sum_{j=1}^{n}\left(\int_{0}^{t}e_{j}(B_{s}^{x})ds\right)^{2},$

is $\mathbb{P}^{B}$-a.s. convergent and thus we have that

$\displaystyle|S_{N}-S_{M}|=\sum_{j=M+1}^{N}\left(\int_{0}^{t}e_{j}(B_{s}^{x})ds\right)^{2}\to
0,\;\mathbb{P}^{B}-a.s.$

when $N,M\to\infty$.

This implies that

$\displaystyle\lim_{N,M\to\infty}\vvvert\Psi_{t,x}^{N}-\Psi_{t,x}^{M}\vvvert_{2}^{2}=0.$

Furthermore notice that conditional on $\mathcal{G}_{T}^{B}$ the random
variable $\Psi_{t,x}^{N}$ is a Gaussian random variable with mean
$-\frac{1}{2}\sum_{j=1}^{K}\left(\int_{0}^{t}e_{j}(B_{s}^{x})ds\right)^{2}$
and variance $\sum_{j=1}^{K}\left(\int_{0}^{t}e_{j}(B_{s}^{x})ds\right)^{2}$
and since the $\mathbb{L}^{2}(P^{W})$ limit preserves the Gaussianity we have
that conditional on $\mathcal{G}_{T}^{B}$ the random variable $\Psi_{t,x}\sim
N\left(-\frac{1}{2}\int_{\mathbb{R}}|L_{a}(t)|^{2}da,\int_{\mathbb{R}}|L_{a}(t)|^{2}da\right)$.

##  7 Proof of theorem 4.3

The proof of our main theorem will be done in several steps:

  1. 1.

Show that the “approximated Feynman-Kac” formula converges in
$\mathbb{L}^{2}(\mathbb{P}^{W})$ to the “formal Feynman-Kac”.

  2. 2.

Obtain the Wiener Chaos expansion of the “approximated Feynman-Kac”.

  3. 3.

Show that the latter converges in $\mathbb{L}^{2}(\mathbb{P}^{W})$ (as
$K\to\infty$) to the solution of (1.1) represented by the formal series given
in [28] and [8].

Then since the limit in $\mathbb{L}^{2}(\mathbb{P}^{W})$ is
$\mathbb{P}^{W}$-a.s. unique, we conclude that the solution given by the
formal Wiener chaos series in [28] and [8] coincides with the “formal Feynman-
Kac” formula.

The previous can be summarized by the following diagram,

$\mathbb{E}^{B}\left[u_{0}(B_{t}^{x})\exp\big{\\{}\Psi_{t,x}^{K}\big{\\}}\right]$$\mathbb{E}^{B}\left[u_{0}(B_{t}^{x})\exp\big{\\{}\Psi_{t,x}\big{\\}}\right]$$\sum_{n=0}^{\infty}I_{n}(f_{n}^{K}(t,x))$$\sum_{n=0}^{\infty}I_{n}(f_{n}(t,x))$Step
1$K\to\infty$Step 3$K\to\infty$Step 2$=$$\mathbf{=}\;(\textit{by uniqueness of
the limit})$

###  Step 1:

We start by showing the convergence of the “approximated Feynman-Kac” formula:

$\displaystyle\mathbb{E}^{W}\left[|u_{t,x}^{K}-u_{t,x}|^{p}\right]$
$\displaystyle=\mathbb{E}^{W}\big{|}\mathbb{E}^{B}\left[u_{0}(B_{t}^{x})\left(\exp\\{\Psi_{t,x}^{K}\\}-\exp\\{\Psi_{t,x}\\}\right)\right]\big{|}^{p}$
$\displaystyle\leq\|u_{0}\|_{\infty}^{p}\mathbb{E}^{W}\mathbb{E}^{B}\left[\big{|}\exp\\{\Psi_{t,x}^{K}\\}-\exp\\{\Psi_{t,x}\\}\big{|}^{p}\right]$

Since $\Psi_{t,x}^{K}\to\Psi_{t,x}$ in $\mathbb{L}^{2}(\mathbb{P}^{W}\otimes
P^{B})$, then $\exp\\{\Psi_{t,x}^{K}\\}\to\exp\\{\Psi_{t,x}\\}$ in
probability, in order to show the desired result we just need to prove that
$\vvvert\exp\\{\Psi_{t,x}^{K}\\}\vvvert_{p}\to\vvvert\exp\\{\Psi_{t,x}\\}\vvvert_{p}$.

Using the Tower rule and the fact that conditional on $\mathcal{G}_{T}^{B}$
the random variables $\Psi_{t,x}$ and $\Psi_{t,x}^{K}$ are Gaussian we have
that

$\displaystyle\vvvert\exp\\{\Psi_{t,x}\\}\vvvert^{p}_{p}$
$\displaystyle=\mathbb{E}^{W}\mathbb{E}^{B}|\exp\\{p\Psi_{t,x}\\}|$
$\displaystyle=\mathbb{E}^{B}\left[\mathbb{E}^{W}\left[\exp\\{p\Psi_{t,x}\\}|\mathcal{G}_{T}^{B}\right]\right]$
$\displaystyle=\mathbb{E}^{B}\left[\exp\bigg{\\{}\frac{p(p-1)}{2}\int_{\mathbb{R}}|L_{a}(t)|^{2}da\bigg{\\}}\right]<\infty,$

and

$\displaystyle\vvvert\exp\\{\Psi_{t,x}^{K}\\}\vvvert^{p}_{p}$
$\displaystyle=\mathbb{E}^{W}\mathbb{E}^{B}|\exp\\{p\Psi_{t,x}^{K}\\}|$
$\displaystyle=\mathbb{E}^{B}\left[\mathbb{E}^{W}\left[\exp\\{p\Psi_{t,x}\\}|\mathcal{F}_{T}^{B}\right]\right]$
$\displaystyle=\mathbb{E}^{B}\left[\exp\bigg{\\{}\frac{p(p-1)}{2}\sum_{j=1}^{K}\left(\int_{0}^{t}e_{j}(B_{s}^{x})ds\right)^{2}\bigg{\\}}\right].$

At this point we can use Monotone convergence theorem to bring the limit
inside the expectation, the continuity of the exponential function and (9.1)
implies the desired result.

###  Step 2:

Now we need to obtain the Wiener chaos decomposition of (4.1) and start by
noticing that conditional on $\mathcal{G}_{T}^{B}$ we can write

$\displaystyle\exp\\{\Psi_{t,x}^{K}\\}=\sum_{n=0}^{\infty}\frac{1}{n!}I_{n}\left(g_{n}^{K}(t,x,\bullet)\right),\;\text{convergence
in }\mathbb{L}^{2}(\mathbb{P}^{W})$

where the $n$-th kernel is given by:

$\displaystyle
g^{K}_{n}(t,x,\bullet)=\sum_{i_{1}=1}^{K}\cdots\sum_{i_{n}=1}^{K}\frac{1}{n!}\left(\int_{[0,t]^{n}}e_{i_{1}}(B_{s_{1}}^{x})\cdots
e_{i_{n}}(B_{s_{n}}^{x})ds_{1}...ds_{n}\right)(e_{i_{1}}\otimes\cdots\otimes
e_{i_{n}})(\bullet).$

From the latter it follows that

$\displaystyle u_{t,x}^{K}$
$\displaystyle=\mathbb{E}^{B}\Bigg{[}u_{0}(B_{t}^{x})\times\sum_{n=0}^{\infty}I_{n}\left(\sum_{i_{1}=1}^{K}\cdots\sum_{i_{n}=1}^{K}\frac{1}{n!}\left(\int_{[0,t]^{n}}e_{i_{1}}(B_{s_{1}}^{x})\cdots
e_{i_{n}}(B_{s_{n}}^{x})ds_{1}...ds_{n}\right)e_{i_{1}}\otimes\cdots\otimes
e_{i_{n}}\right)\Bigg{]}.$

Since the series is convergent in $\mathbb{L}^{2}(\mathbb{P}^{W})$ we can
apply Jensen inequality and monotone convergence to interchange the series
with the expectation yielding

$\displaystyle\sum_{n=0}^{\infty}I_{n}\left(\sum_{i_{1}=1}^{K}\cdots\sum_{i_{n}=1}^{K}\mathbb{E}^{B}\left[u_{0}(B_{t}^{x})\frac{1}{n!}\int_{[0,t]^{n}.}e_{i_{1}}(B_{s_{1}}^{x})\times\cdots\times
e_{i_{n}}(B_{s_{n}}^{x})ds_{1}...ds_{n}\right]e_{i_{1}}\otimes\cdots\otimes
e_{i_{n}}\right).$

Since we are summing over all possible combinations of the indexes
$(i_{1},...,i_{n})$ the expression above equals

$\displaystyle\sum_{n=0}^{\infty}I_{n}\left(\sum_{i_{1}=1}^{K}\cdots\sum_{i_{n}=1}^{K}\mathbb{E}^{B}\left[\int_{\mathbb{T}_{t,n}}u_{0}(B_{t}^{x})e_{i_{1}}(B_{s_{1}}^{x})\times\cdots\times
e_{i_{n}}(B_{s_{n}}^{x})ds_{1}...ds_{n}\right]e_{i_{1}}\otimes\cdots\otimes
e_{i_{n}}\right)$

where the time integrals are taken over the simplex

$\displaystyle\mathbb{T}_{t,n}:=\\{(s_{1},s_{2},...,s_{n});0\leq s_{1}\leq
s_{2}\leq\cdots\leq s_{n}\leq t\\}.$

An application of Fubini-Tonelli lemma shows that the kernel of the $n$-fold
multiple stochastic integral is given by:

$\displaystyle\int_{\mathbf{T}_{t,n}}\sum_{i_{1}=1}^{K}\cdots\sum_{i_{n}=1}^{K}e_{i_{1}}\otimes\cdots\otimes
e_{i_{n}}(\bullet)\mathbb{E}^{B}\left[u_{0}(B_{t}^{x})e_{i_{1}}(B_{s_{1}}^{x})\times\cdots\times
e_{i_{n}}(B_{s_{n}}^{x})\right]ds_{1}...ds_{n},$

the conclusion is stated in the following proposition.

######  Proposition 7.1.

Let $u_{t,x}^{K}$ be given by (4.1) then it holds that:

$\displaystyle u_{t,x}^{K}=\sum_{n=0}^{\infty}I_{n}(f_{n}^{K}(t,x)),$ (7.1)

where

$\displaystyle\begin{cases}f_{0}^{K}(t,x)&=(P_{t}u_{0})(x),\\\
f_{n}^{K}(t,x,\bullet)&=\int_{\mathbb{T}_{t,n}}\sum_{i_{1}=1}^{K}\cdots\sum_{i_{n}=1}^{K}e_{i_{1}}\otimes\cdots\otimes
e_{i_{n}}(\bullet)\\\
&\times\int_{\mathbb{R}^{n+1}}p_{t-s_{n}}(x_{n}-x_{n-1})\cdots
p_{s_{1}}(x_{0}-x)u_{0}(x_{n})e_{i_{1}}(x_{n-1})\times\cdots\times
e_{i_{n}}(x_{0})d\mathbf{x}\;d\mathbf{s}\end{cases}$ (7.2)

where $d\mathbf{x}:=dx_{0}\cdots dx_{n}$, $d\mathbf{s}:=ds_{1}\cdots ds_{n}$.

In [28] the author has shown that if the initial condition is deterministic
and square integrable, then equation (1.1) has a unique weak solution given by
the Wiener Chaos expansion:

$\displaystyle u(t,x)=\sum_{n=0}^{\infty}I_{n}(f_{n}(t,x)),$ (7.3)

where

$\displaystyle\begin{cases}f_{0}(t,x)=(P_{t}u_{0})(x),\\\
f_{n}(t,x;x_{1},...,x_{n})=\int_{\mathbb{T}_{t,n}}\int_{\mathbb{R}}p_{t-s_{n}}(x-x_{n})\cdots
p_{s_{1}}(x_{1}-x_{0})u_{0}(x_{0})dx_{0}d\mathbf{s},\end{cases}$ (7.4)

see also equations $(4.2)$ and $(4.3)$ of [8].

Now the idea is to show that (7.1) converges in
$\mathbb{L}^{2}(\mathbb{P}^{W})$ to (7.3) as $K\to\infty$ and for that we will
need the following:

######  Definition 7.2.

Let $K$ be some fixed positive integer then
$A_{K}:S^{\prime}(\mathbb{R})\to\mathbb{\mathbb{missing}}L^{2}(\mathbb{R})$ is
a self-adjoint projection operator defined by the action

$\displaystyle A_{K}f=A_{K}\left(\sum_{j=1}^{\infty}\langle f,e_{j}\rangle
e_{j}\right)=\sum_{j=1}^{K}\langle f,e_{j}\rangle e_{j},$ (7.5)

for any $f\in S^{\prime}(\mathbb{R})$, i.e. the orthogonal projection on the
linear span of the first $K$ elements of the CONS
$\\{e_{j}\\}_{j\in\mathbb{N}}$.

######  Proposition 7.3.

Let $u(t,x),(t,x)\in[0,T]\times\mathbb{R}$ denote the weak solution of (1.1)
given in [28] (eq. $(3.5)$). Then for any $(t,x)\in[0,T]\times\mathbb{R}$ it
holds that:

$\displaystyle u_{t,x}^{K}=\Gamma(A_{K})u(t,x),$ (7.6)

where $\Gamma(A_{K})$ stands for the second quantization of the projection
operator $A_{K}$.

###### Proof.

In order to prove the latter we need to show that

$\displaystyle f_{n}^{K}(t,x,\bullet)=(A_{K}^{\otimes
n}f_{n}(t,x))(\bullet)=\sum_{i_{1}=1}^{K}\cdots\sum_{i_{n}=1}^{K}\bigg{\langle}f_{n},e_{i_{1}}\otimes\cdots\otimes
e_{i_{n}}\bigg{\rangle}_{\mathbb{\mathbb{missing}}L^{2}(\mathbb{R}^{n})}e_{i_{1}}\otimes\cdots\otimes
e_{i_{n}}(\bullet),$

where $f_{n}^{K}$ and $f_{n}$ are defined by (7.2) and (7.4) respectively.

For the sake of simplicity we consider the case with $n=2$, the general case
does not present particular difficulties besides the more complex notation. In
this case (7.2) takes the form:

$\displaystyle f_{2}^{K}(t,x,\bullet)$
$\displaystyle=\int_{0}^{t}\int_{0}^{s_{2}}\sum_{i_{1}=1}^{K}\sum_{i_{2}=1}^{K}e_{i_{1}}\otimes
e_{i_{2}}(\bullet)$
$\displaystyle\times\int_{\mathbb{R}^{3}}p_{t-s_{2}}(x_{2}-x_{1})p_{s_{2}-s_{1}}(x_{1}-x_{0})p_{s_{1}}(x_{0}-x)u_{0}(x_{2})e_{i_{1}}(x_{1})e_{i_{2}}(x_{0})d\mathbf{x}\;d\mathbf{s}.$

We will define a new set of variables according to the prescription:

$\displaystyle r_{1}$ $\displaystyle:=t-s_{2},$ $\displaystyle r_{2}$
$\displaystyle:=t-s_{1},$ $\displaystyle y_{0}$ $\displaystyle:=x_{2},$
$\displaystyle y_{1}$ $\displaystyle:=x_{1},$ $\displaystyle y_{2}$
$\displaystyle:=x_{0},$

then we can rewrite the expression above as

$\displaystyle\int_{0}^{t}\int_{r_{1}}^{t}\sum_{i_{1}=1}^{K}\sum_{i_{2}=1}^{K}e_{i_{1}}\otimes
e_{i_{2}}(\bullet)\int_{\mathbb{R}^{3}}p_{r_{1}}(y_{1}-y_{0})p_{r_{2}-r_{1}}(y_{2}-y_{1})p_{t-r_{2}}(x-y_{2})u_{0}(y_{0})e_{i_{1}}(y_{1})e_{i_{2}}(y_{2})d\mathbf{y}dr_{2}dr_{1}$
$\displaystyle=\int_{0}^{t}\int_{0}^{r_{2}}\sum_{i_{1}=1}^{K}\sum_{i_{2}=1}^{K}e_{i_{1}}\otimes
e_{i_{2}}(\bullet)\int_{\mathbb{R}^{3}}p_{t-r_{2}}(x-y_{2})p_{r_{2}-r_{1}}(y_{2}-y_{1})p_{r_{1}}(y_{1}-y_{0})u_{0}(y_{0})e_{i_{1}}(y_{1})e_{i_{2}}(y_{2})d\mathbf{y}\;d\mathbf{r},$

which is equal to $(A_{K}^{\otimes 2}f_{2}(t,x))(\bullet)$. From here it’s
easy to see that

$\displaystyle
u_{t,x}^{K}=\sum_{n=0}^{\infty}I_{n}(f_{n}^{K}(t,x))=\sum_{n=0}^{\infty}I_{n}(A_{K}^{\otimes
n}f_{n}(t,x))=\Gamma(A_{K})u(t,x),$

which proves the result. ∎

### Step 3:

It’s straightforward to see that

$\displaystyle\left\|\sum_{n=0}^{\infty}I_{n}(f_{n}^{K}(t,x))-\sum_{n=0}^{\infty}I_{n}(f_{n}(t,x))\right\|_{\mathbb{L}^{2}(\mathbb{P}^{W})}^{2}$
$\displaystyle=\left\|\sum_{n=0}^{\infty}I_{n}(f_{n}(t,x)-A^{\otimes
n}_{K}f_{n}(t,x))\right\|_{\mathbb{L}^{2}(\mathbb{P}^{W})}^{2}$
$\displaystyle=\sum_{n=0}^{\infty}n!\|f_{n}(t,x)-(A^{\otimes
n}_{K}f_{n}(t,x))\|_{\mathbb{L}^{2}(\mathbb{R}^{n})}^{2}\to 0$

as $K\to\infty$. This together with the results obtained in Step 1 and the
unicity of the $\mathbb{L}^{2}(\mathbb{P}^{W})$ limit we conclude that

$\displaystyle\mathbb{E}^{B}\left[u_{0}(B_{t}^{x})\exp\big{\\{}\Psi_{t,x}\big{\\}}\right]=\sum_{n=0}^{\infty}I_{n}(f_{n}(t,x)).$

On the other hand from the propositions 7.3 and 10.1 we see that

$\displaystyle
u_{t,x}^{K}=\mathbb{E}^{W}\left[u(t,x)|\sigma(Z_{1},...,Z_{K})\right],$

and we also notice that
$\sigma(Z_{1},...,Z_{K})\uparrow\mathfrak{B}:=\sigma(\mathfrak{H}(W))$. Then
the martingale convergence theorem (e.g. theorem $35.6$ of [2]) gives us the
$\mathbb{P}^{W}$-a.s. convergence

##  8 Proof of theorem 4.4

The convergence in $\mathbb{L}^{p}(\mathbb{P}^{W}),\;p\in[1,\infty)$ of
$u_{t,x}^{K}$ to the solution of $u_{t,x}$ implies that for any
$q\in\mathbb{N}$ the $q$-th moment of $u_{t,x}^{K}$ converges to that of the
solution.

$\displaystyle\mathbb{E}^{W}\left[\left(u_{t,x}^{K}\right)^{q}\right]$
$\displaystyle=\mathbb{E}^{W}\left[\prod_{i=1}^{q}\mathbb{E}^{B}\left[u_{0}(B_{t}^{(i)}+x)\exp\big{\\{}\Psi_{t,x}^{K,(i)}\big{\\}}\right]\right]$
$\displaystyle=\mathbb{E}^{B}\left[\left(\prod_{i=1}^{q}u_{0}(B_{t}^{(i)}+x)\right)\mathbb{E}^{W}\left[\exp\bigg{\\{}\sum_{i=1}^{q}\Psi_{t,x}^{K,(i)}\bigg{\\}}\bigg{|}\mathcal{F}_{T}^{B}\right]\right]$
$\displaystyle=\mathbb{E}^{B}\bigg{[}\left(\prod_{i=1}^{q}u_{0}(B_{t}^{(i)}+x)\right)\mathbb{E}^{W}\left[\exp\bigg{\\{}\int_{\mathbb{R}}\sum_{i=1}^{q}\sum_{k=1}^{K}\left(\int_{0}^{t}e_{k}(B_{s}^{(i)}+x)ds\right)e_{k}(y)dW_{y}\bigg{\\}}\bigg{|}\mathcal{F}_{T}^{B}\right]$
$\displaystyle\times\exp\bigg{\\{}-\frac{1}{2}\sum_{i=1}^{q}\left(\int_{0}^{t}e_{k}(B_{s}^{(i)}+x)ds\right)^{2}\bigg{\\}}\bigg{]},$

using the fact that conditional on $\mathcal{G}_{T}^{B}$ the stochastic
integral appearing in the exponential is a centered Gaussian random variable
we can see that the latter equals

$\displaystyle=\mathbb{E}^{B}\bigg{[}\left(\prod_{i=1}^{q}u_{0}(B_{t}^{(i)}+x)\right)\exp\bigg{\\{}\left\|\sum_{i=1}^{q}\sum_{k=1}^{K}\left(\int_{0}^{t}e_{k}(B_{s}^{(i)}+x)ds\right)e_{k}(\bullet)\right\|_{\mathbb{\mathbb{missing}}L^{2}(\mathbb{R})}^{2}\bigg{\\}}$
$\displaystyle\times\exp\bigg{\\{}-\frac{1}{2}\sum_{i=1}^{q}\left(\int_{0}^{t}e_{k}(B_{s}^{(i)}+x)ds\right)^{2}\bigg{\\}}\bigg{]}$
$\displaystyle=\mathbb{E}^{B}\left[\left(\prod_{i=1}^{q}u_{0}(B_{t}^{(i)}+x)\right)\exp\bigg{\\{}\sum_{i<j}^{q}\sum_{k=1}^{K}\left(\int_{0}^{t}e_{k}(B_{s}^{(i)}+x)ds\right)\left(\int_{0}^{t}e_{k}(B_{r}^{(j)}+x)dr\right)\bigg{\\}}\right].$

Now we must take the limit for $K\to\infty$ (we can see that the exponential
function is dominated by $\exp\\{q\max_{1\leq i\leq
q}\int_{\mathbb{R}}|L_{a}^{(i)}(t)|^{2}da\\}$ which is integrable) yielding

$\displaystyle\mathbb{E}^{W}\left[\left(u_{t,x}\right)^{q}\right]=\lim_{K\to\infty}\mathbb{E}^{W}\left[\left(u_{t,x}^{K}\right)^{q}\right]=\mathbb{E}^{B}\left[\left(\prod_{i=1}^{q}u_{0}(B_{t}^{(i)}+x)\right)\exp\bigg{\\{}\sum_{i<j}^{q}\int_{0}^{t}\int_{0}^{t}\delta_{0}(B_{s}^{(i)}-B_{r}^{(j)})dsdr\bigg{\\}}\right].$
(8.1)

##  9 Appendix A: Local time

Consider the Brownian local time of a one dimensional Brownian motion
$\\{B_{t}^{x}\\}_{t\in[0,T]}$ starting at $x\in\mathbb{R}$, at level
$a\in\mathbb{R}$ and time $t\in[0,T]$:

$\displaystyle L_{a}^{x}(t)=\int_{0}^{t}\delta_{a}(B_{s}^{x})ds,$

and notice that the latter can be seen as the usual Brownian local time
$L_{a-x}(t)$.

It’s known (e.g. the proof of proposition XIII-2.1. of [27]) that for a fixed
$t$ the map $\mathbb{R}\ni a\mapsto L_{a}(t)$ is a.s. continuous and has
compact support, hence it follows that

$\displaystyle\alpha_{t}=\int_{\mathbb{R}}|L_{a}(t)|^{2}da<\infty,a.s.,$

this together with the invariance of Lebesgue measure implies that $a\mapsto
L_{a}^{x}(t)$ belongs to $\mathbb{\mathbb{missing}}L^{2}(\mathbb{R})$ almost
surely.

Then the following Fourier-like series expansion holds a.s.

$\displaystyle L_{a}^{x}(t)$
$\displaystyle=\sum_{j=1}^{\infty}\left(\int_{\mathbb{R}}L_{y}^{x}(t)e_{j}(y)dy\right)e_{j}(a)$
$\displaystyle=\sum_{j=1}^{\infty}\left(\int_{0}^{t}e_{j}(B_{s}^{x})ds\right)e_{j}(a),$

where in the last equality we’ve used the occupation time formula.

By the Parseval’s identity we have:

$\displaystyle\sum_{j=1}^{\infty}\left(\int_{0}^{t}e_{j}(B_{s}^{x})ds\right)^{2}=\int_{-\infty}^{\infty}|L_{a}^{x}(t)|^{2}da=\int_{-\infty}^{\infty}|L_{a}(t)|^{2}da<\infty\;a.s.$
(9.1)

##  10 Appendix B: Second quantization and Conditional expectation

Let $(\Omega,\mathcal{A},\mathbb{P}^{W})$ be a probability space then it’s
well know that if $X\in\mathbb{L}^{2}(\Omega,\mathcal{A},\mathbb{P}^{W})$ and
$\mathcal{G}\subset\mathcal{A}$ is a sub-sigma-algebra, the conditional
expectation $\mathbb{E}\left[X|\mathcal{G}\right]$ can be seen as the
orthogonal projection of $X$ on
$\mathbb{L}^{2}(\Omega,\mathcal{G},\mathbb{P}^{W})$. In this appendix we will
show an analogous property of the second quantization operator.

######  Proposition 10.1.

Let $A_{K}$ be the projection operator of definition 7.2 then the second
quantization operator $\Gamma(A_{K})$ coincides with the conditional
expectation $\mathbb{E}[\bullet|\sigma(Z_{1},...,Z_{K})]$ where
$\sigma(Z_{1},...,Z_{K})$ is the sigma algebra generated by the family of
i.i.d Gaussian random variables $(Z_{1},...,Z_{K})$.

###### Proof.

We consider again the complete probability space
$(\Omega,\mathfrak{B},\mathbb{P}^{W})$ treated in the introduction. Let
$X\in\mathbb{L}^{2}(\mathbb{P}^{W})$ then a result by Cameron and Martin [3]
tells us that $X$ has a series expansion of the form

$\displaystyle X$
$\displaystyle=\sum_{\alpha\in\mathcal{J}}x_{\alpha}\mathcal{H}_{\alpha},\;\text{convergence
in }\mathbb{L}^{2}(\mathbb{P}^{W})$

where $\mathcal{J}$ is the space of of all sequences
$\alpha=(\alpha_{1},\alpha_{2},...)$ with elements
$\alpha_{i}\in\mathbb{N}_{0}$ and with compact support and

$\displaystyle\mathcal{H}_{\alpha}:=\prod_{j=1}^{\infty}H_{\alpha_{j}}(Z_{j}),$

where $H_{n}(\bullet)$ is the $n$-th Hermite polynomial and
$Z_{j}:=\int_{\mathbb{R}}e_{j}(x)dW_{x}$, are known as “generalized Hermite
polynomials” or “Wick polynomials” (e.g. [7]) and

$\displaystyle
x_{\alpha}=\left(\prod_{j=1}^{\infty}\alpha_{j}!\right)^{-1}\mathbb{E}\left[X\mathcal{H}_{\alpha}\right].$

Now lets take the conditional expectation of $X$ given the sigma algebra
$\sigma_{K}:=\sigma(Z_{1},...,Z_{K})$. It’s well known that we are allowed to
interchange conditional expectation with an $\mathbb{L}^{2}$ convergent
series, yielding

$\displaystyle\mathbb{E}\left[X|\sigma_{K}\right]$
$\displaystyle=\sum_{\alpha\in\mathcal{J}}x_{\alpha}E\left[\mathcal{H}_{\alpha}|\sigma_{K}\right]$
$\displaystyle=\sum_{\alpha\in\mathcal{J}}x_{\alpha}\mathbb{E}\left[\prod_{j=1}^{\infty}H_{\alpha_{j}}\left(Z_{j}\right)\bigg{|}\sigma_{K}\right],$

and at this point we notice that the terms of the product involving $Z_{j}$
for $j\in\\{1,2,...,N\\}$ are $\sigma_{K}$-measurable and hence can be pulled
outside the conditional expectation,

$\displaystyle\mathbb{E}[X|\sigma_{K}]=\sum_{\alpha\in\mathcal{J}}x_{\alpha}\prod_{i=1}^{K}H_{\alpha_{i}}\left(Z_{i}\right)\mathbb{E}\left[\prod_{j=K+1}^{\infty}H_{\alpha_{j}}\left(Z_{j}\right)\bigg{|}\sigma_{K}\right],$

all the remaining terms are independent from $\sigma_{K}$, and mutually
independent which implies that

$\displaystyle\mathbb{E}[X|\sigma_{K}]=\sum_{\alpha\in\mathcal{J}}x_{\alpha}\prod_{i=1}^{K}H_{\alpha_{i}}\left(Z_{i}\right)\prod_{j=K+1}^{\infty}\mathbb{E}\left[H_{\alpha_{j}}\left(Z_{j}\right)\bigg{|}\sigma_{K}\right].$

Furthermore since the Hermite polynomials of a centered Gaussian random
variables can be seen as its Wick power i.e. $H_{n}(Z_{j})=Z_{j}^{\diamond n}$
([15] Theorem $3.19$), and since $\mathbb{E}(Z_{j}^{\diamond
n})=\mathbb{E}(Z_{j})^{n}=0$ we see that the only non-vanishing terms are
those corresponding to the $\alpha$’s containing only positive values in the
first $K$ entries (remember that $H_{0}(\cdot)\equiv 1$). This allows us to
conclude that

$\displaystyle\mathbb{E}[X|\sigma_{K}]=\sum_{\alpha\in\mathcal{J}^{K}}x_{\alpha}\mathcal{H}_{\alpha}$
(10.1)

where $\mathcal{J}^{K}:=\\{\alpha\in\mathcal{J}:\alpha_{i}=0,\forall i>K\\}$.

On the other hand we could write the Chaos decomposition in terms of multiple
Wiener integrals, i.e.

$\displaystyle X=\sum_{n=0}^{\infty}I_{n}(f_{n}),$

where the kernel $f_{n}$ is a symmetric function in
$\mathbb{\mathbb{missing}}L^{2}(\mathbb{R})$. Then by definition of the second
quantization operator we have

$\displaystyle\Gamma(A_{K})X$
$\displaystyle=\sum_{n=0}^{\infty}I_{n}(\Gamma(A_{K})^{\otimes n}f_{n})$
$\displaystyle=\sum_{n=0}^{\infty}\sum_{\alpha\in\mathcal{J}^{K}_{n}}\bigg{\langle}f_{n},\bigodot_{j=1}^{K}e_{j}^{\odot\alpha_{j}}\bigg{\rangle}_{\mathbb{\mathbb{missing}}L^{2}(\mathbb{R})^{\odot
n}}I_{n}\left(\bigodot_{j=1}^{K}e_{j}^{\odot\alpha_{j}}\right)$
$\displaystyle=\sum_{n=0}^{\infty}\sum_{\alpha\in\mathcal{J}^{K}_{n}}x_{\alpha}\mathcal{H}_{\alpha}$
$\displaystyle=\sum_{n=0}^{\infty}\sum_{\alpha\in\mathcal{J}^{K}_{n}}x_{\alpha}\mathcal{H}_{\alpha}$
$\displaystyle=\sum_{\alpha\in\mathcal{J}^{K}}x_{\alpha}\mathcal{H}_{\alpha}$
$\displaystyle=\mathbb{E}[X|\sigma_{K}],$

where we have used the following identity proved by Itô [14]

$\displaystyle\mathcal{H}_{\alpha}=I_{n}\left(\bigodot_{j=1}^{K}e_{j}^{\odot\alpha_{j}}\right),$

and for
$\alpha\in\mathcal{J}_{n}^{K}:=\\{\alpha\in\mathcal{J}:|\alpha|=n,\alpha_{i}=0,\forall
i>K\\}$ we let

$\displaystyle
x_{\alpha}=\bigg{\langle}f_{n},\bigodot_{j=1}^{K}e_{j}^{\odot\alpha_{j}}\bigg{\rangle}_{\mathbb{\mathbb{missing}}L^{2}(\mathbb{R})^{\odot
n}}=n!\bigg{\langle}f_{n},\bigodot_{j=1}^{K}e_{j}^{\odot\alpha_{j}}\bigg{\rangle}_{\mathbb{\mathbb{missing}}L^{2}(\mathbb{R}^{n})}.$

∎

## References

  * [1] Lorenzo Bertini and Nicoletta Cancrini  “The stochastic heat equation: Feynman-Kac formula and intermittence”  In _Journal of statistical Physics_ 78.5 Springer, 1995, pp. 1377–1401
  * [2] Patrick Billingsley  “Probability and measure”  John Wiley & Sons, 2008
  * [3] Robert H Cameron and William T Martin  “The orthogonal development of non-linear functionals in series of Fourier-Hermite functionals”  In _Annals of Mathematics_ JSTOR, 1947, pp. 385–392
  * [4] Mark Iosifovich Freidlin  “Functional Integration and Partial Differential Equations.(AM-109), Volume 109”  Princeton university press, 2016
  * [5] Håkon Gjessing et al.  “The Wick product”  In _Proceedings of the Third Finnish-Soviet Symposium on Probability Theory and Mathematical Statistics, Turku, Finland, August 13–16, 1991_ , 2020, pp. 29–67  De Gruyter
  * [6] AJ Heunis  “On the stochastic differential equations of filtering theory”  In _Applied mathematics and computation_ 37.3 Elsevier, 1990, pp. 185–218
  * [7] H. Holden, B. Øksendal, J. Ubøe and T. Zhang  “Stochastic Partial Differential Equations: A Modeling, White Noise Functional Approach”, Universitext  Springer New York, 2009
  * [8] Yaozhong Hu  “Chaos expansion of heat equations with white noise potentials”  In _Potential Analysis_ 16.1 Springer, 2002, pp. 45–66
  * [9] Yaozhong Hu  “Heat equations with fractional white noise potentials”  In _Applied Mathematics and Optimization_ 43.3 Springer, 2001, pp. 221–243
  * [10] Yaozhong Hu, Jingyu Huang, David Nualart and Samy Tindel  “Stochastic heat equations with general multiplicative Gaussian noises: Hölder continuity and intermittency”  In _Electronic Journal of Probability_ 20 Institute of Mathematical StatisticsBernoulli Society, 2015, pp. 1–50
  * [11] Yaozhong Hu and David Nualart  “Stochastic heat equation driven by fractional noise and local time”  In _Probability Theory and Related Fields_ 143.1 Springer, 2009, pp. 285–328
  * [12] Yaozhong Hu, David Nualart and Jian Song  “Feynman–Kac formula for heat equation driven by fractional white noise”  In _The Annals of Probability_ 39.1 Institute of Mathematical Statistics, 2011, pp. 291–326
  * [13] Yaozhong Hu and Bernt Øksendal  “Wick approximation of quasilinear stochastic differential equations”  In _Stochastic Analysis and Related Topics V_ Springer, 1996, pp. 203–231
  * [14] Kiyosi Itô  “Multiple wiener integral”  In _Journal of the Mathematical Society of Japan_ 3.1 The Mathematical Society of Japan, 1951, pp. 157–169
  * [15] Svante Janson  “Gaussian hilbert spaces”  Cambridge university press, 1997
  * [16] H-J Kim and Sergey V Lototsky  “Time-homogeneous parabolic Wick–Anderson model in one space dimension: regularity of solution”  In _Stochastics and Partial Differential Equations: Analysis and Computations_ 5.4 Springer, 2017, pp. 559–591
  * [17] Hyun-Jung Kim  “Stochastic parabolic Anderson model with time-homogeneous generalized potential: Mild formulation of solution”  In _Communications on Pure & Applied Analysis_ 18.2 American Institute of Mathematical Sciences, 2019, pp. 795
  * [18] Hui-Hsiung Kuo  “White noise distribution theory”  CRC press, 2018
  * [19] Alberto Lanconelli and Ramiro Scorolli  “A small time approximation for the solution to the Zakai Equation”  In _arXiv preprint arXiv:2105.12416_ , 2021
  * [20] Alberto Lanconelli and Ramiro Scorolli  “Wong-Zakai approximations for quasilinear systems of Itô’s type stochastic differential equations”  In _Stochastic Processes and their Applications_ , 2021  URL: https://doi.org/10.1016/j.spa.2021.07.007.
  * [21] Jean-François Le Gall  “Exponential moments for the renormalized self-intersection local time of planar Brownian motion”  In _Séminaire de probabilités de Strasbourg_ 28, 1994, pp. 172–180
  * [22] Sergey Lototsky, Remigijus Mikulevicius and Boris L Rozovskii  “Nonlinear filtering revisited: a spectral approach”  In _SIAM Journal on Control and Optimization_ 35.2 SIAM, 1997, pp. 435–461
  * [23] R Mikulevicius and BL Rozovskii  “Separation of observations and parameters in nonlinear filtering”  In _Proceedings of 32nd IEEE Conference on Decision and Control_ , 1993, pp. 1564–1569  IEEE
  * [24] David Nualart  “The Malliavin calculus and related topics”  Springer, 2006
  * [25] Rainer H Picard  “Hilbert spaces of tempered distributions, Hermite expansions and sequence spaces”  In _Proceedings of the Edinburgh Mathematical Society_ 34.2 Cambridge University Press, 1991, pp. 271–293
  * [26] Jürgen Potthoff, Gjermund Våge and Hisao Watanabe  “Generalized solutions of linear parabolic stochastic partial differential equations”  In _Applied Mathematics and Optimization_ 38.1 Springer, 1998, pp. 95–107
  * [27] Daniel Revuz and Marc Yor  “Continuous martingales and Brownian motion”  Springer Science & Business Media, 2013
  * [28] Hideaki Uemura  “Construction of the solution of 1-dimensional heat equation with white noise potential and its asymptotic behaviour”  In _Stochastic Analysis and Applications_ 14.4 Taylor & Francis, 1996, pp. 487–506
  * [29] Moshe Zakai  “On the optimal filtering of diffusion processes”  In _Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete_ 11.3 Springer, 1969, pp. 230–243