# Stochastic evolution equations with multiplicative noise

Tijana Levajković , Stevan Pilipović , Dora Seleši , Milica Žigić University
of Belgrade, Serbia<EMAIL_ADDRESS>of Sciences, University
of Novi Sad, Serbia<EMAIL_ADDRESS>of Sciences,
University of Novi Sad, Serbia<EMAIL_ADDRESS>of Sciences,
University of Novi Sad, Serbia<EMAIL_ADDRESS>
###### Abstract

We study parabolic stochastic partial differential equations (SPDEs), driven
by two types of operators: one linear closed operator generating a
$C_{0}-$semigroup and one linear bounded operator with Wick-type
multiplication, all of them set in the infinite dimensional space framework of
white noise analysis. We prove existence and uniqueness of solutions for this
class of SPDEs. In particular, we also treat the stationary case when the
time-derivative is equal to zero.

##  1 Introduction and definitions

We consider a stochastic Cauchy problem of the form

$\begin{split}\frac{d}{dt}U(t,x,\omega)&={\mathbf{A}}U(t,x,\omega)+{\mathbf{B}}\lozenge
U(t,x,\omega)+F(t,x,\omega)\\\ U(0,x,\omega)&=U^{0}(x,\omega),\end{split}$ (1)

where $t\in(0,T]$, $\omega\in\Omega$, and $U(t,\cdot,\omega)$ belongs to some
Banach space $X$. The operator $\mathbf{A}$ is densely defined, generating a
$C_{0}-$semigroup and $\mathbf{B}$ is a linear bounded operator which combined
with the Wick product $\lozenge$ introduces convolution-type perturbations
into the equation. All stochastic processes are considered in the setting of
Wiener-Itô chaos expansions. A comprehensive explanation of the action of the
operators $\mathbf{A}$ and $\mathbf{B}$ in this framework will be provided in
Section 2.

Our investigations in this paper are inspired by [12] where the authors
provide a comprehensive analysis of equations of the form

$\frac{d}{dt}u(t,x,\omega)={\mathbf{A}}u(t,x,\omega)+\delta({\mathbf{M}}u(t,x,\omega))={\mathbf{A}}u(t,x,\omega)+\int{\mathbf{M}}u(t,x,\omega)\lozenge
W(x,\omega)\;dx,$

where $\delta$ denotes the Skorokhod integral and $W$ denotes the spatial
white noise process. In Proposition 2.8 we prove that for every operator
${\mathbf{M}}$ there exists a corresponding operator $\mathbf{B}$ such that
${\mathbf{B}}\lozenge u=\delta({\mathbf{M}}u)$. On the other hand, the class
of operators $\mathbf{B}$ is much larger. This holds also for the class of
operators $\mathbf{A}$ we consider (a comprehensive analysis of all operators
is given in Section 2.1). Thus, we extend the results of [12] and [13] to a
more general class of stochastic differential equations which are driven by
two linear multiplicative operators: $\mathbf{A}$ acting with ordinary
multiplication, while $\mathbf{B}\lozenge$ is acting with the convolution-type
Wick product.

We have studied elliptic SPDEs, in particular the stochastic Dirichlet problem
of the form ${\mathbf{L}}\lozenge u+f=0$ in our previous papers [11], [18],
[19]. As a conclusion to this series of papers we study parabolic SPDEs of the
form (3). Such equations also include as a special case equations of the form
$\frac{d}{dt}u={\mathbf{L}}u+f$ and $\frac{d}{dt}u={\mathbf{L}}\lozenge u+f$,
where $L$ is a strictly elliptic second order partial differential operator.
These equations describe the heat conduction in random media (inhomogeneous
and anisotropic materials), where the properties of the material are modeled
by a positively definite stochastic matrix.

Other special cases of (3) include the heat equation with random potential
$\frac{d}{dt}u=\Delta u+{\mathbf{B}}\lozenge u$, the Schrödinger equation
$(i\hbar)\frac{d}{dt}u=\Delta u+{\mathbf{B}}\lozenge u+f$, the transport
equation $\frac{d}{dt}u=\frac{d^{2}}{dx^{2}}u+W\lozenge\frac{d}{dx}u$ driven
by white noise as in [20], the generalized Langevin equation
$\frac{d}{dt}u={\mathbf{J}}u+{\mathbf{C}}(Y^{\prime}),$ where $Y$ is a Lévy
process, ${\mathbf{J}}$ the infinitesimal generator of a $C_{0}-$semigroup and
${\mathbf{C}}$ a bounded operator, which was studied in [1], as well as the
equation $\frac{d}{dt}u={\mathbf{L}}u+W\lozenge u,$ where ${\mathbf{L}}$ is a
strictly elliptic partial differential operator as studied in [3] and [8].

Equations of the form $\frac{d}{dt}u={\mathbf{A}}u+{\mathbf{B}}W$ were also
studied in [14] and [15], where ${\mathbf{A}}$ is not necessarily generating a
$C_{0}-$semigroup, but an $r$-integrated or a convolution semigroup.

In order to solve (3) we apply the method of Wiener-Itô chaos expansions, also
known as the propagator method. With this method we reduce the SPDE to an
infinite triangular system of PDEs, which can be solved by induction. Summing
up all coefficients of the expansion and proving convergence in an appropriate
weight space, one obtains the solution of the initial SPDE.

We also consider the case of stationary equations
$\mathbf{A}U+\mathbf{B}\lozenge U+F=0$. In particular, elliptic SPDEs have
been studied in [11], [13], [18] and [19]. With the method of chaos expansions
one can also treat hyperbolic SPDEs [9] and SPDEs with singularities [21]. One
of its advantages is that it provides explicit solutions in terms of a series
expansion, which can be easily implemented also to numerical approximations
and computational simulations.

###  1.1 $C_{0}-$semigroups

We recall some well-known facts which will be used in the sequel (see [16]).
Let $X$ be a Banach space. If $B$ is a bounded linear operator on $X$ and $A$
is the infinitesimal generator of a $C_{0}-$semigroup $\\{T_{t}\\}_{t\geq 0}$
satisfying $||T_{t}||_{L(X)}\leq Me^{wt},\;t\geq 0,\mbox{ for some }\;M,w>0,$
then $A+B$ is the infinitesimal generator of a $C_{0}-$semigroup
$\\{S_{t}\\}_{t\geq 0},$ on $X$ satisfying

$\|S_{t}\|_{L(X)}\leq Me^{(w+M\|B\|_{L(X)})t},\;t\geq 0.$

Let $u(0)=u^{0}\in D=Dom(A)$ and $f\in C([0,\infty),X)$. Recall that
$u:[0,T]\to X$ is a _(classical) solution_ on $[0,T]$ to

$\frac{d}{dt}u(t)=Au(t)+f(t),\;t\in(0,T],\quad u(0)=u^{0},$ (2)

if $u$ is continuous on $[0,T]$, continuously differentiable on $(0,T]$,
$u(t)\in D,\;t\in(0,T]$ and the equation is satisfied on $(0,T]$. If $f\in
L^{1}((0,T),X)$, then $u(t)=T_{t}u^{0}+\int_{0}^{t}T_{t-s}f(s)ds,t\in[0,T]$
belongs to $C([0,T],X)$, and it is called a _mild solution._ Clearly, a mild
solution that is continuously differentiable on $(0,T]$ is a classical
solution.

Let $f\in L^{1}((0,T),X)\cap C((0,T],X)$ and
$v(t)=\int_{0}^{t}T_{t-s}f(s)ds,t\in[0,T].$ The initial value problem has a
solution $u$ for every $u^{0}\in D$ if one of the following conditions is
satisfied (see [16]):

  * (i)

$v$ is continuously differentiable on $(0,T)$.

  * (ii)

$v(t)\in D$ for $0<t\leq T$ and $Av(t)$ is continuous on $(0,T]$.

If the initial value problem has a solution on $[0,T]$ for some $u^{0}\in D$,
then $v(t)$ satisfies both (i) and (ii). Note that if $f\in C^{1}([0,T],X)$
then conditions (i) and (ii) are fulfilled. Moreover, if $f\in C^{1}([0,T],X)$
and $u^{0}\in D(A)$, then for the solution $u$ of (2) we have that $u\in
C^{1}([0,T],X)$ and $\frac{d}{dt}u(0)=Au^{0}+f(0)$.

###  1.2 Generalized stochastic processes

Denote by $(\Omega,\mathcal{F},P)$ the Gaussian white noise probability space
$(S^{\prime}(\mathbb{R}),\mathcal{B},\mu),$ where $S^{\prime}(\mathbb{R})$
denotes the space of tempered distributions, $\mathcal{B}$ the Borel sigma-
algebra generated by the weak topology on $S^{\prime}(\mathbb{R})$ and $\mu$
the Gaussian white noise measure corresponding to the characteristic function

$\int_{S^{\prime}(\mathbb{R})}\,e^{{i\langle\omega,\phi\rangle}}d\mu(\omega)=\exp\left[-\frac{1}{2}\|\phi\|^{2}_{L^{2}(\mathbb{R})}\right],\quad\quad\phi\in
S(\mathbb{R}),$

given by the Bochner-Minlos theorem.

We recall the notions related to $L^{2}(\Omega,\mu)$ (see [7]) where
$\Omega=S^{\prime}(\mathbb{R})$ and $\mu$ is Gaussian white noise measure.
Define the set of multi-indices $\mathcal{I}$ to be
$(\mathbb{N}_{0}^{\mathbb{N}})_{c}$, i.e. the set of sequences of non-negative
integers which have only finitely many nonzero components. Especially, we
denote by $\mathbf{0}=(0,0,0,\ldots)$ the multi-index with all entries equal
to zero. The length of a multi-index is
$|\alpha|=\sum_{i=1}^{\infty}\alpha_{i}$ for
$\alpha=(\alpha_{1},\alpha_{2},\ldots)\in\mathcal{I}$, and it is always
finite. Similarly, $\alpha!=\prod_{i=1}^{\infty}\alpha_{i}!$, and all other
operations are also carried out componentwise. We will use the convention that
$\alpha-\beta$ is defined if $\alpha_{n}-\beta_{n}\geq 0$ for all
$n\in\mathbb{N}$, i.e., if $\alpha-\beta\geq\mathbf{0}$, and leave
$\alpha-\beta$ undefined if $\alpha_{n}<\beta_{n}$ for some $n\in\mathbb{N}$.

The Wiener-Itô theorem (sometimes also referred to as the Cameron-Martin
theorem) states that one can define an orthogonal basis
$\\{H_{\alpha}\\}_{\alpha\in\mathcal{I}}$ of $L^{2}(\Omega,\mu)$, where
$H_{\alpha}$ are constructed by means of Hermite orthogonal polynomials
$h_{n}$ and Hermite functions $\xi_{n}$,

$H_{\alpha}(\omega)=\prod_{n=1}^{\infty}h_{\alpha_{n}}(\langle\omega,\xi_{n}\rangle),\quad\alpha=(\alpha_{1},\alpha_{2},\ldots,\alpha_{n}\ldots)\in\mathcal{I},\quad\omega\in\Omega=S^{\prime}(\mathbb{R}).$

Then, every $F\in L^{2}(\Omega,\mu)$ can be represented via the so called
_chaos expansion_

$F(\omega)=\sum_{\alpha\in\mathcal{I}}f_{\alpha}H_{\alpha}(\omega),\quad\omega\in
S^{\prime}(\mathbb{R}),\quad\sum_{\alpha\in\mathcal{I}}|f_{\alpha}|^{2}\alpha!<\infty,\quad
f_{\alpha}\in\mathbb{R},\quad\alpha\in\mathcal{I}.$

Denote by $\varepsilon_{k}=(0,0,\ldots,1,0,0,\ldots),\;k\in\mathbb{N}$ the
multi-index with the entry 1 at the $k$th place. Denote by $\mathcal{H}_{1}$
the subspace of $L^{2}(\Omega,\mu)$, spanned by the polynomials
$H_{\varepsilon_{k}}(\cdot)$, $k\in\mathbb{N}$. The subspace $\mathcal{H}_{1}$
contains Gaussian stochastic processes, e.g. Brownian motion is given by the
chaos expansion
$B(t,\omega)=\sum_{k=1}^{\infty}\int_{0}^{t}\xi_{k}(s)ds\;H_{\varepsilon_{k}}(\omega).$

Denote by $\mathcal{H}_{m}$ the $m$th order chaos space, i.e. the closure of
the linear subspace spanned by the orthogonal polynomials $H_{\alpha}(\cdot)$
with $|\alpha|=m$, $m\in\mathbb{N}_{0}$. Then the Wiener-Itô chaos expansion
states that $L^{2}(\Omega,\mu)=\bigoplus_{m=0}^{\infty}\mathcal{H}_{m}$, where
$\mathcal{H}_{0}$ is the set of constants in $L^{2}(\Omega,\mu)$.

It is well-known that the time-derivative of Brownian motion (white noise
process) does not exist in the classical sense. However, changing the topology
on $L^{2}(\Omega,\mu)$ to a weaker one, T. Hida [6] defined spaces of
generalized random variables containing the white noise as a weak derivative
of the Brownian motion. We refer to [6], [7] for white noise analysis (as an
infinite dimensional analogue of the Schwartz theory of deterministic
generalized functions).

Let
$(2\mathbb{N})^{\alpha}=\prod_{n=1}^{\infty}(2n)^{\alpha_{n}},\quad\alpha=(\alpha_{1},\alpha_{2},\ldots,\alpha_{n},\ldots)\in\mathcal{I}.$
We will often use the fact that the series
$\sum_{\alpha\in\mathcal{I}}(2\mathbb{N})^{-p\alpha}$ converges for $p>1$.
Define the Banach spaces

$(S)_{1,p}=\\{F=\sum_{\alpha\in\mathcal{I}}f_{\alpha}{H_{\alpha}}\in
L^{2}(\Omega,\mu):\;\|F\|^{2}_{(S)_{1,p}}=\sum_{\alpha\in\mathcal{I}}(\alpha!)^{2}|f_{\alpha}|^{2}(2\mathbb{N})^{p\alpha}<\infty\\},\quad
p\in\mathbb{N}_{0}.$

Their topological dual spaces are given by

$(S)_{-1,-p}=\\{F=\sum_{\alpha\in\mathcal{I}}f_{\alpha}{H_{\alpha}}:\;\|F\|^{2}_{(S)_{-1,-p}}=\sum_{\alpha\in\mathcal{I}}|f_{\alpha}|^{2}(2\mathbb{N})^{-p\alpha}<\infty\\},\quad
p\in\mathbb{N}_{0}.$

The Kondratiev space of generalized random variables is
$(S)_{-1}=\bigcup_{p\in\mathbb{N}_{0}}(S)_{-1,-p}$ endowed with the inductive
topology. It is the strong dual of
$(S)_{1}=\bigcap_{p\in\mathbb{N}_{0}}(S)_{1,p}$, called the Kondratiev space
of test random variables which is endowed with the projective topology. Thus,

$(S)_{1}\subseteq L^{2}(\Omega,\mu)\subseteq(S)_{-1}$

forms a Gelfand triplet.

The time-derivative of the Brownian motion exists in the generalized sense and
belongs to the Kondratiev space $(S)_{-1,-p}$ for $p\geq\frac{5}{12}$. We
refer to it as to _white noise_ and its formal expansion is given by
$W(t,\omega)=\sum_{k=1}^{\infty}\xi_{k}(t)H_{\varepsilon_{k}}(\omega).$

We extended in [17] the definition of stochastic processes also to processes
of the chaos expansion form
$U(t,\omega)=\sum_{\alpha\in\mathcal{I}}u_{\alpha}(t){H_{\alpha}}(\omega)$,
where the coefficients $u_{\alpha}$ are elements of some Banach space $X$. We
say that $U$ is an _$X$ -valued generalized stochastic process_, i.e.
$U(t,\omega)\in X\otimes(S)_{-1}$ if there exists $p>0$ such that
$\|U\|_{X\otimes(S)_{-1,-p}}^{2}=\sum_{\alpha\in\mathcal{I}}\|u_{\alpha}\|_{X}^{2}(2\mathbb{N})^{-p\alpha}<\infty$.

The _Wick product_ of stochastic processes
$F=\sum_{\alpha\in\mathcal{I}}f_{\alpha}H_{\alpha},G=\sum_{\beta\in\mathcal{I}}g_{\beta}H_{\beta}\in
X\otimes(S)_{-1}$ is

$F\lozenge
G=\sum_{\gamma\in\mathcal{I}}\sum_{\alpha+\beta=\gamma}f_{\alpha}g_{\beta}H_{\gamma}=\sum_{\alpha\in\mathcal{I}}\sum_{\beta\leq\alpha}f_{\beta}g_{\alpha-\beta}H_{\alpha},$

and the $n$th Wick power is defined by $F^{\lozenge
n}=F^{\lozenge(n-1)}\lozenge F$, $F^{\lozenge 0}=1$. Note that
$H_{n\varepsilon_{k}}=H_{\varepsilon_{k}}^{\lozenge n}$ for
$n\in\mathbb{N}_{0}$, $k\in\mathbb{N}$.

For example, let $X=C^{k}[0,T]$, $k\in\mathbb{N}$. In [18] we proved that
differentiation of a stochastic process can be carried out componentwise in
the chaos expansion, i.e. due to the fact that $(S)_{-1}$ is a nuclear space
it holds that $C^{k}([0,T],(S)_{-1})=C^{k}[0,T]\otimes(S)_{-1}$. This means
that a stochastic process $U(t,\omega)$ is $k$ times continuously
differentiable if and only if all of its coefficients $u_{\alpha}(t)$,
$\alpha\in\mathcal{I}$ are in $C^{k}[0,T]$.

The same holds for Banach space valued stochastic processes i.e. elements of
$C^{k}([0,T],X)\otimes(S)_{-1}$, where $X$ is an arbitrary Banach space. By
the nuclearity of $(S)_{-1}$, these processes can be regarded as elements of
the tensor product space

$C^{k}([0,T],X\otimes(S)_{-1})=C^{k}([0,T],X)\otimes(S)_{-1}=\bigcup_{p=0}^{\infty}C^{k}([0,T],X)\otimes(S)_{-1,-p}.$

##  2 Stochastic operators

######  Definition 2.1.

Let $X$ be a Banach space and $\mathbf{O}:X\otimes(S)_{-1}\rightarrow
X\otimes(S)_{-1}$ an operator acting on the space of stochastic processes. We
will call $\mathbf{O}$ to be a _coordinatewise operator_ if there exists a
family of operators $o_{\alpha}:X\rightarrow X$, $\alpha\in\mathcal{I}$, such
that
$\mathbf{O}(\sum_{\alpha\in\mathcal{I}}f_{\alpha}H_{\alpha})=\sum_{\alpha\in\mathcal{I}}o_{\alpha}(f_{\alpha})H_{\alpha}$
for all $F=\sum_{\alpha\in\mathcal{I}}f_{\alpha}H_{\alpha}\in
X\otimes(S)_{-1}$.

Clearly, not all operators are coordinatewise, for example
$\mathbf{O}(F)=F^{\lozenge 2}$ can not be written in this form.

######  Definition 2.2.

The subclass of _simple coordinatewise operators_ consists of operators for
which $o_{\alpha}=o_{\beta}=o$, $\alpha,\beta\in\mathcal{I}$, that is, they
can be written in form of
$\mathbf{O}(\sum_{\alpha\in\mathcal{I}}f_{\alpha}H_{\alpha})=\sum_{\alpha\in\mathcal{I}}o(f_{\alpha})H_{\alpha}$
for some operator $o:X\rightarrow X$.

For example, the operator of differentiation [18] and the Fourier transform
[21] are simple coordinatewise operators. The Ornstein-Uhlenbeck operator is a
coordinatewise operator but it is not a simple coordinatewise operator.

Note that even if all $o_{\alpha}$, $\alpha\in\mathcal{I}$, are bounded linear
operators, the coordinatewise operator $\mathbf{O}$ itself does not need to be
bounded. If $o_{\alpha}$, $\alpha\in\mathcal{I}$, are uniformly bounded by
some $C>0$, then $\mathbf{O}$ is also a bounded operator. This follows from

$\begin{split}\|\mathbf{O}(F)\|^{2}_{X\otimes(S)_{-1,-p}}&\leq\sum_{\alpha\in\mathcal{I}}\|o_{\alpha}\|^{2}_{L(X)}\|f_{\alpha}\|_{X}^{2}(2\mathbb{N})^{-p\alpha}\\\
&\leq
C^{2}\sum_{\alpha\in\mathcal{I}}\|f_{\alpha}\|_{X}^{2}(2\mathbb{N})^{-p\alpha}=C^{2}\|F\|^{2}_{X\otimes(S)_{-1,-p}}<\infty,\end{split}$

for $F\in X\otimes(S)_{-1,-p}$.

This condition is sufficient, but not necessary, and can be loosened by the
embedding $(S)_{-1,-p}\subseteq(S)_{-1,-q}$, $q\geq p$.

######  Lemma 2.3.

Let $\mathbf{O}$ be a coordinatewise operator for which all $o_{\alpha}$,
$\alpha\in\mathcal{I}$, are polynomially bounded i.e.
$\|o_{\alpha}\|_{L(X)}\leq R(2\mathbb{N})^{r\alpha}$ for some $r,R>0$. Then,
there exists $q\geq p$ such that $\mathbf{O}:X\otimes(S)_{-1,-p}\rightarrow
X\otimes(S)_{-1,-q}$ is bounded.

###### Proof.

Let $q\geq p+2r$. Then,

$\displaystyle\|\mathbf{O}(F)\|^{2}_{X\otimes(S)_{-1,-q}}$ $\displaystyle\leq
R^{2}\sum_{\alpha\in\mathcal{I}}(2\mathbb{N})^{2r\alpha}\|f_{\alpha}\|_{X}^{2}(2\mathbb{N})^{-q\alpha}=R^{2}\sum_{\alpha\in\mathcal{I}}\|f_{\alpha}\|_{X}^{2}(2\mathbb{N})^{-(q-2r)\alpha}$
$\displaystyle\leq
R^{2}\sum_{\alpha\in\mathcal{I}}\|f_{\alpha}\|_{X}^{2}(2\mathbb{N})^{-p\alpha}=R^{2}\|F\|^{2}_{X\otimes(S)_{-1,-p}}<\infty.$

Thus, $\|\mathbf{O}\|_{L(X)\otimes(S)_{-1}}\leq R$. ∎

Note that the condition $\|o_{\alpha}\|_{L(X)}\leq R(2\mathbb{N})^{r\alpha}$
for some $r,R>0$ is actually equivalent to stating that there exists $r>0$
such that
$\sum_{\alpha\in\mathcal{I}}\|o_{\alpha}\|^{2}_{L(X)}(2\mathbb{N})^{-r\alpha}<\infty$.

Throughout the paper we will consider the equation

$\begin{split}\frac{d}{dt}U(t,\omega)&={\mathbf{A}}U(t,\omega)+{\mathbf{B}}\lozenge
U(t,\omega)+F(t,\omega),\quad t\in(0,T],\omega\in\Omega,\\\
U(0,\omega)&=U^{0}(\omega),\end{split}$ (3)

where both operators $\mathbf{A}$ and $\mathbf{B}$ are assumed to be
coordinatewise operators, i.e. composed out of a family of operators
$\\{A_{\alpha}\\}_{\alpha\in\mathcal{I}}$,
$\\{B_{\alpha}\\}_{\alpha\in\mathcal{I}}$, respectively. The operators
$A_{\alpha}$, $\alpha\in\mathcal{I}$, are assumed to be infinitesimal
generators of $C_{0}-$semigroups with a common domain $D$ dense in $X$ and the
action of $\mathbf{A}$ is given by
$\mathbf{A}(U)=\sum_{\alpha\in\mathcal{I}}A_{\alpha}(u_{\alpha})H_{\alpha}$,
for $U=\sum_{\alpha\in\mathcal{I}}u_{\alpha}H_{\alpha}\in
Dom(\mathbf{A})\subseteq D\otimes(S)_{-1}$, where

$Dom(\mathbf{A})=\\{U=\sum_{\alpha\in\mathcal{I}}u_{\alpha}H_{\alpha}\in
D\otimes(S)_{-1}:\;\exists
p_{U}>0,\;\sum_{\alpha\in\mathcal{I}}\|A_{\alpha}(u_{\alpha})\|^{2}_{X}(2\mathbb{N})^{-p_{U}\alpha}<\infty\\}.$

The operators $B_{\alpha}$, $\alpha\in\mathcal{I}$, are assumed to be bounded
and linear on $X$, and the action of the operator
$\mathbf{B}\lozenge:X\otimes(S)_{-1}\to X\otimes(S)_{-1}$ is defined by

$\mathbf{B}\lozenge(U)=\sum_{\alpha\in\mathcal{I}}\sum_{\beta\leq\alpha}B_{\beta}(u_{\alpha-\beta})H_{\alpha}=\sum_{\gamma\in\mathcal{I}}\sum_{\alpha+\beta=\gamma}B_{\alpha}(u_{\beta})H_{\gamma}.$

In the next two lemmas we provide two sufficient conditions that ensure the
operator $\mathbf{B}\lozenge$ to be well-defined. Both conditions are actually
equivalent to the fact that $B_{\alpha}$, $\alpha\in\mathcal{I}$, are
polynomially bounded, but they provide finer estimates on the stochastic order
(Kondratiev weight) of the domain and codomain of $\mathbf{B}\lozenge$.

######  Lemma 2.4.

If the operators $B_{\alpha}$, $\alpha\in\mathcal{I}$, satisfy
$\sum_{\alpha\in\mathcal{I}}\|B_{\alpha}\|^{2}_{L(X)}(2\mathbb{N})^{-r\alpha}<\infty$,
then $\mathbf{B}\lozenge$ is well-defined as a mapping
$\mathbf{B}\lozenge:X\otimes(S)_{-1,-p}\rightarrow X\otimes(S)_{-1,-(p+r+m)}$,
$m>1$.

###### Proof.

For $U\in X\otimes(S)_{-1,-p}$ and $q=p+r+m$ we have

$\begin{split}\sum_{\gamma\in\mathcal{I}}\|\sum_{\alpha+\beta=\gamma}B_{\alpha}(u_{\beta})\|_{X}^{2}&(2\mathbb{N})^{-q\gamma}\leq\sum_{\gamma\in\mathcal{I}}\Big{[}\sum_{\alpha+\beta=\gamma}\|B_{\alpha}\|_{L(X)}\|u_{\beta}\|_{X}\Big{]}^{2}(2\mathbb{N})^{-(p+r+m)\gamma}\\\
&=\sum_{\gamma\in\mathcal{I}}(2\mathbb{N})^{-m\gamma}\left(\sum_{\alpha+\beta=\gamma}\|B_{\alpha}\|_{L(X)}^{2}(2\mathbb{N})^{-r\gamma}\right)\left(\sum_{\alpha+\beta=\gamma}\|u_{\beta}\|_{X}^{2}(2\mathbb{N})^{-p\gamma}\right)\\\
&\leq
M\left(\sum_{\alpha\in\mathcal{I}}\|B_{\alpha}\|_{L(X)}^{2}(2\mathbb{N})^{-r\alpha}\right)\left(\sum_{\beta\in\mathcal{I}}\|u_{\beta}\|_{X}^{2}(2\mathbb{N})^{-p\beta}\right)<\infty,\end{split}$

where $M=\sum_{\gamma\in\mathcal{I}}(2\mathbb{N})^{-m\gamma}<\infty$, for
$m>1$. ∎

######  Lemma 2.5.

If the operators $B_{\alpha}$, $\alpha\in\mathcal{I}$, satisfy
$\sum_{\alpha\in\mathcal{I}}\|B_{\alpha}\|_{L(X)}(2\mathbb{N})^{-\frac{r}{2}\alpha}<\infty$,
for some $r>0$, then $\mathbf{B}\lozenge$ is well-defined as a mapping
$\mathbf{B}\lozenge:X\otimes(S)_{-1,-r}\rightarrow X\otimes(S)_{-1,-r}$.

###### Proof.

For $U\in X\otimes(S)_{-1,-r}$, we have by the generalized Minkowski
inequality that

$\displaystyle\sum_{\gamma\in\mathcal{I}}\|\sum_{\alpha+\beta=\gamma}B_{\alpha}(u_{\beta})\|_{X}^{2}(2\mathbb{N})^{-r\gamma}$
$\displaystyle\leq$
$\displaystyle\sum_{\gamma\in\mathcal{I}}\Big{[}\sum_{\alpha+\beta=\gamma}\|B_{\alpha}\|_{L(X)}\|u_{\beta}\|_{X}\Big{]}^{2}(2\mathbb{N})^{-r\gamma}$
$\displaystyle\leq$
$\displaystyle\sum_{\gamma\in\mathcal{I}}\Big{[}\sum_{\alpha+\beta=\gamma}\|B_{\alpha}\|_{L(X)}(2\mathbb{N})^{-\frac{r}{2}\alpha}\|u_{\beta}\|_{X}(2\mathbb{N})^{-\frac{r}{2}\beta}\Big{]}^{2}$
$\displaystyle\leq$
$\displaystyle\left(\sum_{\alpha\in\mathcal{I}}\|B_{\alpha}\|_{L(X)}(2\mathbb{N})^{-\frac{r}{2}\alpha}\right)^{2}\sum_{\beta\in\mathcal{I}}\|u_{\beta}\|_{X}^{2}(2\mathbb{N})^{-r\beta}<\infty.$

∎

###  2.1 Special cases and relationship to other works

Some of the most important operators of stochastic calculus are the operators
of the Malliavin calculus. We recall their definitions in the generalized
${S^{\prime}(\mathbb{R})}$ setting [10].

  * $\bullet$

The _Malliavin derivative_ , $\mathbb{D}$, as a stochastic gradient in the
direction of white noise, is a linear and continuous mapping
$\mathbb{D}:X\otimes(S)_{-1}\rightarrow
X\otimes{S^{\prime}(\mathbb{R})}\otimes(S)_{-1}$ given by

$\mathbb{D}u=\sum_{\alpha\in\mathcal{I}}\sum_{k\in\mathbb{N}}\,\alpha_{k}\,u_{\alpha}\,\otimes\,\xi_{k}\,\otimes{H}_{\alpha-\varepsilon_{k}},\quad\mbox{
for }u=\sum_{\alpha\in\mathcal{I}}u_{\alpha}\otimes H_{\alpha}.$

In terms of quantum theory it corresponds to the annihilation operator
reducing the order of the chaos space (
$\mathbb{D}:\mathcal{H}_{m}\rightarrow\mathcal{H}_{m-1}$).

  * $\bullet$

The _Skorokhod integral_ , $\delta$, as an extension of the Itô integral to
non-anticipating processes, is a linear and continuous mapping
$\delta:X\otimes{S^{\prime}(\mathbb{R})}\otimes(S)_{-1}\rightarrow
X\otimes(S)_{-1}$ given by

$\delta(F)=\sum_{\alpha\in\mathcal{I}}\sum_{k\in\mathbb{N}}f_{\alpha}\otimes
v_{\alpha,k}\otimes H_{\alpha+\varepsilon_{k}},\quad\mbox{ for
}F=\sum_{\alpha\in\mathcal{I}}f_{\alpha}\otimes\left(\sum_{k\in\mathbb{N}}v_{\alpha,k}\,\xi_{k}\right)\otimes
H_{\alpha}.$

It is the adjoint operator of the Malliavin derivative and in terms of quantum
theory it corresponds to the creation operator increasing the order of the
chaos space ($\delta:\mathcal{H}_{m}\rightarrow\mathcal{H}_{m+1}$).

  * $\bullet$

The _Ornstein-Uhlenbeck operator_ , $\mathcal{R}$, as the composition of the
previous ones $\delta\circ\mathbb{D}$, is the stochastic analogue of the
Laplacian. It is a linear and continuous mapping
$\mathcal{R}:X\otimes(S)_{-1}\rightarrow X\otimes(S)_{-1}$ given by

$\mathcal{R}(u)=\sum_{\alpha\in\mathcal{I}}|\alpha|u_{\alpha}\otimes
H_{\alpha},\quad\mbox{ for }u=\sum_{\alpha\in\mathcal{I}}u_{\alpha}\otimes
H_{\alpha}.$

In terms of quantum theory it corresponds to the number operator. It is a
selfadjoint operator $\mathcal{R}:\mathcal{H}_{m}\rightarrow\mathcal{H}_{m}$
with eigenvectors equal to the basis elements $H_{\alpha}$,
$\alpha\in\mathcal{I}$, i.e. $\mathcal{R}(H_{\alpha})=|\alpha|H_{\alpha}$,
$\alpha\in\mathcal{I}$. Thus, Gaussian processes with zero expectation are the
only fixed points for the Ornstein-Uhlenbeck operator.

Clearly, the Ornstein-Uhlenbeck operator is a coordinatewise operator, while
the Malliavin derivative and the Skorokhod integral are not coordinatewise
operators.

The Ornstein-Uhlenbeck operator is the infinitesimal generator of the
semigroup $T_{t}=e^{t\mathcal{R}}$, $t\geq 0$, given by
$T_{t}(u)=\sum_{\alpha\in\mathcal{I}}e^{-|\alpha|t}u_{\alpha}\otimes
H_{\alpha}$, for $u=\sum_{\alpha\in\mathcal{I}}u_{\alpha}\otimes H_{\alpha}\in
X\otimes(S)_{-1}$.

It is also closely connected to the Ornstein-Uhlenbeck process. The Ornstein-
Uhlenbeck process is the solution of the SDE
$du(t,\omega)=-u(t,\omega)dt+dB(t,\omega)$, $u(0,\omega)=u_{0}(x,\omega)$, and
it is given by
$u(t,\omega)=e^{-t}u_{0}(\omega)+\int_{0}^{t}e^{t-s}dB(s,\omega)$. It is a
Markov process with transition semigroup $\\{T_{t}\\}_{t\geq 0}$ [2]. The
solution of the generalized heat equation $\frac{d}{dt}u+\mathcal{R}(u)=0$,
$u(0)=u_{0}$, is given by $u=T_{t}(u_{0})$, i.e. $u(t,x)=(T_{t}u_{0})(x)$ and
$(T_{t}\varphi)(x)=E(\varphi(u(t,x))$ for any $\varphi\in C_{b}(\mathbb{R})$
and $u$ is the Ornstein-Uhlenbeck process.

Now we turn to our equation

$\frac{d}{dt}U(t,\omega)={\mathbf{A}}U(t,\omega)+{\mathbf{B}}\lozenge
U(t,\omega)+F(t,\omega),$ (4)

where $\mathbf{A}$ and $\mathbf{B}$ are coordinatewise operators as described
in Section 2, composed out of a family of operators
$\\{A_{\alpha}\\}_{\alpha\in\mathcal{I}}$,
$\\{B_{\alpha}\\}_{\alpha\in\mathcal{I}}$, respectively, where $A_{\alpha}$
are infinitesimal generators on $X$ and $B_{\alpha}$ are bounded linear
operators on $X$, both families being polynomially bounded, and their actions
given by

${\mathbf{A}}U=\sum_{\alpha\in\mathcal{I}}A_{\alpha}(u_{\alpha})H_{\alpha},\qquad\mbox{
for}\;\;U=\sum_{\alpha\in\mathcal{I}}u_{\alpha}H_{\alpha},$ (5)
${\mathbf{B}}\lozenge
U=\sum_{\alpha\in\mathcal{I}}\sum_{\beta\leq\alpha}B_{\beta}(u_{\alpha-\beta})H_{\alpha},\qquad\mbox{
for}\;\;U=\sum_{\alpha\in\mathcal{I}}u_{\alpha}H_{\alpha}.$ (6)

Some important special cases include the following:

  1. I)

Special cases for $\mathbf{A}$:

    1. 1)

$\mathbf{A}$ is a simple coordinatewise operator, i.e.
$A_{\alpha}=A,\alpha\in\mathcal{I},$ where $A$ is the infinitesimal generator
of a $C_{0}-$semigroup on $X$. Such operators are, for example the Laplacian
$\Delta$ on $X=W^{2,2}(\mathbb{R}^{n})$ or any strictly elliptic linear
partial differential operator of even order $P(x,D)=\sum_{|\iota|\leq
2m}a_{\iota}(x)D^{\iota}$. For example, second order elliptic operators can be
written in divergence form $L=\nabla\cdot(Q\nabla\cdot+b)+c\nabla\cdot$, where
$Q$ is a positively definite function matrix.

    2. 2)

$A_{\alpha}=A+R_{\alpha},\alpha\in\mathcal{I}$, where $A$ is as in 1), while
$R_{\alpha},\alpha\in\mathcal{I},$ are bounded linear operators on $X$ so that
$\mathbf{R}$ is a coordinatewise operator

${\mathbf{R}}U(t,\omega)=\sum_{\alpha\in\mathcal{I}}R_{\alpha}u_{\alpha}(t)H_{\alpha}(\omega).$

Especially, if we take $A=0$ and $R_{\alpha}$ to be multiplication operators
$R_{\alpha}(x)=r_{\alpha}\cdot x$, $x\in X$, then the resulting operator
$\mathbf{R}$ is a self-adjoint operator with eigenvalues $r_{\alpha}$
corresponding to the eigenvectors $H_{\alpha}$ and thus represents a natural
generalization of the Ornstein-Uhlenbeck operator. For $r_{\alpha}=|\alpha|$,
$\alpha\in\mathcal{I}$, we retrieve the Ornstein-Uhlenbeck operator
$\mathcal{R}$.

Finally, we note that every bounded linear coordinatewise operator
$\mathbf{R}$ can be written in the form ${\mathbf{R}}u=\delta(\mathbf{M}u)$
where $\mathbf{M}$ is a generalization of the Malliavin derivative. This will
be done in Proposition 2.6.

  2. II)

Special cases for $\mathbf{B}$:

    1. 1)

$\mathbf{B}$ is an operator acting as a multiplication operator with a
deterministic function, i.e. $B_{\alpha}=b$ for $\alpha=(0,0,0,0,\ldots)$ and
$B_{\alpha}=0$ for all other $\alpha\in\mathcal{I}$. Its action is thus

$\mathbf{B}\lozenge U(t,\omega)=\sum_{\alpha\in\mathcal{I}}b\cdot
u_{\alpha}(t)H_{\alpha}(\omega).$

For example, we may take $X=L^{2}(\mathbb{R}^{n})$ and $b=b(x)$,
$x\in\mathbb{R}^{n}$, for an essentially bounded function $b$.

    2. 2)

$\mathbf{B}$ is multiplication with spatial white noise on
$X=L^{2}(\mathbb{R}^{n})$. Let $B_{k}:=B_{\varepsilon_{k}}=\xi_{k}$,
$k\in\mathbb{N}$, and $B_{\alpha}=0$ for $\alpha\neq\varepsilon_{k}$, i.e.
$B_{k}(v(x))=\xi_{k}(x)\cdot v(x)$, $k\in\mathbb{N}$. Then,

$\mathbf{B}\lozenge U(t,\omega)=W(x,\omega)\lozenge U(t,\omega).$

Clearly,

$\displaystyle\mathbf{B}\lozenge U$ $\displaystyle=$
$\displaystyle\sum_{\gamma\in\mathcal{I}}\sum_{k\in\mathbb{N}}B_{k}(u_{\alpha-\varepsilon_{k}})H_{\gamma}=\sum_{\gamma\in\mathcal{I}}\sum_{k\in\mathbb{N}}u_{\alpha-\varepsilon_{k}}\xi_{k}H_{\gamma}$
$\displaystyle=$
$\displaystyle\sum_{\gamma\in\mathcal{I}}\sum_{\alpha+\varepsilon_{k}=\gamma}u_{\alpha}\xi_{k}H_{\gamma}=W\lozenge
U.$

Multiplication with spatial white noise is important for applications since it
describes stationary perturbations.

    3. 3)

$\mathbf{B}$ is of the form $B_{\varepsilon_{k}}=B_{k}$, $k\in\mathbb{N}$, and
$B_{\alpha}=0$ for $\alpha\neq\varepsilon_{k}$, where $B_{k}:X\rightarrow X$,
$k\in\mathbb{N}$, are bounded linear operators.

Note that in this case there is a one-to-one correspondence between operators
of the form $\mathbf{B}\lozenge$ and operators of the form
$\delta(\mathbf{M}u)$ where $\mathbf{M}$ is a simple coordinatewise operator.
This will be done in Proposition 2.8.

    4. 4)

$\mathbf{B}$ is a simple coordinatewise operator, i.e.
$B_{\alpha}=B,\alpha\in\mathcal{I},$ where $B$ is a bounded linear operator on
$X$. Alternatively, we may also regard operators as $B:X\to X^{\prime}$ in
order to make them bounded; such operators are for example the divergence
$\nabla\cdot$ as a mapping from $X=W^{1,2}(\mathbb{R}^{n})$ to
$X^{\prime}=W^{-1,2}(\mathbb{R}^{n})$.

    5. 5)

$\mathbf{B}\lozenge=\nabla\cdot\lozenge(Q\lozenge\nabla\cdot+b\lozenge)+c\lozenge\nabla\cdot$
as a strictly elliptic second order operator with random coefficients. This
operator is obtained for
$B_{\alpha}=\nabla\cdot(Q_{\alpha}\nabla\cdot+b_{\alpha})+c_{\alpha}\nabla\cdot$,
$\alpha\in\mathcal{I}$, and was studied in [18] and [19].

######  Proposition 2.6.

Let $\mathbf{R}:X\otimes(S)_{-1}\rightarrow X\otimes(S)_{-1}$ be a bounded
linear coordinatewise operator defined by
${\mathbf{R}}u(t,\omega)=\sum_{\alpha\in\mathcal{I}}R_{\alpha}u_{\alpha}(t)H_{\alpha}(\omega)$.

  1. 1.

There exists an operator $\mathbf{M}:X\otimes(S)_{-1}\rightarrow
X\otimes{S^{\prime}(\mathbb{R})}\otimes(S)_{-1}$ of the form

$\mathbf{M}u=\sum_{k=1}^{\infty}\mathbf{M}_{k}u\otimes\xi_{k},\quad u\in
X\otimes(S)_{-1},$

for some coordinatewise operators $\mathbf{M}_{k}:X\otimes(S)_{-1}\rightarrow
X\otimes(S)_{-1}$, $k\in\mathbb{N}$, such that

${\mathbf{R}}u=\delta(\mathbf{M}u)$

holds.

  2. 2.

Especially, if $\mathbf{R}$ is a selfadjoint operator, then $\mathbf{M}$ is a
generalization of the Malliavin derivative.

###### Proof.

a) In [10] we proved that the Skorokhod integral is invertible, i.e. there
exists a unique solution to equations of the form $\delta(v)=f$. Considering
the equation
$\delta(\mathbf{M}u)=\sum_{\alpha\in\mathcal{I}}R_{\alpha}u_{\alpha}\;H_{\alpha}$
and applying the result from [10], we obtain $\mathbf{M}u$ in the form

$\mathbf{M}u=\sum_{\alpha\in\mathcal{I}}\sum_{k\in\mathbb{N}}(\alpha_{k}+1)\frac{R_{\alpha+\varepsilon_{k}}(u_{\alpha+\varepsilon_{k}})}{|\alpha+\varepsilon_{k}|}\otimes\xi_{k}\otimes
H_{\alpha}.$

By defining

$\mathbf{M}_{k}u=\sum_{\alpha\in\mathcal{I}}(\alpha_{k}+1)\frac{R_{\alpha+\varepsilon_{k}}(u_{\alpha+\varepsilon_{k}})}{|\alpha+\varepsilon_{k}|}\otimes
H_{\alpha},\quad k\in\mathbb{N},$

we obtain the assertion.

b) Let $\mathbf{R}$ be a self-adjoint operator with eigenvalues $r_{\alpha}$
and eigenvectors $H_{\alpha}$, $\alpha\in\mathcal{I}$, i.e., an operator of
the form
${\mathbf{R}}u=\sum_{\alpha\in\mathcal{I}}r_{\alpha}u_{\alpha}H_{\alpha}$.
Assume that $r_{\alpha}=\sum_{k\in\mathbb{N}}r_{k,\alpha}$ for some
$r_{k,\alpha}\in\mathbb{R}$, $k\in\mathbb{N}$, $\alpha\in\mathcal{I}$, is an
arbitrary decomposition of the value $r_{\alpha}$.

Define

$\mathbf{M}_{k}u=\sum_{\alpha\in\mathcal{I}}r_{k,\alpha}u_{\alpha}\otimes
H_{\alpha-\varepsilon_{k}}.$

Then
$\mathbf{M}u=\sum_{k\in\mathbb{N}}\mathbf{M}_{k}u\otimes\xi_{k}=\sum_{k\in\mathbb{N}}\sum_{\alpha\in\mathcal{I}}r_{k,\alpha}u_{\alpha}\otimes
H_{\alpha-\varepsilon_{k}}\otimes\xi_{k}$ and

$\delta(\mathbf{M}u)=\sum_{k\in\mathbb{N}}\sum_{\alpha\in\mathcal{I}}r_{k,\alpha}u_{\alpha}\otimes
H_{\alpha}=\sum_{\alpha\in\mathcal{I}}r_{\alpha}u_{\alpha}\otimes H_{\alpha}.$

∎

######  Remark 2.7.

The converse is not true. Even if each $\mathbf{M}_{k}$, $k\in\mathbb{N}$, is
a simple coordinatewise operator (and so is $\mathbf{M}$),
${\mathbf{R}}:=\delta\circ\mathbf{M}$ does not need to be a coordinatewise
operator. This would require that the system
$R_{\alpha}(u_{\alpha})=\sum_{k\in\mathbb{N}}m_{k}(u_{\alpha-\varepsilon_{k}})$,
$\alpha\in\mathcal{I}$, is solvable for $R_{\alpha}(\cdot)$ given the
functions $m_{k}(\cdot)$, $k\in\mathbb{N}$, which is not true in general.

######  Proposition 2.8.

Let $\mathbf{M}:X\otimes(S)_{-1}\rightarrow
X\otimes{S^{\prime}(\mathbb{R})}\otimes(S)_{-1}$ be of the form

$\mathbf{M}u=\sum_{k=1}^{\infty}\mathbf{M}_{k}u\otimes\xi_{k},\quad u\in
X\otimes(S)_{-1},$ (7)

for some simple coordinatewise operators
$\mathbf{M}_{k}:X\otimes(S)_{-1}\rightarrow X\otimes(S)_{-1}$,
$k\in\mathbb{N}$. Then, there exists a coordinatewise operator $\mathbf{B}$
such that $B_{\alpha}=0$ for $\alpha\neq\varepsilon_{k}$, $k\in\mathbb{N}$,
and

$\delta(\mathbf{M}u)={\mathbf{B}}\lozenge u$

holds.

Conversely, for any coordinatewise operator $\mathbf{B}$ such that
$B_{\alpha}=0$ for $\alpha\neq\varepsilon_{k}$, $k\in\mathbb{N}$, there exists
an operator $\mathbf{M}$ of the form
$\mathbf{M}u=\sum_{k=1}^{\infty}\mathbf{M}_{k}u\otimes\xi_{k}$ for some simple
coordinatewise operators $\mathbf{M}_{k}$, $k\in\mathbb{N}$, such that
$\delta(\mathbf{M}u)={\mathbf{B}}\lozenge u$ holds.

###### Proof.

Let $\mathbf{M}$ be an operator as stated above and since $\mathbf{M}_{k}$ are
simple coordinatewise operators, we can write them as

$\mathbf{M}_{k}(u)=\sum_{\alpha\in\mathcal{I}}m_{k}(u_{\alpha})H_{\alpha},\qquad
u=\sum_{\alpha\in\mathcal{I}}u_{\alpha}H_{\alpha},$

for some operators $m_{k}:X\rightarrow X$, $k\in\mathbb{N}$. Thus,

$\mathbf{M}u=\sum_{k=1}^{\infty}\sum_{\alpha\in\mathcal{I}}m_{k}(u_{\alpha})H_{\alpha}\otimes\xi_{k}$

which further implies

$\delta(\mathbf{M}u)=\sum_{k=1}^{\infty}\sum_{\alpha\in\mathcal{I}}m_{k}(u_{\alpha})H_{\alpha+\varepsilon_{k}}=\sum_{k=1}^{\infty}\sum_{\alpha\in\mathcal{I}}m_{k}(u_{\alpha-\varepsilon_{k}})H_{\alpha}.$
(8)

On the other hand, if $\mathbf{B}$ is such that $B_{\alpha}=0$ for
$\alpha\neq\varepsilon_{k}$, $k\in\mathbb{N}$, and we denote by
$B_{k}:=B_{\varepsilon_{k}}$, $k\in\mathbb{N}$, the operators acting on $X$,
then

$\mathbf{B}\lozenge
u=\sum_{\alpha\in\mathcal{I}}\sum_{k=1}^{\infty}B_{k}(u_{\alpha-\varepsilon_{k}})H_{\alpha}.$
(9)

From (8) and (9) it follows that $\delta(\mathbf{M}u)=\mathbf{B}\lozenge u$ if
and only if $m_{k}=B_{k}$ for all $k\in\mathbb{N}$. Thus, there is a one-to-
one correspondence between the operators $\mathbf{B}\lozenge$ and
$\delta\circ\mathbf{M}$. ∎

######  Remark 2.9.

In [12] and [13] Rozovskii and Lototsky considered the equation
$\frac{d}{dt}=\mathbf{A}u+\delta(\mathbf{M}u)+f$, where $\mathbf{M}$ is of the
form (7). They implicitly assumed that all their operators $\mathbf{A}$ and
$\mathbf{M}_{k}$, $k\in\mathbb{N}$, belong to our class of simple
coordinatewise operators. This corresponds to our special cases I-1) and
II-3).

Some special cases of stochastic differential equations covered by (4) include
the following:

  * $\bullet$

The heat equation with random potential

$\frac{d}{dt}u=\Delta u+{\mathbf{B}}\lozenge u.$

In particular, if the random potential is modeled by stationary perturbations,
we may take spatial white noise as a model and obtain $\frac{d}{dt}u=\Delta
u+W\lozenge u$. This corresponds to the special choice of operators I-1) and
II-2).

  * $\bullet$

The heat equation in random (inhomogeneous and anisotropic) media, where the
physical properties of the medium are modeled by a stochastic matrix $Q$. This
corresponds to the case I-1) with $\mathbf{A}=0$ and II-5) leading to an
equation of the form

$\frac{d}{dt}u=\nabla\cdot\lozenge(Q\lozenge\nabla\cdot u+b\lozenge
u)+c\lozenge\nabla\cdot u+f.$

  * $\bullet$

Taking $\mathbf{A}=0$ and $B_{k}:=B_{\varepsilon_{k}}=\xi_{k}\nabla\cdot$,
$k\in\mathbb{N}$, (see special cases II-2) and II-4)) we obtain the transport
equation driven by white noise

$\frac{d}{dt}u=\Delta u+W\lozenge\nabla\cdot u.$

  * $\bullet$

The Langevin equation

$\frac{d}{dt}u=-\lambda u+W(t),$

$\lambda>0$, corresponding to the case I-1) with $A=-\lambda$, $f=W$ and
$\mathbf{B}=0$. Its solution is the Ornstein-Uhlenbeck process describing the
spatial position of a Brownian particle in a fluid with viscosity $\lambda$.

In [1] the authors considered the generalized Langevin equation leading to
generalized Ornstein-Uhlenbeck operators driven by Lévy processes

$\frac{d}{dt}u=Ju+C(\frac{d}{dt}Y),$

where $Y$ is a Lévy process, $J$ the infinitesimal generator of a
$C_{0}-$semigroup and $C$ a bounded operator. All processes are Hilbert space
valued. This corresponds to our case with $X$ being this Hilbert space,
$\mathbf{A}=J$, $\mathbf{B}=0$ and $f=C(Y^{\prime})$.

  * $\bullet$

The equation $\frac{d}{dt}=\mathbf{A}u+\delta(\mathbf{M}u)+f$, that was
extensively studied in [12] and [13]. This corresponds to our special cases
I-1) and II-3).

  * $\bullet$

The equation

$\frac{d}{dt}u=Lu+W\lozenge u,$

where $L$ is a strictly elliptic partial differential operator as studied in
[3] and [8]. This corresponds to the special case I-1) and II-2).

##  3 Stochastic evolution equations

Now we turn to the general case of stochastic Cauchy problems of the form
$\frac{d}{dt}U(t,\omega)={\mathbf{A}}U(t,\omega)+{\mathbf{B}}\lozenge
U(t,\omega)+F(t,\omega),$ $t\in(0,T]$, $\omega\in\Omega$, with initial value
$U(0,\omega)=U^{0}(\omega)$, $\omega\in\Omega$, and all processes are
$X$-valued for a Banach space $X$.

######  Definition 3.1.

It is said that $U$ is a solution to (3) if $U\in
C([0,T],X)\otimes(S)_{-1}\cap C^{1}((0,T],X)\otimes(S)_{-1}$ and $U$ satisfies
(3).

######  Theorem 3.2.

Let $\mathbf{A}$ be a coordinatewise operator of the form (5), where the
operators $A_{\alpha},\;\alpha\in\mathcal{I},$ defined on the same domain $D$
dense in $X,$ are infinitesimal generators of $C_{0}-$semigroups
$(T_{t})_{\alpha},\;t\geq 0,\;\alpha\in\mathcal{I},$ uniformly bounded by

$\|(T_{t})_{\alpha}\|_{L(X)}\leq Me^{wt},\;t\geq 0,\quad\mbox{ for some
}\;M,w>0.$ (10)

Let $\mathbf{B}\lozenge$ be of the form (6), where
$B_{\alpha},\;\alpha\in\mathcal{I},$ are bounded linear operators on $X$ so
that there exists $p>0$ such that

$K:=\sum_{\alpha\in\mathcal{I}}\|B_{\alpha}\|(2\mathbb{N})^{-p\frac{\alpha}{2}}<\infty.$
(11)

Let the initial value $U^{0}\in X\otimes(S)_{-1}$ be such that $U^{0}\in
Dom(\mathbf{A})$ i.e.

$\begin{split}U^{0}(\omega)=\sum_{\alpha\in\mathcal{I}}u^{0}_{\alpha}H_{\alpha}(\omega)\in
X\otimes(S)_{-1,-p},\;\mbox{satisfies}\quad\sum_{\alpha\in\mathcal{I}}\|u_{\alpha}^{0}\|_{X}^{2}(2\mathbb{N})^{-p\alpha}<\infty;\end{split}$
(12)

and

$\begin{split}\mathbf{A}U^{0}(\omega)=\sum_{\alpha\in\mathcal{I}}A_{\alpha}u^{0}_{\alpha}H_{\alpha}(\omega)\in
X\otimes(S)_{-1,-p},\;\mbox{satisfies}\quad\sum_{\alpha\in\mathcal{I}}\|A_{\alpha}u_{\alpha}^{0}\|_{X}^{2}(2\mathbb{N})^{-p\alpha}<\infty.\end{split}$
(13)

Moreover, let

$\begin{split}&F(t,\omega)=\sum_{\alpha\in\mathcal{I}}f_{\alpha}(t)H_{\alpha}(\omega)\in
C^{1}([0,T],X)\otimes(S)_{-1},\;\;\;t\mapsto f_{\alpha}(t)\in
C^{1}([0,T],X),\;\alpha\in\mathcal{I},\\\ &\mbox{so
that}\;\sum_{\alpha\in\mathcal{I}}\|f_{\alpha}\|_{C^{1}([0,T],X)}^{2}(2\mathbb{N})^{-p\alpha}=\sum_{\alpha\in\mathcal{I}}\Big{(}\sup_{t\in[0,T]}\|f_{\alpha}(t)\|_{X}+\sup_{t\in[0,T]}\|f^{\prime}_{\alpha}(t)\|_{X}\Big{)}^{2}(2\mathbb{N})^{-p\alpha}<\infty.\end{split}$
(14)

Then, the stochastic Cauchy problem (3) has a unique solution $U$ in
$C^{1}([0,T],X)\otimes(S)_{-1,-p}$.

###### Proof.

We seek for the solution in form of
$U(t,\omega)=\sum_{\alpha\in\mathcal{I}}u_{\alpha}(t)H_{\alpha}(\omega)$.
Then, the Cauchy problem (3) is equivalent to the infinite system:

$\begin{split}\frac{d}{dt}\;u_{\alpha}(t)&=A_{\alpha}u_{\alpha}(t)+\sum_{\beta\leq\alpha}B_{\beta}u_{\alpha-\beta}(t)+f_{\alpha}(t),\quad
t\in(0,T],\\\ u_{\alpha}(0)&=u_{\alpha}^{0}\in
D,\quad\alpha\in\mathcal{I}.\end{split}$ (15)

Let $\mathbf{0}$ be the multi-index $\mathbf{0}=(0,0,...)$. We rewrite (15) as

$\begin{split}\frac{d}{dt}\;u_{\alpha}(t)&=(A_{\alpha}+B_{\mathbf{0}})u_{\alpha}(t)+\sum_{\mathbf{0}<\beta\leq\alpha}B_{\beta}u_{\alpha-\beta}(t)+f_{\alpha}(t),\quad
t\in(0,T],\\\ u_{\alpha}(0)&=u_{\alpha}^{0}\in
D,\quad\alpha\in\mathcal{I}.\end{split}$ (16)

Next, $A_{\alpha}+B_{\mathbf{0}}$ are infinitesimal generators of
$C_{0}-$semigroups $(S_{t})_{\alpha}$ in $X$ such that

$\|(S_{t})_{\alpha}\|\leq Me^{(w+M\|B_{\mathbf{0}}\|)t},\;t\geq
0,\;\alpha\in\mathcal{I}.$ (17)

According to Subsection 1.1, if $f_{\alpha},\;\alpha\in\mathcal{I},$ fulfills
condition (i) or (ii), the inhomogeneous initial value problem (16) has a
solution $u_{\alpha}(t)\in C([0,T],X)\cap C^{1}((0,T],X)$,
$\alpha\in\mathcal{I}$, given by

$\begin{split}u_{\mathbf{0}}(t)&=(S_{t})_{\mathbf{0}}u_{\mathbf{0}}^{0}+\int_{0}^{t}(S_{t-s})_{\mathbf{0}}f_{\mathbf{0}}(s)ds,\quad
t\in[0,T]\\\
u_{\alpha}(t)&=(S_{t})_{\alpha}u_{\alpha}^{0}+\int_{0}^{t}(S_{t-s})_{\alpha}\Big{(}\sum_{\mathbf{0}<\beta\leq\alpha}B_{\beta}u_{\alpha-\beta}(s)+f_{\alpha}(s)\Big{)}ds,\quad
t\in[0,T].\\\ \end{split}$ (18)

Since $f_{\alpha}\in C^{1}([0,T],X)$ it follows by induction on $\alpha$ that

$\sum_{\mathbf{0}<\beta\leq\alpha}B_{\beta}u_{\alpha-\beta}(s)+f_{\alpha}(s)\in
C^{1}([0,T],X),\quad\mbox{ for all }\quad\alpha\in\mathcal{I}.$

Thus, $u_{\alpha}\in C^{1}([0,T],X)$ and
$\frac{d}{dt}u_{\alpha}(0)=(A_{\alpha}+B_{\mathbf{0}})u^{0}_{\alpha}+\sum_{\mathbf{0}<\beta\leq\alpha}B_{\beta}u_{\alpha-\beta}^{0}+f_{\alpha}(0),\;\alpha\in\mathcal{I}$.

Note that for each fixed $\alpha\in\mathcal{I}$, $u_{\alpha}(t)$ exists for
all $t\in[0,T]$ and it is the unique (classical) solution on the whole
interval $[0,T]$. It remains to prove that
$\sum_{\alpha\in\mathcal{I}}u_{\alpha}(t)H_{\alpha}(\omega)$ converges in
$C^{1}([0,T],X)\otimes(S)_{-1,-p}$.

First, we show that
$U(t,\omega)=\sum_{\alpha\in\mathcal{I}}u_{\alpha}(t)H_{\alpha}(\omega)\in
C^{1}([0,T_{0}],X)\otimes S_{-1,-p}$ for appropriate $T_{0}\in(0,T]$, i.e. we
show that

$\sum_{\alpha\in\mathcal{I}}\|u_{\alpha}\|_{C^{1}([0,T_{0}],X)}^{2}(2\mathbb{N})^{-p\alpha}=\sum_{\alpha\in\mathcal{I}}\Big{(}\sup_{t\in[0,T_{0}]}\|u_{\alpha}(t)\|_{X}+\sup_{t\in[0,T_{0}]}\|\frac{d}{dt}u_{\alpha}(t)\|_{X}\Big{)}^{2}(2\mathbb{N})^{-p\alpha}<\infty.$
(19)

Later on we will prove that the same holds if we take in (19) supremums over
the intervals $[T_{0},2T_{0}]$, $[2T_{0},3T_{0}],...$ etc. Since $[0,T]$ can
be covered by finitely many intervals of the form
$[kT_{0},(k+1)T_{0}],\;k\in\mathbb{N}_{0},$ we conclude that

$\sum_{\alpha\in\mathcal{I}}\|u_{\alpha}\|_{C^{1}([0,T],X)}^{2}(2\mathbb{N})^{-p\alpha}=\sum_{\alpha\in\mathcal{I}}\Big{(}\sup_{t\in[0,T]}\|u_{\alpha}(t)\|_{X}+\sup_{t\in[0,T]}\|\frac{d}{dt}u_{\alpha}(t)\|_{X}\Big{)}^{2}(2\mathbb{N})^{-p\alpha}<\infty.$
(20)

In order to do this, we introduce a notation for subsets of multi-indices

$\mathcal{I}_{n,m}=\\{\alpha\in\mathcal{I}:\;|\alpha|\leq n\wedge{\rm
Index}(\alpha)\leq m\\},\;n,m\in\mathbb{N},$

where, for
$\alpha=(\alpha_{1},\alpha_{2},\dots,\alpha_{m},0,0,\dots)\in\mathcal{I}$, we
have $|\alpha|=\alpha_{1}+\alpha_{2}+\dots+\alpha_{m}$ and ${\rm
Index}(\alpha)$ is last coordinate where $\alpha$ has a nonzero entry. For
later reference, we introduce the function

$C(t)=\frac{M^{2}}{(w+M\|B_{\mathbf{0}}\|)^{2}}(e^{(w+M\|B_{\mathbf{0}}\|)t}-1)^{2}$
(21)

and fix $T_{0}\in(0,T]$ such that $C(T_{0})<\frac{1}{5K^{2}}$.

First, we show that

$\sum_{\alpha\in\mathcal{I}}\|u_{\alpha}(t)\|_{C([0,T_{0}],X)}^{2}(2\mathbb{N})^{-p\alpha}=\sum_{\alpha\in\mathcal{I}}\sup_{t\in[0,T_{0}]}\|u_{\alpha}(t)\|_{X}^{2}(2\mathbb{N})^{-p\alpha}<\infty,$

by proving that partial sums
$\sum_{\alpha\in\mathcal{I}_{n,m}}\sup_{t\in[0,T_{0}]}\|u_{\alpha}(t)\|_{X}^{2}(2\mathbb{N})^{-p\alpha},\;n,m\in\mathbb{N},$
are bounded from above.

Using (18) we obtain

$\displaystyle\frac{1}{3}\sum_{\alpha\in\mathcal{I}_{n,m}}\|u_{\alpha}(t)\|_{X}^{2}(2\mathbb{N})^{-p\alpha}$
$\displaystyle\leq\sum_{\alpha\in\mathcal{I}_{n,m}}\|(S_{t})_{\alpha}\|^{2}\|u_{\alpha}^{0}\|_{X}^{2}(2\mathbb{N})^{-p\alpha}$
$\displaystyle+\sum_{\alpha\in\mathcal{I}_{n,m}}\Big{[}\int_{0}^{t}\|(S_{t-s})_{\alpha}\|\sum_{\mathbf{0}<\beta\leq\alpha}\|B_{\beta}u_{\alpha-\beta}(s)\|_{X}ds\Big{]}^{2}(2\mathbb{N})^{-p\alpha}$
$\displaystyle+\sum_{\alpha\in\mathcal{I}_{n,m}}\Big{[}\int_{0}^{t}\|(S_{t-s})_{\alpha}\|\|f_{\alpha}(s)\|_{X}ds\Big{]}^{2}(2\mathbb{N})^{-p\alpha}.$

The first term on the right-hand side, for all $t\in[0,T_{0}]$, having in mind
(12) and (17), satisfies

$\displaystyle\sum_{\alpha\in\mathcal{I}_{n,m}}\|(S_{t})_{\alpha}\|^{2}\|u_{\alpha}^{0}\|_{X}^{2}(2\mathbb{N})^{-p\alpha}$
$\displaystyle\leq\sum_{\alpha\in\mathcal{I}}\|(S_{t})_{\alpha}\|^{2}\|u_{\alpha}^{0}\|_{X}^{2}(2\mathbb{N})^{-p\alpha}$
$\displaystyle\leq
M^{2}e^{2(w+M\|B_{\mathbf{0}}\|)T_{0}}\sum_{\alpha\in\mathcal{I}}\|u_{\alpha}^{0}\|_{X}^{2}(2\mathbb{N})^{-p\alpha}:=Q_{1}<\infty.$
(22)

Similarly, for all $t\in[0,T_{0}]$, using (14) and (17), the third term
satisfies

$\displaystyle\sum_{\alpha\in\mathcal{I}_{n,m}}$
$\displaystyle\Big{[}\int_{0}^{t}\|(S_{t-s})_{\alpha}\|\|f_{\alpha}(s)\|_{X}ds\Big{]}^{2}(2\mathbb{N})^{-p\alpha}\leq\sum_{\alpha\in\mathcal{I}}\Big{[}\int_{0}^{t}\|(S_{t-s})_{\alpha}\|\|f_{\alpha}(s)\|_{X}ds\Big{]}^{2}(2\mathbb{N})^{-p\alpha}$
$\displaystyle\leq\Big{[}\int_{0}^{t}Me^{(w+M\|B_{\mathbf{0}}\|)(t-s)}ds\Big{]}^{2}\sum_{\alpha\in\mathcal{I}}\sup_{s\in[0,t]}\|f_{\alpha}(s)\|_{X}^{2}(2\mathbb{N})^{-p\alpha}$
$\displaystyle\leq\frac{M^{2}}{(w+M\|B_{\mathbf{0}}\|)^{2}}\Big{(}e^{(w+M\|B_{\mathbf{0}}\|)T_{0}}-1\Big{)}^{2}\sum_{\alpha\in\mathcal{I}}\sup_{t\in[0,T]}\|f_{\alpha}(t)\|_{X}^{2}(2\mathbb{N})^{-p\alpha}:=G<\infty.$
(23)

Note that in (3) we took the supremum over the whole interval $[0,T]$.

For the second term, using (11), (17), (21) and the generalized Minkowski
inequality, we obtain

$\displaystyle\sum_{\alpha\in\mathcal{I}_{n,m}}$
$\displaystyle\Big{[}\int_{0}^{t}\|(S_{t-s})_{\alpha}\|\sum_{\beta+\gamma=\alpha}\|B_{\beta}\|\|u_{\gamma}(s)\|_{X}ds\Big{]}^{2}(2\mathbb{N})^{-p\alpha}$
$\displaystyle\leq\frac{M^{2}}{(w+M\|B_{\mathbf{0}}\|)^{2}}\Big{(}e^{(w+M\|B_{\mathbf{0}}\|)t}-1\Big{)}^{2}\sum_{\alpha\in\mathcal{I}_{n,m}}\Big{[}\sum_{\beta+\gamma=\alpha}\sup_{s\in[0,t]}\|B_{\beta}\|\|u_{\gamma}(s)\|_{X}\Big{]}^{2}(2\mathbb{N})^{-p\alpha}$
$\displaystyle\leq
C(T_{0})\Big{(}\sum_{\beta\in\mathcal{I}_{n,m}}\|B_{\beta}\|(2\mathbb{N})^{-p\frac{\beta}{2}}\Big{)}^{2}\Big{(}\sum_{\gamma\in\mathcal{I}_{n,m}}\sup_{t\in[0,T_{0}]}\|u_{\gamma}(t)\|_{X}^{2}(2\mathbb{N})^{-p\gamma}\Big{)}$
$\displaystyle\leq
C(T_{0})K^{2}\sum_{\alpha\in\mathcal{I}_{n,m}}\sup_{t\in[0,T_{0}]}\|u_{\alpha}(t)\|_{X}^{2}(2\mathbb{N})^{-p\alpha}.$
(24)

Finally, for all $n,m\in\mathbb{N}$, we obtain

$\displaystyle\frac{1}{3}\sum_{\alpha\in\mathcal{I}_{n,m}}\sup_{t\in[0,T_{0}]}\|u_{\alpha}(t)\|_{X}^{2}(2\mathbb{N})^{-p\alpha}\leq
Q_{1}+G+C(T_{0})K^{2}\sum_{\alpha\in\mathcal{I}_{n,m}}\sup_{t\in[0,T_{0}]}\|u_{\alpha}(t)\|_{X}^{2}(2\mathbb{N})^{-p\alpha}.$

Since $\frac{1}{3}-C(T_{0})K^{2}>\frac{1}{5}-C(T_{0})K^{2}>0$, we have

$\displaystyle\sum_{\alpha\in\mathcal{I}_{n,m}}\sup_{t\in[0,T_{0}]}\|u_{\alpha}(t)\|_{X}^{2}(2\mathbb{N})^{-p\alpha}\leq\frac{Q_{1}+G}{\frac{1}{3}-C(T_{0})K^{2}}.$
(25)

Let $(m_{n})_{n\in\mathbb{N}}$ be an arbitrary sequence of positive integers
tending to infinity. Then,

$\sum_{\alpha\in\mathcal{I}}\sup_{t\in[0,T_{0}]}\|u_{\alpha}(t)\|_{X}^{2}(2\mathbb{N})^{-p\alpha}=\lim_{n\to\infty}\sum_{\alpha\in\mathcal{I}_{n,m_{n}}}\sup_{t\in[0,T_{0}]}\|u_{\alpha}(t)\|_{X}^{2}(2\mathbb{N})^{-p\alpha}\leq\frac{Q_{1}+G}{\frac{1}{3}-C(T_{0})K^{2}},$

since it is a series of positive numbers and thus does not depend on the order
of summation.

Now we show that

$\sum_{\alpha\in\mathcal{I}}\|\frac{d}{dt}u_{\alpha}(t)\|_{C([0,T_{0}],X)}^{2}(2\mathbb{N})^{-p\alpha}=\sum_{\alpha\in\mathcal{I}}\sup_{t\in[0,T_{0}]}\|\frac{d}{dt}u_{\alpha}(t)\|_{X}^{2}(2\mathbb{N})^{-p\alpha}<\infty.$

In order to acomplish that, we differentiate (18) with respect to $t$, and
obtain

$\begin{split}\frac{d}{dt}u_{\mathbf{0}}(t)&=(S_{t})_{\mathbf{0}}(A_{\mathbf{0}}+B_{\mathbf{0}})u_{\mathbf{0}}^{0}+\int_{0}^{t}(S_{t-s})_{\mathbf{0}}\frac{d}{ds}f_{\mathbf{0}}(s)ds+(S_{t})_{\mathbf{0}}f(0),\quad
t\in[0,T],\\\
\frac{d}{dt}u_{\alpha}(t)&=(S_{t})_{\alpha}(A_{\alpha}+B_{\mathbf{0}})u_{\alpha}^{0}+\int_{0}^{t}(S_{t-s})_{\alpha}\Big{(}\sum_{\mathbf{0}<\beta\leq\alpha}B_{\beta}\frac{d}{ds}u_{\alpha-\beta}(s)+\frac{d}{ds}f_{\alpha}(s)\Big{)}ds\\\
&+(S_{t})_{\alpha}\Big{(}\sum_{\mathbf{0}<\beta\leq\alpha}B_{\beta}u_{\alpha-\beta}(0)+f_{\alpha}(0)\Big{)},\quad
t\in[0,T],\quad\alpha\in\mathcal{I}.\end{split}$ (26)

In the sequel we estimate partial sums of
$\sum_{\alpha\in\mathcal{I}}\sup_{t\in[0,T_{0}]}\|\frac{d}{dt}u_{\alpha}(t)\|_{X}^{2}(2\mathbb{N})^{-p\alpha}$.
So,

$\displaystyle\frac{1}{5}\sum_{\alpha\in\mathcal{I}_{n,m}}\|\frac{d}{dt}u_{\alpha}(t)\|_{X}^{2}(2\mathbb{N})^{-p\alpha}$
$\displaystyle\leq\sum_{\alpha\in\mathcal{I}_{n,m}}\|(S_{t})_{\alpha}\|^{2}\|(A_{\alpha}+B_{\mathbf{0}})u_{\alpha}^{0}\|_{X}^{2}(2\mathbb{N})^{-p\alpha}$
$\displaystyle+\sum_{\alpha\in\mathcal{I}_{n,m}}\Big{[}\int_{0}^{t}\|(S_{t-s})_{\alpha}\|\sum_{\mathbf{0}<\beta\leq\alpha}\|B_{\beta}\frac{d}{ds}u_{\alpha-\beta}(s)\|_{X}ds\Big{]}^{2}(2\mathbb{N})^{-p\alpha}$
$\displaystyle+\sum_{\alpha\in\mathcal{I}_{n,m}}\Big{[}\int_{0}^{t}\|(S_{t-s})_{\alpha}\|\|\frac{d}{ds}f_{\alpha}(s)\|_{X}ds\Big{]}^{2}(2\mathbb{N})^{-p\alpha}$
$\displaystyle+\sum_{\alpha\in\mathcal{I}_{n,m}}\|(S_{t})_{\alpha}\|^{2}\Big{[}\sum_{\mathbf{0}<\beta\leq\alpha}\|B_{\beta}u_{\alpha-\beta}(0)\|_{X}\Big{]}^{2}(2\mathbb{N})^{-p\alpha}$
$\displaystyle+\sum_{\alpha\in\mathcal{I}_{n,m}}\|(S_{t})_{\alpha}\|^{2}\|f_{\alpha}(0)\|_{X}^{2}(2\mathbb{N})^{-p\alpha}.$

According to (12) and (13), we obtain
$\sum_{\alpha\in\mathcal{I}}(A_{\alpha}+B_{\mathbf{0}})u_{\alpha}^{0}H_{\alpha}(\omega)\in
X\otimes(S)_{-1,-p}$. So the first term on the right-hand side can be
evaluated by

$\displaystyle\sum_{\alpha\in\mathcal{I}_{n,m}}$
$\displaystyle\|(S_{t})_{\alpha}\|^{2}\|(A_{\alpha}+B_{\mathbf{0}})u_{\alpha}^{0}\|_{X}^{2}(2\mathbb{N})^{-p\alpha}\leq\sum_{\alpha\in\mathcal{I}}\|(S_{t})_{\alpha}\|^{2}\|(A_{\alpha}+B_{\mathbf{0}})u_{\alpha}^{0}\|_{X}^{2}(2\mathbb{N})^{-p\alpha}$
$\displaystyle\leq
M^{2}e^{2(w+M\|B_{\mathbf{0}}\|)T_{0}}\sum_{\alpha\in\mathcal{I}}\|(A_{\alpha}+B_{\mathbf{0}})u_{\alpha}^{0}\|_{X}^{2}(2\mathbb{N})^{-p\alpha}:=Q^{\prime}_{1}<\infty.$
(27)

The third term, for all $t\in[0,T_{0}]$, satisfies

$\displaystyle\sum_{\alpha\in\mathcal{I}_{n,m}}$
$\displaystyle\Big{[}\int_{0}^{t}\|(S_{t-s})_{\alpha}\|\|\frac{d}{ds}f_{\alpha}(s)\|ds\Big{]}^{2}(2\mathbb{N})^{-p\alpha}\leq\sum_{\alpha\in\mathcal{I}}\Big{[}\int_{0}^{t}\|(S_{t-s})_{\alpha}\|\|\frac{d}{ds}f_{\alpha}(s)\|_{X}ds\Big{]}^{2}(2\mathbb{N})^{-p\alpha}$
$\displaystyle\leq\frac{M^{2}}{(w+M\|B_{\mathbf{0}}\|)^{2}}\Big{(}e^{(w+M\|B_{\mathbf{0}}\|)T_{0}}-1\Big{)}^{2}\sum_{\alpha\in\mathcal{I}}\sup_{t\in[0,T]}\|\frac{d}{ds}f_{\alpha}(t)\|_{X}^{2}(2\mathbb{N})^{-p\alpha}:=G^{\prime}<\infty.$
(28)

The fourth term, using (11), (12), (17) and the generalized Minkowski
inequality, can be estimated by

$\displaystyle\sum_{\alpha\in\mathcal{I}_{n,m}}$
$\displaystyle\|(S_{t})_{\alpha}\|^{2}\Big{[}\sum_{\mathbf{0}<\beta\leq\alpha}\|B_{\beta}u_{\alpha-\beta}(0)\|_{X}\Big{]}^{2}(2\mathbb{N})^{-p\alpha}\leq\sum_{\alpha\in\mathcal{I}}\|(S_{t})_{\alpha}\|^{2}\Big{[}\sum_{\beta+\gamma=\alpha}\|B_{\beta}u_{\gamma}^{0}\|_{X}\Big{]}^{2}(2\mathbb{N})^{-p\alpha}$
$\displaystyle\leq
M^{2}e^{2(w+M\|B_{\mathbf{0}}\|)t}\sum_{\alpha\in\mathcal{I}}\Big{[}\sum_{\beta+\gamma=\alpha}\|B_{\beta}\|\|u_{\gamma}^{0}\|_{X}\Big{]}^{2}(2\mathbb{N})^{-p\alpha}$
$\displaystyle\leq
M^{2}e^{2(w+M\|B_{\mathbf{0}}\|)T_{0}}\Big{(}\sum_{\beta\in\mathcal{I}}\|B_{\beta}\|(2\mathbb{N})^{-p\frac{\beta}{2}}\Big{)}^{2}\Big{(}\sum_{\gamma\in\mathcal{I}}\|u_{\gamma}^{0}\|_{X}^{2}(2\mathbb{N})^{-p\gamma}\Big{)}:=H_{1}^{\prime}<\infty.$
(29)

For the fifth term, using (14) and (17), we have

$\displaystyle\sum_{\alpha\in\mathcal{I}_{n,m}}$
$\displaystyle\|(S_{t})_{\alpha}\|^{2}\|f_{\alpha}(0)\|_{X}^{2}(2\mathbb{N})^{-p\alpha}\leq\sum_{\alpha\in\mathcal{I}}\|(S_{t})_{\alpha}\|^{2}\|f_{\alpha}(0)\|_{X}^{2}(2\mathbb{N})^{-p\alpha}$
$\displaystyle\leq
M^{2}e^{2(w+M\|B_{\mathbf{0}}\|)T_{0}}\sum_{\alpha\in\mathcal{I}}\sup_{t\in[0,T]}\|f_{\alpha}(t)\|_{X}^{2}(2\mathbb{N})^{-p\alpha}:=N^{\prime}<\infty.$
(30)

Finally, for the second term, using (11), (17), (21) and the generalized
Minkowski inequality, we obtain

$\displaystyle\sum_{\alpha\in\mathcal{I}_{n,m}}$
$\displaystyle\Big{[}\int_{0}^{t}\|(S_{t-s})_{\alpha}\|\sum_{\beta+\gamma=\alpha}\|B_{\beta}\|\|\frac{d}{ds}u_{\gamma}(s)\|_{X}ds\Big{]}^{2}(2\mathbb{N})^{-p\alpha}$
$\displaystyle\leq\frac{M^{2}}{(w+M\|B_{\mathbf{0}}\|)^{2}}\Big{(}e^{(w+M\|B_{\mathbf{0}}\|)t}-1\Big{)}^{2}\sum_{\alpha\in\mathcal{I}_{n,m}}\Big{[}\sum_{\beta+\gamma=\alpha}\sup_{s\in[0,t]}\|B_{\beta}\|\|\frac{d}{ds}u_{\gamma}(s)\|_{X}\Big{]}^{2}(2\mathbb{N})^{-p\alpha}$
$\displaystyle\leq
C(t)\Big{(}\sum_{\beta\in\mathcal{I}_{n,m}}\|B_{\beta}\|(2\mathbb{N})^{-p\frac{\beta}{2}}\Big{)}^{2}\Big{(}\sum_{\gamma\in\mathcal{I}_{n,m}}\sup_{s\in[0,t]}\|\frac{d}{dt}u_{\gamma}(s)\|_{X}^{2}(2\mathbb{N})^{-p\gamma}\Big{)}$
$\displaystyle\leq
C(T_{0})K^{2}\sum_{\alpha\in\mathcal{I}_{n,m}}\sup_{t\in[0,T_{0}]}\|\frac{d}{dt}u_{\alpha}(t)\|_{X}^{2}(2\mathbb{N})^{-p\alpha}.$
(31)

Finally, for all $n,m\in\mathbb{N}$, we obtain

$\displaystyle\frac{1}{5}\sum_{\alpha\in\mathcal{I}_{n,m}}\sup_{t\in[0,T_{0}]}\|\frac{d}{dt}u_{\alpha}(t)\|_{X}^{2}(2\mathbb{N})^{-p\alpha}\leq$
$\displaystyle Q^{\prime}_{1}+G^{\prime}+H^{\prime}_{1}+N^{\prime}$
$\displaystyle+C(T_{0})K^{2}\sum_{\alpha\in\mathcal{I}_{n,m}}\sup_{t\in[0,T_{0}]}\|\frac{d}{dt}u_{\alpha}(t)\|_{X}^{2}(2\mathbb{N})^{-p\alpha}.$

Since $\frac{1}{5}-C(T_{0})K^{2}>0$, we have

$\displaystyle\sum_{\alpha\in\mathcal{I}_{n,m}}\sup_{t\in[0,T_{0}]}\|\frac{d}{dt}u_{\alpha}(t)\|_{X}^{2}(2\mathbb{N})^{-p\alpha}\leq\frac{Q^{\prime}_{1}+G^{\prime}+H^{\prime}_{1}+N^{\prime}}{\frac{1}{5}-C(T_{0})K^{2}}.$
(32)

Again, taking $(m_{n})_{n\in\mathbb{N}}$ to be an arbitrary sequence of
positive integers tending to infinity, we have

$\sum_{\alpha\in\mathcal{I}}\sup_{t\in[0,T_{0}]}\|\frac{d}{dt}u_{\alpha}(t)\|_{X}^{2}(2\mathbb{N})^{-p\alpha}=\lim_{n\to\infty}\sum_{\alpha\in\mathcal{I}_{n,m_{n}}}\sup_{t\in[0,T_{0}]}\|\frac{d}{dt}u_{\alpha}(t)\|_{X}^{2}(2\mathbb{N})^{-p\alpha}\leq\frac{Q^{\prime}_{1}+G^{\prime}+H^{\prime}_{1}+N^{\prime}}{\frac{1}{5}-C(T_{0})K^{2}}.$

Therefore, we obtain

$\begin{split}&U(t,\omega)\in
C^{1}([0,T_{0}],X)\otimes(S)_{-1,-p},\;\mbox{i.e.}\\\
&\sum_{\alpha\in\mathcal{I}}\Big{(}\sup_{t\in[0,T_{0}]}\|u_{\alpha}(t)\|_{X}+\sup_{t\in[0,T_{0}]}\|\frac{d}{dt}u_{\alpha}(t)\|_{X}\Big{)}^{2}(2\mathbb{N})^{-p\alpha}\leq\\\
&2\sum_{\alpha\in\mathcal{I}}\Big{(}\sup_{t\in[0,T_{0}]}\|u_{\alpha}(t)\|^{2}_{X}+\sup_{t\in[0,T_{0}]}\|\frac{d}{dt}u_{\alpha}(t)\|^{2}_{X}\Big{)}(2\mathbb{N})^{-p\alpha}<\infty.\end{split}$
(33)

Next, we consider in (33) supremums over the interval $[T_{0},2T_{0}]$. On
$[T_{0},2T_{0}]$ one can rewrite the initial value problem (15) in the
following equivalent form:

$\begin{split}\frac{d}{dt}\;v_{\alpha}(t)&=A_{\alpha}v_{\alpha}(t)+\sum_{\beta\leq\alpha}B_{\beta}v_{\alpha-\beta}(t)+f_{\alpha}(T_{0}+t),\quad
t\in(0,T_{0}]\\\
v_{\alpha}(0)&=v_{\alpha}^{0}:=u_{\alpha}(T_{0}),\quad\alpha\in\mathcal{I}.\end{split}$
(34)

The semigroup corresponding to the generator $A_{\alpha}+B_{\mathbf{0}}$ in
(34) is again the semigroup $(S_{t})_{\alpha},\;t\geq 0$. Using (15) and (33),
we have that $U(t,\omega)\in Dom(\mathbf{A}),$ for all $t\in[0,T_{0}],$ and
$\mathbf{A}U(t,\omega)\in X\otimes(S)_{-1,-p},\;t\in[0,T_{0}].$ According to
this we have that
$V^{0}(\omega)=U(T_{0},\omega)=\sum_{\alpha\in\mathcal{I}}v^{0}_{\alpha}H_{\alpha}(\omega)\in
Dom(\mathbf{A})$ and $\mathbf{A}V^{0}(\omega)\in X\otimes(S)_{-1,-p}.$ Thus,

$v_{\alpha}(t)=(S_{t})_{\alpha}v_{\alpha}^{0}+\int_{0}^{t}(S_{t-s})_{\alpha}\Big{(}\sum_{\mathbf{0}<\beta\leq\alpha}B_{\beta}v_{\alpha-\beta}(s)+f_{\alpha}(T_{0}+s)\Big{)}ds,\quad
t\in[0,T_{0}],$

and clearly
$v_{\alpha}(t)=u_{\alpha}(T_{0}+t),\;t\in[0,T_{0}],\;\alpha\in\mathcal{I}$.

When approximating partial sums of
$\sum_{\alpha\in\mathcal{I}}\sup_{t\in[0,T_{0}]}\|v_{\alpha}(t)\|_{X}^{2}(2\mathbb{N})^{-p\alpha}$,
comparing to the previous calculations for $u_{\alpha}(t)$, only the constant
$Q_{1}$ will be different, and here, we denote it by $Q_{2}$, so we again
obtain

$\sum_{\alpha\in\mathcal{I}}\sup_{t\in[0,T_{0}]}\|v_{\alpha}(t)\|_{X}^{2}(2\mathbb{N})^{-p\alpha}=\sum_{\alpha\in\mathcal{I}}\sup_{t\in[T_{0},2T_{0}]}\|u_{\alpha}(t)\|_{X}^{2}(2\mathbb{N})^{-p\alpha}\leq\frac{Q_{2}+G}{\frac{1}{3}-C(T_{0})K^{2}}.$

Similarly, for the derivative $\frac{d}{dt}V(t,\omega)$ we obtain

$\sum_{\alpha\in\mathcal{I}}\sup_{t\in[0,T_{0}]}\|\frac{d}{dt}v_{\alpha}(t)\|_{X}^{2}(2\mathbb{N})^{-p\alpha}\leq\frac{Q^{\prime}_{2}+G^{\prime}+H^{\prime}_{2}+N^{\prime}}{\frac{1}{5}-C(T_{0})K^{2}},$

where, comparing to the estimates of $\frac{d}{dt}U(t,\omega)$, only the
constants $Q^{\prime}_{1}$ and $H^{\prime}_{1}$ have changed and we denoted
them here by $Q^{\prime}_{2}$ and $H^{\prime}_{2}$.

For arbitrary $T>0$, one can cover the interval $[0,T]$ by intervals of the
form $[kT_{0},(k+1)T_{0}],\;k\in\mathbb{N}_{0},$ in finitely many steps (say
in $l$ steps). So we have

$\sum_{\alpha\in\mathcal{I}}\sup_{t\in[0,T]}\|u_{\alpha}(t)\|_{X}^{2}(2\mathbb{N})^{-p\alpha}\leq\frac{Q+G}{\frac{1}{3}-C(T_{0})K^{2}},$

where $Q=\max_{1\leq k\leq l}\\{Q_{k}\\}$. Thus,

$U(t,\omega)=\sum_{\alpha\in\mathcal{I}}u_{\alpha}(t)H_{\alpha}(\omega)\in
C([0,T],X)\otimes(S)_{-1,-p}.$

Also,

$\sum_{\alpha\in\mathcal{I}}\sup_{t\in[0,T]}\|\frac{d}{dt}u_{\alpha}(t)\|_{X}^{2}(2\mathbb{N})^{-p\alpha}\leq\frac{Q^{\prime}+G^{\prime}+H^{\prime}+N^{\prime}}{\frac{1}{5}-C(T_{0})K^{2}},$

where $Q^{\prime}=\max_{1\leq k\leq l}\\{Q^{\prime}_{k}\\}$,
$H^{\prime}=\max_{1\leq k\leq l}\\{H^{\prime}_{k}\\}$. Since
$\frac{d}{dt}u_{\alpha}(t)\in C([0,T],X)$, $\alpha\in\mathcal{I}$, we have

$\frac{d}{dt}U(t,\omega)=\sum_{\alpha\in\mathcal{I}}\frac{d}{dt}u_{\alpha}(t)H_{\alpha}(\omega)\in
C([0,T],X)\otimes(S)_{-1,-p}.$

Therefore, $U(t,\omega)\in C^{1}([0,T],X)\otimes(S)_{-1,-p}$ and thus, $U$ is
a solution of (3) in the sense of Definition 3.1.

The solution $U$ is unique due to the uniqueness of the coordinatewise
(classical) solutions $u_{\alpha}$ in (18) and due to uniqueness in the
Wiener-Itô chaos expansion.

∎

Note that according to the previous theorem the solution $U$ remains in the
same stochastic order space $(S)_{-1,-p}$ where the input data
$U^{0},\;\mathbf{A}U^{0}$ and $F$ belong to.

######  Example 3.3.

We provide three examples of equation (3) where $\mathbf{A}$ is a uniformly
bounded (not a simple) coordinatewise operator. Consider the Banach space
$X=L^{p}(\mathbb{R}),\;1\leq p<\infty,$ and the stochastic Cauchy problem

$\begin{split}\frac{d}{dt}U(t,x,\omega)&=\mathbf{A}U(t,x,\omega)+W\lozenge
U(t,x,\omega)+F(t,x,\omega),\\\ U(0,x,\omega)&=U^{0}(x,\omega),\end{split}$
(35)

where the operator $\mathbf{A}:Dom(\mathbf{A})\to X\otimes(S)_{-1}$ is a
coordinatewise operator composed out of a family of closed operators
$\\{A_{\alpha}\\}_{\alpha\in\mathcal{I}}$ of the form
$A_{\alpha}=a_{\alpha}D,\;\alpha\in\mathcal{I}$, where the functions
$a_{\alpha}\in L^{\infty}(\mathbb{R}),\;\alpha\in\mathcal{I},$ are uniformly
bounded, i.e. $\sup_{x\in\mathbb{R}}|a_{\alpha}(x)|\leq
M,\;\alpha\in\mathcal{I},$ for some $M>0,$ and $D$ is one of the following
differential operators: $\frac{\partial}{\partial
x},\;\frac{\partial^{2}}{\partial x^{2}}$ or $\frac{\partial^{2}}{\partial
x^{2}}+\frac{\partial}{\partial x}$, and
$W=\sum_{k\in\mathbb{N}}\xi_{k}H_{\varepsilon_{k}}$ represents spatial white
noise. Then, (35) is equivalent to the infinite system

$\begin{split}\frac{d}{dt}u_{\alpha}(t,x)&=A_{\alpha}u_{\alpha}(t,x)+\sum_{k\in\mathbb{N}}\xi_{k}(x)u_{\alpha-\varepsilon_{k}}(t,x)+f_{\alpha}(t,x)\\\
u_{\alpha}(0,x)&=u_{\alpha}^{0}(x),\quad\alpha\in\mathcal{I}.\end{split}$

The $C_{0}-$semigroup that corresponds to the closed operator $D$, denoted by
$T_{t},\;t\geq 0,$ is, respectively,

$\displaystyle T_{t}g(x)$ $\displaystyle=g(t+x),\;\;g\in
L^{p}(\mathbb{R}),\qquad\mbox{for }\;D=\frac{\partial}{\partial x},$
$\displaystyle T_{t}g(x)$ $\displaystyle=\frac{1}{\sqrt{4\pi
t}}\int_{\mathbb{R}}g(x-y)e^{-\frac{y^{2}}{4t}}dy,\;\;g\in
L^{p}(\mathbb{R}),\qquad\mbox{for }\;D=\frac{\partial^{2}}{\partial x^{2}},$
$\displaystyle T_{t}g(x)$ $\displaystyle=\frac{1}{\sqrt{4\pi
t}}\int_{\mathbb{R}}g(x-y)e^{-\frac{(y+t)^{2}}{4t}}dy,\;\;g\in
L^{p}(\mathbb{R}),\qquad\mbox{for }\;D=\frac{\partial^{2}}{\partial
x^{2}}+\frac{\partial}{\partial x}.$

In all cases, we have, using Young’s inequality, that $\|T_{t}\|\leq 1,\;t\geq
0.$ The $C_{0}-$semigroups corresponding to the operators
$A_{\alpha},\;\alpha\in\mathcal{I},$ are of the form
$(S_{t})_{\alpha}=a_{\alpha}T_{t}$. Thus, $\|(S_{t})_{\alpha}\|\leq
M,\;\alpha\in\mathcal{I}$. The operators $B_{\alpha},\;\alpha\in\mathcal{I},$
are given by $B_{\varepsilon_{k}}=\xi_{k},\;k\in\mathbb{N}$ and
$B_{\alpha}=0,\;\alpha\neq\varepsilon_{k}.$ Thus,
$\|B_{\alpha}\|\leq\sup_{k\in\mathbb{N}}\|\xi_{k}\|_{L^{\infty}(\mathbb{R})}\leq
1,\;\alpha\in\mathcal{I}$. Now, according to Theorem 3.2, equation (35) has a
unique solution
$U(t,x,\omega)=\sum_{\alpha\in\mathcal{I}}u_{\alpha}(t,x)H_{\alpha}(\omega),$
where

$u_{\alpha}(t,x)=(S_{t})_{\alpha}u_{\alpha}^{0}(x)+\int_{0}^{t}(S_{t-s})_{\alpha}(\sum_{k}\xi_{k}(x)u_{\alpha-\varepsilon_{k}}(s,x)+f_{\alpha}(s,x))ds,\;\alpha\in\mathcal{I}.$

######  Example 3.4.

Consider the Cauchy problem

$\begin{split}\frac{d}{dt}U(t,\omega)&=\mathbf{A}U(t,\omega)+\mathbf{B}\lozenge
U(t,\omega)+F(t,\omega)\\\ U(0,\omega)&=U^{0}(\omega),\end{split}$

where $\mathbf{A}$ is a simple coordinatewise operator $A_{\alpha}=A$,
$\alpha\in\mathcal{I}$, generating a $C_{0}-$semigroup, $B_{\alpha}\neq 0$
only for $\alpha=\varepsilon_{k},\;k\in\mathbb{N},$ are such that
$\sum_{k\in\mathbb{N}}\|B_{\varepsilon_{k}}\|(2k)^{-\frac{p}{2}}<\infty$, and
$U^{0}$ and $F$ are deterministic functions, i.e. $u^{0}_{\alpha}=0$ and
$f_{\alpha}=0$ for all $\alpha\in\mathcal{I}\setminus\\{\mathbf{0}\\}$.

The solution of this system, according to Theorem 3.2, is

$\begin{split}u_{\mathbf{0}}(t)&=T_{t}u^{0}_{\mathbf{0}}+\int_{0}^{t}T_{t-s}f_{\mathbf{0}}(s)ds,\\\
u_{\alpha}(t)&=\int_{0}^{t}T_{t-s}\Big{(}\sum_{k\in\mathbb{N}}B_{\varepsilon_{k}}u_{\alpha-\varepsilon_{k}}(s)\Big{)}ds,\quad\alpha\in\mathcal{I}\setminus{\mathbf{0}},\end{split}$

the same form as it was obtained in [12].

We provide two generalisations of Theorem 3.2: one possibility is to allow the
operators $B_{\alpha}$ to depend on the time variable $t$ (except for
$B_{\mathbf{0}}$ which must be free of $t$). This embraces for example SPDEs
driven by space-time noises which have zero expectation (and are thus free of
$t$). The other possibility is to allow $B_{\mathbf{0}}$ to be unbounded but
satisfying certain properties so that $A_{\alpha}+B_{\mathbf{0}}$ are
infinitesimal generators of $C_{0}-$semigroups. For example, if
$A_{\alpha}=\frac{\partial^{2}}{\partial x^{2}}$ and
$B_{\mathbf{0}}=\frac{\partial}{\partial x}$, then although $B_{0}$ is
unbounded, $A_{\alpha}+B_{\mathbf{0}}$ is the generator of a contraction
semigroup. Following [4] we will enlist some sufficient conditions which
ensure that $A_{\alpha}+B_{\mathbf{0}}$ is the generators of a
$C_{0}-$semigroup.

######  Remark 3.5.

In Theorem 3.2 one can consider operators
$B_{\alpha}(t),\;\alpha\in\mathcal{I}\setminus\\{\mathbf{0}\\}$, depending on
$t$, so that $B_{\alpha}\in
C^{1}([0,T],L(X)),\;\alpha\in\mathcal{I}\setminus\\{\mathbf{0}\\}$,
$B_{\mathbf{0}}(t)=B_{\mathbf{0}}\in L(X),$ for all $t\in[0,T],$ and

$\displaystyle K:$
$\displaystyle=\sum_{\alpha\in\mathcal{I},\atop\alpha>\mathbf{0}}\|B_{\alpha}\|_{C^{1}([0,T],L(X))}(2\mathbb{N})^{-p\frac{\alpha}{2}}$
$\displaystyle=\sum_{\alpha\in\mathcal{I},\atop\alpha>\mathbf{0}}\Big{(}\sup_{t\in[0,T]}\|B_{\alpha}(t)\|_{L(X)}+\sup_{t\in[0,T]}\|\frac{d}{dt}B_{\alpha}(t)\|_{L(X)}\Big{)}(2\mathbb{N})^{-p\frac{\alpha}{2}}<\infty.$
(36)

Replacing (11) by (3.5) and retaining all other assumptions of Theorem 3.2,
one can again obtain a unique solution $U$ in
$C^{1}([0,T],X)\otimes(S)_{-1,-p}$ of the corresponding Cauchy problem (3).

The solution is
$U(t,\omega)=\sum_{\alpha\in\mathcal{I}}u_{\alpha}(t)H_{\alpha}(\omega),$
$u_{\alpha}(t)\in C^{1}([0,T],X)$, $\alpha\in\mathcal{I}$, where (see (18))

$\begin{split}u_{\mathbf{0}}(t)&=(S_{t})_{\mathbf{0}}u_{\mathbf{0}}^{0}+\int_{0}^{t}(S_{t-s})_{\mathbf{0}}f_{\mathbf{0}}(s)ds,\quad
t\in[0,T],\\\
u_{\alpha}(t)&=(S_{t})_{\alpha}u_{\alpha}^{0}+\int_{0}^{t}(S_{t-s})_{\alpha}\Big{(}\sum_{\mathbf{0}<\beta\leq\alpha}B_{\beta}(s)u_{\alpha-\beta}(s)+f_{\alpha}(s)\Big{)}ds,\quad
t\in[0,T].\\\ \end{split}$ (37)

Its derivative is
$\frac{d}{dt}U(t,\omega)=\sum_{\alpha\in\mathcal{I}}\frac{d}{dt}u_{\alpha}(t)H_{\alpha}(\omega),$
where (see (26))

$\begin{split}\frac{d}{dt}u_{\mathbf{0}}(t)&=(S_{t})_{\mathbf{0}}(A_{\mathbf{0}}+B_{\mathbf{0}})u_{\mathbf{0}}^{0}+\int_{0}^{t}(S_{t-s})_{\mathbf{0}}\frac{d}{ds}f_{\mathbf{0}}(s)ds+(S_{t})_{\mathbf{0}}f(0),\quad
t\in[0,T],\\\
\frac{d}{dt}u_{\alpha}(t)&=(S_{t})_{\alpha}(A_{\alpha}+B_{\mathbf{0}})u_{\alpha}^{0}\\\
&+\int_{0}^{t}(S_{t-s})_{\alpha}\Big{(}\sum_{\mathbf{0}<\beta\leq\alpha}\Big{(}B_{\beta}(s)\frac{d}{ds}u_{\alpha-\beta}(s)+\frac{d}{ds}B_{\beta}(s)u_{\alpha-\beta}(s)\Big{)}+\frac{d}{ds}f_{\alpha}(s)\Big{)}ds\\\
&+(S_{t})_{\alpha}\Big{(}\sum_{\mathbf{0}<\beta\leq\alpha}B_{\beta}(0)u_{\alpha-\beta}(0)+f_{\alpha}(0)\Big{)},\quad
t\in[0,T],\quad\alpha\in\mathcal{I}.\end{split}$ (38)

The proof can be performed in the same manner as in Theorem 3.2, now taking
$T_{0}\in(0,T]$ to be small enough so that $C(T_{0})<\frac{1}{6K^{2}},$ since
now we have six summands in (38) instead of the previous five in (26).

######  Remark 3.6.

In Theorem 3.2 one can consider the operator $B_{\mathbf{0}}$ to be unbounded,
densely defined on $D$ (the same domain which is common for all $A_{\alpha}$)
so that either of the following holds:

  * (i)

$A_{\alpha}$, $\alpha\in\mathcal{I}$, are generating contraction semigroups
(i.e. $M=1$, $w=0$), and $B_{\mathbf{0}}$ is dissipative, $A_{\alpha}-$bounded
with $a_{\alpha}^{0}<1$ (i.e. there exist $a_{\alpha},b_{\alpha}>0$ such that
$\|B_{\mathbf{0}}x\|\leq a_{\alpha}\|A_{\alpha}x\|+b_{\alpha}\|x\|$, $x\in D$,
and $a_{\alpha}^{0}=\inf\\{a_{\alpha}>0:\exists b_{\alpha}>0,\forall x\in
D,\|B_{\mathbf{0}}x\|\leq a_{\alpha}\|A_{\alpha}x\|+b_{\alpha}\|x\|\\}$), for
all $\alpha\in\mathcal{I}$,

  * (ii)

$B_{\mathbf{0}}$ is closable, dissipative and $A_{\alpha}-$compact (i.e.
$B:(D,\|\cdot\|_{A_{\alpha}})\rightarrow X$ is compact where
$\|\cdot\|_{A_{\alpha}}$ denotes the graph norm), for all
$\alpha\in\mathcal{I}$,

  * (iii)

$A_{\alpha}$ are generating analytic semigroups (i.e. $w<0$),
$\alpha\in\mathcal{I}$, and $B_{\mathbf{0}}$ is closable and
$A_{\alpha}-$compact .

Then, $A_{\alpha}+B_{\mathbf{0}}$ is the infinitesimal generator of a
$C_{0}-$semigroup (denote it $(S_{t})_{\alpha}$) for all
$\alpha\in\mathcal{I}$. If the semigroups $(T_{t})_{\alpha}$ corresponding to
$A_{\alpha}$ are uniformly bounded in $\alpha$, then so will be
$(S_{t})_{\alpha}$. Retaining all other assumptions of Theorem 3.2, now we
follow the same proof pattern with the semigroup $(S_{t})_{\alpha}$,
$\|(S_{t})_{\alpha}\|\leq\tilde{M}e^{\tilde{w}t}$, for some $\tilde{M}\geq 1$,
$\tilde{w}\in\mathbb{R}$, independent of $\alpha$.

Finally we note that in case (i) and (ii) $A_{\alpha}+B_{\mathbf{0}}$ will be
generating contraction semigroups, while in case (iii) they will be generating
analytic semigroups.

##  4 Stationary equations

In this section we consider stationary equations of the form

$\mathbf{A}U+\mathbf{B}\lozenge U+F=0,$ (39)

where $\mathbf{A}:\,X\otimes(S)_{-1}\rightarrow X\otimes(S)_{-1}$ and
$\mathbf{B}\lozenge:\,X\otimes(S)_{-1}\rightarrow X\otimes(S)_{-1}$ are
coordinatewise operators as in (5) and (6). We assume that
$\\{A_{\alpha}\\}_{\alpha\in\mathcal{I}}$ and
$\\{B_{\alpha}\\}_{\alpha\in\mathcal{I}}$ are bounded operators and that
$A_{\alpha}$ are of the form

$A_{\alpha}=\widetilde{A}_{\alpha}+C_{\alpha},\;\;\;\alpha\in\mathcal{I},$

where $B_{\mathbf{0}}$ and $\widetilde{A}_{\alpha}$, $\alpha\in\mathcal{I}$
are compact operators and $C_{\alpha}$ are self adjoint operators for all
$\alpha\in\mathcal{I}$. Denote by $r_{\alpha}$ the eigenvalue corresponding to
the orthogonal family of eigenvectors $H_{\alpha}$, i.e.
$C_{\alpha}(H_{\alpha})=r_{\alpha}H_{\alpha}$, $\alpha\in\mathcal{I}$. Using
classical results of elliptic PDEs and the Fredholm alternative (see [5]) we
prove existence and uniqueness of the solution to (39).

######  Theorem 4.1.

Let $X$ be a Banach space. Let $\mathbf{A}:\,X\otimes(S)_{-1}\rightarrow
X\otimes(S)_{-1}$ and $\mathbf{B}\lozenge:\,X\otimes(S)_{-1}\rightarrow
X\otimes(S)_{-1}$ be coordinatewise operators, for which the following
assumptions hold:

  1. 1.

$\mathbf{A}$ is of the form $\mathbf{A}=\mathbf{\widetilde{A}}+\mathbf{C}$,
where
$\mathbf{\widetilde{A}}(U)=\sum\limits_{\alpha\in\mathcal{I}}\widetilde{A}_{\alpha}(u_{\alpha})H_{\alpha}$
and $\widetilde{A}_{\alpha}:X\rightarrow X$ are compact operators for all
$\alpha\in\mathcal{I}$,
$\mathbf{C}(U)=\sum\limits_{\alpha\in\mathcal{I}}\,r_{\alpha}u_{\alpha}H_{\alpha}$,
$r_{\alpha}\in\mathbb{R}$, $\alpha\in\mathcal{I}$, and $\mathbf{B}$ is of the
form (6), where $B_{\mathbf{0}}:X\rightarrow X$ is a compact operator. Assume
there exists $K>0$ such that:

$-\|\widetilde{A}_{\alpha}\|-\|B_{\mathbf{0}}\|-r_{\alpha}\geq 0,\quad\mbox{
for all }\;\;\alpha\in\mathcal{I},$ (40)

and

$\sup\limits_{\alpha\in\mathcal{I}}\left(\frac{1}{-r_{\alpha}-\|\widetilde{A}_{\alpha}\|-\|B_{\mathbf{0}}\|}\right)<K.$
(41)

  2. 2.

$\mathbf{B}$ is of the form (6), where $B_{\beta}:X\rightarrow X$,
$\beta\in\mathcal{I}\setminus\\{\mathbf{0}\\}$, are bounded operators and
there exists $p>0$ such that

$K\sum\limits_{\beta\in\mathcal{I}\atop\beta>\mathbf{0}}\|B_{\beta}\|\,(2\mathbb{N})^{\frac{-p\beta}{2}}<\frac{1}{\sqrt{2}}.$
(42)

  3. 3.

For every $\alpha\in\mathcal{I}$

${\rm Ker}\left(\widetilde{A}_{\alpha}+(1+r_{\alpha}){\rm
Id}+B_{\mathbf{0}}\right)=\\{0\\}.$ (43)

Then, for every $F\in X\otimes(S)_{-1,-p}$ there exists a unique solution
$U\in X\otimes(S)_{-1,-p}$ to equation (39).

###### Proof.

Equation (39) is equivalent to
$U-(\mathbf{\widetilde{A}}(U)+\mathbf{C}U+U+\mathbf{B}\lozenge U)=F$ and

$\sum\limits_{\gamma\in\mathcal{I}}\left(u_{\gamma}-\widetilde{A}_{\gamma}u_{\gamma}-(1+r_{\gamma})\,u_{\gamma}-\sum\limits_{\alpha+\beta=\gamma}B_{\alpha}(u_{\beta})\right)H_{\gamma}=\sum\limits_{\gamma\in\mathcal{I}}f_{\gamma}H_{\gamma}.$

Due to uniqueness of the Wiener-Itô chaos expansion this is equivalent to

$u_{\gamma}-\left(\widetilde{A}_{\gamma}+(1+r_{\gamma})Id+B_{\mathbf{0}}\right)u_{\gamma}=f_{\gamma}+\sum\limits_{\mathbf{0}<\beta\leq\gamma}B_{\beta}(u_{\gamma-\beta}),\quad\gamma\in\mathcal{I}.$
(44)

By (43) it follows that for each $\gamma\in\mathcal{I}$ the homogeneous
equation

$u_{\gamma}-\left(\widetilde{A}_{\gamma}+(1+r_{\gamma})Id+B_{\mathbf{0}}\right)u_{\gamma}=0$

has only trivial solution $u_{\gamma}=0$. Since the operator
$\widetilde{A}_{\gamma}+(1+r_{\gamma})Id+B_{\mathbf{0}}$ is compact, the
classical Fredholm alternative implies that for each $\gamma\in\mathcal{I}$
there exists a unique $u_{\gamma}$ that solves (44) and it is of the form

$u_{\gamma}=(Id-((r_{\gamma}+1)\,Id+\widetilde{A}_{\gamma}+B_{\mathbf{0}}))^{-1}\left(f_{\gamma}+\sum\limits_{\beta>\mathbf{0}}B_{\beta}(u_{\gamma-\beta})\right),\,\,\,\quad\gamma\in\mathcal{I},$

so that

$\|u_{\gamma}\|_{X}\leq\frac{1}{-r_{\gamma}-\|\widetilde{A}_{\gamma}\|-\|B_{\mathbf{0}}\|}\cdot\left(\|f_{\gamma}\|_{X}+\sum\limits_{\beta>\mathbf{0}}\|B_{\beta}\|\|u_{\gamma-\beta}\|_{X}\right),\quad\gamma\in\mathcal{I}.$

We will prove that $\sum\limits_{\gamma\in\mathcal{I}}u_{\gamma}\otimes
H_{\gamma}$ converges in $X\otimes(S)_{-1}$. Indeed,

$\displaystyle\sum\limits_{\gamma\in\mathcal{I}}\|u_{\gamma}\|_{X}^{2}(2\mathbb{N})^{-p\gamma}$
$\displaystyle\leq$ $\displaystyle
K^{2}\sum\limits_{\gamma\in\mathcal{I}}\left(\|f_{\gamma}\|_{X}+\sum_{\gamma=\alpha+\beta,\alpha>\mathbf{0}}\,\|B_{\alpha}\|\|u_{\beta}\|_{X}\right)^{2}(2\mathbb{N})^{-p\gamma}$
$\displaystyle\leq$ $\displaystyle
2K^{2}\left(\sum_{\gamma\in\mathcal{I}}\|f_{\gamma}\|_{X}^{2}(2\mathbb{N})^{-p\gamma}+\sum_{\gamma\in\mathcal{I}}(\sum_{\gamma=\alpha+\beta,\alpha>\mathbf{0}}\|B_{\alpha}\|\|u_{\beta}\|_{X})^{2}(2\mathbb{N})^{-p\gamma}\right)$
$\displaystyle\leq$ $\displaystyle
2K^{2}\left(\sum_{\gamma\in\mathcal{I}}\|f_{\gamma}\|_{X}^{2}(2\mathbb{N})^{-p\gamma}+(\sum_{\alpha>\mathbf{0}}\|B_{\alpha}\|(2\mathbb{N})^{-\frac{p\alpha}{2}})^{2}\sum\limits_{\beta\in\mathcal{I}}\|u_{\beta}\|_{X}^{2}(2\mathbb{N})^{-p\beta}\right).$

Therefore,

$(1-2K^{2}(\sum\limits_{\alpha>\mathbf{0}}\|B_{\alpha}\|(2\mathbb{N})^{-\frac{p\alpha}{2}})^{2})\cdot\sum\limits_{\gamma\in\mathcal{I}}\|u_{\gamma}\|_{X}^{2}(2\mathbb{N})^{-p\gamma}\leq
2K^{2}\sum_{\gamma\in\mathcal{I}}\|f_{\gamma}\|_{X}^{2}(2\mathbb{N})^{-p\gamma}.$

By assumption (42) we have that
$M=1-2K^{2}(\sum\limits_{\alpha>\mathbf{0}}\|B_{\alpha}\|(2\mathbb{N})^{-\frac{p\alpha}{2}})^{2}>0$.
This implies

$\sum_{\gamma\in\mathcal{I}}\|u_{\gamma}\|_{X}^{2}(2\mathbb{N})^{-p\gamma}\leq\frac{2K^{2}}{M}\sum_{\gamma\in\mathcal{I}}\|f_{\gamma}\|_{X}^{2}(2\mathbb{N})^{-p\gamma}<\infty.$

∎

######  Example 4.2.

We provide some special cases of equation (39).

  * 1.

If $A_{\alpha}=0$ for all $\alpha\in\mathcal{I}$ and $B_{\alpha}$,
$\alpha\in\mathcal{I}$ are second order strictly elliptic partial differential
operators in divergent form

$B_{\alpha}=\sum_{i=1}^{n}D_{i}(\sum_{j=1}^{n}a^{ij}_{\alpha}(x)D_{j}+b^{i}_{\alpha}(x))+\sum_{i=1}^{n}c^{i}_{\alpha}(x)D_{i}+d_{\alpha}(x)$
(45)

with essentially bounded coefficients, then equation (39) reduces to the
elliptic equation

$\mathbf{B}\lozenge U=F,$

which was solved in [18] and [19].

  * 2.

Let $\widetilde{A}_{\alpha}=0$ for all $\alpha\in\mathcal{I}$ and let
$B_{\alpha}$, $\alpha\in\mathcal{I}$, be second order strictly elliptic
partial differential operators in divergent form (45). Let
$\mathbf{C}=c\,P(\mathcal{R})$, for some $c\in\mathbb{R}$, where $\mathcal{R}$
is the Ornstein-Uhlenbeck operator, $P$ a polynomial of degree $m$ with real
coefficients and $P(\mathcal{R})$ the differential operator
$P(\mathcal{R})=p_{m}\mathcal{R}^{m}+p_{m-1}\mathcal{R}^{m-1}+...+p_{1}\mathcal{R}+p_{0}Id$.
Then, the corresponding eigenvalues are $r_{\alpha}=cP(|\alpha|)$,
$\alpha\in\mathcal{I}$. Hence, equation (39) transforms to the elliptic
equation with a perturbation term driven by the polynomial of the Ornstein-
Uhlenbeck operator

$\mathbf{B}\lozenge U+cP(\mathcal{R})U=F,$

that was solved in [11].

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## Acknowledgement

The paper was supported by the projects Modeling and harmonic analysis methods
and PDEs with singularities, No. 174024, and Modeling and research methods of
operational control of traffic based on electric traction vehicles optimized
by power consumption criterion, No. TR36047, both financed by the Ministry of
Science, Republic of Serbia and project No. 114-451-3605/2013 financed by the
Provincial Secretariat for Science of Vojvodina.