# Stochastic evolution equations with Wick-analytic nonlinearities

Tijana LEVAJKOVIĆ and Stevan PILIPOVIĆ and Dora SELEŠI and Milica ŽIGIĆ
Applied Statistics Research Unit, Institute of Statistics and Mathematical
Methods in Economics, TU Vienna, Austria.  Department of Mathematics and
Informatics, Faculty of Sciences, University of Novi Sad, Serbia.
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
(Date: 05.02.2021.)

###### Abstract.

We study nonlinear stochastic partial differential equations with Wick-
analytic type nonlinearities set in the framework of white noise analysis.
These equations include the stochastic Fisher–KPP equations, stochastic
Allen–Cahn, stochastic Newell–Whitehead–Segel, and stochastic Fujita–Gelfand
equations. By implementing the theory of $C_{0}-$semigroups and evolution
systems into the chaos expansion theory in infinite dimensional spaces, we
prove existence and uniqueness of solutions for this class of stochastic
partial differential equations.

###### Key words and phrases:

weighted Hida spaces; stochastic nonlinear evolution equations; Wick product;
$C_{0}-$semigroup; infinitesimal generator

###### 2020 Mathematics Subject Classification:

60H15; 60H40; 60G20; 37L55; 47J35

Corresponding author: Milica Žigić<EMAIL_ADDRESS>
##  1\. Introduction

As a conclusion to our previous papers [22] and [23] we perform a follow-up
study to extend the results obtained therein. In [22] we treated the case of
stochastic evolution equations with Wick-multiplicative noise of the form
$u_{t}={\mathbf{A}}u+{\mathbf{B}}\lozenge u+f$, where ${\mathbf{A}}$ was a
densely defined unbounded operator generating a $C_{0}$-semigroup (e.g. a
strictly elliptic second order partial differential operator), while
${\mathbf{B}}$ was a linear bounded operator modeling convolution-type
perturbations via the Wick product in the equation. Examples of these
equations included the heat equation with random potential, the Schrödinger
equation, the transport equation driven by white noise, and many others.
Afterwards, in [23] we studied stochastic evolution equations with Wick-
polynomial nonlinearities of the form $u_{t}={\mathbf{A}}u+u^{\lozenge n}+f$,
including several reaction–diffusion examples such as the Fisher–KPP
equations, FitzHugh–Nagumo equations, and many more. In both papers we proved
existence and uniqueness of the solutions and also developed an exact
algorithm for obtaining explicit solutions in terms of generators of a
corresponding $C_{0}$-semigroup and polynomial chaos expansion. In the current
paper we study fully nonlinear stochastic evolution equations of the form
$u_{t}={\mathbf{A}}u+\Phi^{\lozenge}(u)+f$, where the nonlinearity is modeled
by an analytic (entire) function $\Phi$, and $\Phi^{\lozenge}$ denotes its
corresponding Wick-version as it will be described in detail below. Examples
of these equations involve stochastic versions of the Allen–Cahn,
Newell–Whitehead–Segel equations, and various other equations.

Let $X$ be a certain Banach algebra. Let $\Phi$ be an analytic function on $X$
with its Maclaurin expansion $\displaystyle\Phi(\xi)=\sum_{n\geq
0}a_{n}\xi^{n},\;\xi\in X,$ where $a_{n},\;n\geq 0$ are either constants or
deterministic functions (i.e. elements of $X$). Here we implicitly assumed
that $X$ is a Banach algebra of deterministic functions over some field of
constants. We study stochastic nonlinear evolution equations of the form

$\displaystyle u_{t}(t,\omega)$
$\displaystyle=\mathbf{A}\,u(t,\omega)+\Phi^{\lozenge}\big{(}u(t,\omega)\big{)}+f(t,\omega),\quad
t\in(0,T],\quad\omega\in\Omega,$
$\displaystyle=\mathbf{A}\,u(t,\omega)+\sum_{n=0}^{\infty}a_{n}u^{\lozenge
n}(t,\omega)+f(t,\omega),\quad t\in(0,T],\quad\omega\in\Omega,$ (1)
$\displaystyle u(0,\omega)$
$\displaystyle=u^{0}(\omega),\quad\omega\in\Omega,$

where $u(t,\omega)$, $t\in[0,T]$, $\omega\in\Omega$ is an $X-$valued
generalized stochastic process and $\mathbf{A}$ corresponds to a densely
defined infinitesimal generator of a $C_{0}-$semigroup. The nonlinear part
$\Phi^{\lozenge}\big{(}u(t,\omega)\big{)}$ is given in terms of Wick-powers
$u^{\lozenge n}=u^{\lozenge n-1}\lozenge u=u\lozenge\dots\lozenge
u,\;n\in\mathbb{N}$, where $\lozenge$ denotes the Wick product, instead of a
classical product considered in the definition of the function $\Phi$. The
Wick product is involved due to the fact that we allow random terms to be
present both in the initial condition $u_{0}$ and the driving force $f$, hence
the solution $u$ will inherit these random properties. This leads to singular
solutions that do not allow to use ordinary multiplication, but require a
renormalization of the multiplication, which is done by introducing the Wick
product into the equation. The Wick product is known to represent the highest
order stochastic approximation of the ordinary product [27], while some better
approximations may also be achieved [36] in the framework of Malliavin
calculus. Although the Wick product of random variables as a modelling tool in
SPDEs [9] has received some critiques [20] for having non-intuitive properties
such as factorizing expectations and forcing independence it has been used in
many papers not just as a simplification tool, but as a necessity for dealing
with products and nonlinear functions of distributions where the ordinary
product would fail to be well defined. Several other theories allow for a
formal treatment of nonlinearities of generalized stochastic processes such as
Colombeau algebras of generalized stochastic processes [1],[29], also
combinations of the latter with Wick calculus [34], rough-paths theory [26]
together with its extensions, Hairer’s celebrated model of regularity
structures [11]. Several papers deal with a comparison of the Wick product and
the ordinary product, e.g. in terms of an error estimate [36], numerical
simulations [35], as a scaling limit process [10] etc. in those cases where
the ordinary product is meaningful. Generally, the Wick product has shown to
be a satisfactory approximation of the ordinary product from a modelling point
of view.

Various problems appearing in quantum mechanics, plasma physics, mathematical
biology, ecology, and medicine can be formulated in the form (1). In
particular, in stochastic Fisher–KPP equations, Fujita-type equations,
FitzHugh–Nagumo equations nonlinearities can be generated by Wick-polynomial
nonlinearities. In our previous paper [23] we proved existence and uniqueness
of solutions of nonlinear parabolic stochastic partial differential equations
with Wick-power and Wick-polynomial type nonlinearities.

Stochastic reaction-diffusion equation perturbed by multiplicative noise with
a polynomial nonlinearity of the form

$\Phi(s)=\sum_{i=1}^{2p-1}a_{i}\,s^{i},\quad\text{with}\,\,a_{2p-1}<0,$ (2)

in a bounded domain, where $p$ is a positive integer and
$a_{1},\dots,a_{2p-1}$ are real numbers was considered in [7]. Based on
semigroups techniques, the author proved global existence and uniqueness of
such problems. Stochastic nonlinear equations of the form (1) include also the
problems $u_{t}=\triangle u+\Phi^{\lozenge}(u)+u\lozenge W_{t}$, with
$\Phi^{\lozenge}$ being the Wick version of the polynomial nonlinearity (2).

Polynomial nonlinearities of the form (2) appear in a special reaction-
diffusion equation, the stochastic Allen–Cahn equation (also called the
stochastic Ginzburg–Landau equation), $u_{t}=\triangle u+u-u^{\lozenge 3}$.
The deterministic Allen-Cahn equation is used to describe phase separation and
the evolution of interfaces between the phases for systems without mass
conservation [6, 8], while the corresponding stochastic Allen–Cahn equations
take into account also the effects due to the thermal fluctuations of the
system [3]. The stochastic Newell-Whitehead equation $u_{t}=\triangle u+au-
bu^{\lozenge n}$, $a,b>0$, $n>1$, is also a reaction-diffusion equation which
belongs to the class (1). The deterministic Newell–Whitehead–Segel equation
models the interaction of the effect of the diffusion term with the nonlinear
effect of the reaction term. It has found various applications in mechanical
and chemical engineering, ecology, biology and bioengineering.

Exponential type nonlinearities appear in the stochastic Fujita–Gelfand
equation in combustion, i.e. the stochastic exponential reaction-diffusion
equation of the form $u_{t}=\triangle u+\lambda e^{\lozenge u}$, $\lambda>0$.
The corresponding deterministic problems are used to model convection in a
heat transfer process when exothermic chemical reactions occur in the fluid
body [19].

The nonlinear Schrödinger equation with logarithmic nonlinearity possesses
solitonlike solutions of a Gaussian shape, also called Gaussons, in any number
of dimensions. These solutions in nonlinear wave mechanics describe the
propagation of wave packets of freely moving particles and were explained for
example in [4].

Existence and uniqueness of nonnegative solutions of the heat equation with a
logarithmic nonlinearity was proven in [2]. In the current paper, as an
application of the general class of problems (1), we solve stochastic heat
equations with logarithmic nonlinearity of the form $u_{t}=\triangle
u+u\lozenge\log^{\lozenge}|u|+f$. Other types of nonlinearities $\Phi$ which
are analytic functions, such as trigonometric or hyperbolic functions also fit
to our setting. Namely, the nonlinear stochastic equation of the form
$u_{t}=\triangle u+\cos^{\lozenge}u+\cosh^{\lozenge}u+f$ belongs to the
considered class of stochastic equations (1).

We implement the Wiener-Itô chaos expansion method from white noise analysis
combined with $C_{0}-$ operator semigroup theory in order to prove the
existence and the uniqueness of a solution to (1). Using the chaos expansion
method the initial stochastic partial differential equation (SPDE) is
transformed into a lower triangular infinite system of PDEs (propagator
system) that can be solved recursively. Solving this system, one obtains the
coefficients of the solution to (1). Finally, convergence of the series needs
to be proven, and for this purpose we will define a new class of weighted
Hida-type spaces and utilize some estimates for growth rates of a factorial
function.

The propagator system method has been successfully implemented in [22]–[24],
[27], [32]. Our approach provides a way to solve nonlinear stochastic
equations (1) explicitly. It is important to note that SPDEs with Wick-
multiplication have been considered since the very beginning of the foundation
of Hida’s white noise analysis [13] and continued in the papers of Holden,
Øksendal and their coworkers in [14], [15], [17], etc. as summed up in [16].
Our approach differs in one important fact, namely in [16] the solution is
provided in terms of the Hermite transform and its inverse. For the existence
of the inverse Hermite transform the corresponding (deterministic) PDE needs
to satisfy some very strict analytical properties. In our approach there is no
need to apply the Hermite transform (or its counterpart the $S-$transform),
one can just follow a recursive pattern to calculate the coefficients of the
solution. This is an algorithmic procedure that provides an
explicit/computable form of the solution and therefore it can be also used as
an efficient method for numerical implementations. In the existing literature,
certain types of these equations were treated only numerically. For example,
an efficient numerical approximation of nonlinear parabolic SPDEs with
additive noise was proposed in [18]. The authors obtained higher convergence
order in comparison to convergence results of classical schemes such as the
linear implicit Euler scheme. An optimal strong convergence rate of a fully
discrete numerical scheme for the stochastic Allen–Cahn equation driven by an
additive space-time white noise was established in [25]. An accurate and
efficient Haar transform or Haar wavelet method was introduced in [12] for
solving deterministic nonlinear parabolic partial differential equations such
as for example Allen–Cahn or Newell–Whitehead–Segel equations.

The paper is organized as follows: In the introductory section we recall upon
basic notions of $C_{0}-$semigroups, evolution systems, entire functions
defined on Banach spaces and white noise theory including chaos expansions of
generalized stochastic processes. Here we will introduce a space of weighted
Hida generalized random variables. In Section 2, which represents the main
part of the paper, we prove existence and uniqueness of the solution to (1).
In Section 3 we provide some concrete examples of nonlinear problems (1).

###  1.1. Generalized stochastic processes

Denote by $(\Omega,\mathcal{F},\mu)$ the Gaussian white noise probability
space $(S^{\prime}(\mathbb{R}),\mathcal{B},\mu),$ where
$\Omega=S^{\prime}(\mathbb{R})$ denotes the space of tempered distributions,
$\mathcal{B}$ is 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 $L^{2}(\Omega,\mu)$, the space of square
integrable random variables over the underlying probability space
$(\Omega,\mathcal{F},\mu)$, see [16]. The set of multiindices $\mathcal{I}$ is
$(\mathbb{N}_{0}^{\mathbb{N}})_{c}$, i.e. the set of sequences of non-negative
integers which have only finitely many non-zero components. Especially, we
denote by $\mathbf{0}=(0,0,0,\ldots)$ the zero multiindex with all entries
equal to zero, the length of a multiindex is
$|\alpha|=\sum_{i=1}^{\infty}\alpha_{i}$ for
$\alpha=(\alpha_{1},\alpha_{2},\ldots)\in\mathcal{I}$ and
$\alpha!=\prod_{i=1}^{\infty}\alpha_{i}!.$ 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 $\alpha-\beta\geq\mathbf{0}.$

The Wiener-Itô 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 $\xi_{n}$,

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

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
multiindex 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
$B(t,\omega)=\sum_{k=1}^{\infty}\int_{0}^{t}\xi_{k}(s)ds\;H_{\varepsilon_{k}}(\omega).$

Changing the topology on $L^{2}(\Omega,\mu)$ to a weaker one, T. Hida [13]
defined spaces of generalized random variables containing the white noise as a
weak derivative of the Brownian motion. We refer to [13, 16] for white noise
analysis.

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 $p>1$.
Using the same technique as in [16, Chapter 2] one can define Banach spaces
$(FS)_{r,p}$ of test functions and their topological duals $(FS)_{-r,-p}$ of
stochastic distributions for all $r\geq 2$ and $p\geq 0.$

######  Definition 1.1.

Let $r\geq 2,\;p\geq 0.$ We define the stochastic test function space by

$(FS)_{r,p}=\\{F=\sum_{\alpha\in\mathcal{I}}f_{\alpha}{H_{\alpha}}\in
L^{2}(\Omega,\mu):\;\|F\|^{2}_{(FS)_{r,p}}=\sum_{\alpha\in\mathcal{I}}\alpha!|f_{\alpha}|^{2}r^{|\alpha|^{3}}!(2\mathbb{N})^{p\alpha}<\infty\\},$

and its topological dual, the stochastic distribution space, by formal sums

$(FS)_{-r,-p}=\\{F=\sum_{\alpha\in\mathcal{I}}f_{\alpha}{H_{\alpha}}:\;\|F\|^{2}_{(FS)_{-r,-p}}=\sum_{\alpha\in\mathcal{I}}\alpha!|f_{\alpha}|^{2}(r^{|\alpha|^{3}}!)^{-1}(2\mathbb{N})^{-p\alpha}<\infty\\}.$

The space of test random variables is

$(FS)=\bigcap_{r\geq 2,\;p\geq 0}(FS)_{r,p}$

endowed with the projective topology. Its dual, the space of generalized
random variables is

$(FS)^{\prime}=\bigcup_{r\geq 2,\;p\geq 0}(FS)_{-r,-p}$

endowed with the inductive topology.

The action of
$F=\sum_{\alpha\in\mathcal{I}}b_{\alpha}H_{\alpha}\in(FS)^{\prime}$ onto
$f=\sum_{\alpha\in\mathcal{I}}c_{\alpha}H_{\alpha}\in(FS)$ is given by
$\langle F,f\rangle=\sum_{\alpha\in\mathcal{I}}\alpha!b_{\alpha}c_{\alpha},$
where $b_{\alpha},c_{\alpha}\in\mathbb{R}$. Thus, they form a Gel’fand triplet

$(FS)\subset L^{2}(\Omega,\mu)\subset(FS)^{\prime}.$

Particularly, it also holds $(FS)_{-r,-p}\subseteq(FS)_{-r,-q}$ for $0\leq
p\leq q$, $r\geq 2$. We will refer to $(FS)$ and $(FS)^{\prime}$ as to
factorial-weighted Hida spaces. Note that for $r=1$ one would retrieve the
classical Hida spaces which are usually denoted as $(S)$ and $(S)^{*}$.

####  1.1.1. Nuclearity of the space of test random variables $(FS)$ and its
dual $(FS)^{\prime}$

The spaces $(FS)_{r,p}$ and $(FS)_{-r,-p},\;r\geq 2,\;p\geq 0$ are separable
Hilbert spaces. Moreover, $(FS)$ and $(FS)^{\prime}$ are nuclear spaces as one
can infer form the following lemma.

######  Lemma 1.1.

The space of the stochastic test functions $(FS)$ is nuclear.

###### Proof.

The space $(FS)_{r,p}$ is a Hilbert space with the inner product

$(f,g)_{r,p}=\sum_{\alpha\in\mathcal{I}}f_{\alpha}\,g_{\alpha}\,\alpha!\,\,r^{|\alpha|^{3}}!\,\,\,\,(2\mathbb{N})^{p\alpha}$

for all $f=\sum_{\alpha\in\mathcal{I}}f_{\alpha}\,H_{\alpha}\in(FS)_{r,p}$ and
$g=\sum_{\beta\in\mathcal{I}}g_{\beta}\,H_{\beta}\in(FS)_{r,p}$, $r\geq 2$,
$p\geq 0$. Thus, the family of functions

$K_{\alpha,r,p}=\frac{1}{\sqrt{\alpha!\;r^{|\alpha|^{3}}}!}\,\cdot(2\mathbb{N})^{-\frac{p\alpha}{2}}\,H_{\alpha},\quad\alpha\in\mathcal{I},\;$

forms an orthonormal basis for $(FS)_{r,p}$. By definition, $(FS)$ is the
projective limit of $(FS)_{r,p}$. If $s\geq r$ and $q>p+1$ then

$\displaystyle\sum_{\alpha\in\mathcal{I}}\|K_{\alpha,s,q}\|^{2}_{(FS)_{r,p}}$
$\displaystyle=\sum_{\alpha\in\mathcal{I}}\Big{(}\frac{1}{\sqrt{\alpha!s^{|\alpha|^{3}}}!}\,\cdot(2\mathbb{N})^{-\frac{q\alpha}{2}}\Big{)}^{2}\,\,\cdot\,\|H_{\alpha}\|^{2}_{(FS)_{r,p}}$
$\displaystyle=\sum_{\alpha\in\mathcal{I}}\frac{1}{\alpha!\;s^{|\alpha|^{3}}!}\,\cdot(2\mathbb{N})^{-q\alpha}\,\cdot\,\alpha!\,r^{|\alpha|^{3}}!\,\,\,\,(2\mathbb{N})^{p\alpha}$
$\displaystyle=\sum_{\alpha\in\mathcal{I}}\frac{r^{|\alpha|^{3}}!}{s^{|\alpha|^{3}}!}\,\cdot(2\mathbb{N})^{-(q-p)\alpha}$
$\displaystyle\leq\sum_{\alpha\in\mathcal{I}}(2\mathbb{N})^{-(q-p)\alpha}<\infty.$

Therefore, the embedding $(FS)_{s,q}\subset(FS)_{r,p}$, $s>r$ and $q>p+1$ is
Hilbert–Schmidt and we conclude that $(FS)$ is a nuclear space. ∎

Since $(FS)=\bigcap_{r\geq 2,p\geq 0}(FS)_{r,p}$ is nuclear, then the
inductive limit of the spaces $(FS)_{-r,-p},$ namely,
$(FS)^{\prime}=\bigcup_{r\geq 2,p\geq 0}(FS)_{-r,-p}$ is nuclear too. We used
the property that the dual of a Fréchet space is nuclear if and only if the
initial space is nuclear.

####  1.1.2. Closedness of $(FS)^{\prime}$ under the Wick product

The _Wick product_ of the stochastic functions
$f=\sum_{\alpha\in\mathcal{I}}f_{\alpha}H_{\alpha}$ and
$g=\sum_{\beta\in\mathcal{I}}g_{\beta}H_{\beta}$ is given by

$f\lozenge
g=\sum_{\gamma\in\mathcal{I}}\sum_{\alpha+\beta=\gamma}f_{\alpha}g_{\beta}H_{\gamma}=\sum_{\alpha\in\mathcal{I}}\sum_{\beta\leq\alpha}f_{\beta}g_{\alpha-\beta}H_{\alpha}.$

In the following theorem we prove that the weighted Hida space $(FS)^{\prime}$
is closed under the Wick multiplication.

######  Theorem 1.1.

Let the stochastic functions $f\in(FS)_{-r,-p}$ and $g\in(FS)_{-r,-p}$, $r\geq
2$, $p\geq 0$, be given in their chaos expansion forms

$f=\sum_{\alpha\in\mathcal{I}}f_{\alpha}H_{\alpha}\qquad\text{and}\qquad
g=\sum_{\beta\in\mathcal{I}}g_{\beta}H_{\beta}.$

Then, the Wick product $f\lozenge g$ is a well defined element in
$(FS)_{-s,-q}$ for $s\geq r\geq 2$ and $q>p$ (i.e. for $q\geq p-1+k$, where
$k>1$).

###### Proof.

First we note that for $s\geq 2$ and positive numbers $a$ and $b$ it holds
that

$\displaystyle s^{a^{3}}!\cdot s^{b^{3}}!\leq(s^{a^{3}}+s^{b^{3}})!\leq
s^{a^{3}+b^{3}}!\leq s^{(a+b)^{3}}!$ (3)

which then implies

$(s^{(a+b)^{3}}!)^{-1}\leq(s^{a^{3}}!)^{-1}\cdot(s^{b^{3}}!)^{-1}.$

Thus, in the proof we will use the inequality

$(s^{(a+b)^{3}}!)^{-1}\leq(r^{a^{3}}!)^{-1}\cdot(r^{b^{3}}!)^{-1},\qquad\text{for}\quad
r\leq s.$

Also, we use the estimate

$(\alpha+\beta)!\leq\alpha!\beta!(2\mathbb{N})^{\alpha+\beta},\quad\alpha,\beta\in\mathcal{I}.$

By the Cauchy–Schwartz inequality, for $q=p-1+k$ and $s\geq r\geq 2$ the
following holds

$\displaystyle\|f\lozenge g\|^{2}_{(FS)_{-s,-q}}=$
$\displaystyle=\sum_{\gamma\in\mathcal{I}}\Big{\|}\sum_{\alpha+\beta=\gamma}f_{\alpha}g_{\beta}\Big{\|}^{2}\,\gamma!\,(s^{|\gamma|^{3}}!)^{-1}\,(2\mathbb{N})^{-q\gamma}$
$\displaystyle=\sum_{\gamma\in\mathcal{I}}\Big{\|}\sum_{\alpha+\beta=\gamma}f_{\alpha}g_{\beta}(\alpha+\beta)!^{\frac{1}{2}}\Big{\|}^{2}\,(s^{|\gamma|^{3}}!)^{-1}\,(2\mathbb{N})^{-(p-1+k)\gamma}$
$\displaystyle\leq\sum_{\gamma\in\mathcal{I}}\Big{\|}\sum_{\alpha+\beta=\gamma}f_{\alpha}\,\alpha!^{\frac{1}{2}}\,(s^{|\alpha|^{3}}!)^{-\frac{1}{2}}\,\,(2\mathbb{N})^{-\frac{(p-1)\alpha}{2}+\frac{\alpha}{2}}\,g_{\beta}\,\beta!^{\frac{1}{2}}\,(s^{|\beta|^{3}}!)^{-\frac{1}{2}}\,\,(2\mathbb{N})^{-\frac{(p-1)\beta}{2}+\frac{\beta}{2}}\Big{\|}^{2}(2\mathbb{N})^{-k\gamma}$
$\displaystyle\leq\sum_{\gamma\in\mathcal{I}}\Big{(}\sum_{\alpha+\beta=\gamma}\,|f_{\alpha}|^{2}\,\alpha!\,(r^{|\alpha|^{3}}!)^{-1}\,\,(2\mathbb{N})^{-p\alpha}\Big{)}\,\,\Big{(}\sum_{\alpha+\beta=\gamma}|g_{\beta}|^{2}\,\beta!\,(r^{|\beta|^{3}}!)^{-1}\,\,(2\mathbb{N})^{-p\beta}\Big{)}\,\,(2\mathbb{N})^{-k\gamma}$
$\displaystyle\leq\,\Big{(}\sum_{\gamma\in\mathcal{I}}(2\mathbb{N})^{-k\gamma}\Big{)}\,\Big{(}\sum_{\alpha\in\mathcal{I}}\,|f_{\alpha}|^{2}\,\alpha!\,(r^{|\alpha|^{3}}!)^{-1}\,\,(2\mathbb{N})^{-p\alpha}\Big{)}\,\,\Big{(}\sum_{\beta\in\mathcal{I}}|g_{\beta}|^{2}\,\beta!\,(r^{|\beta|^{3}!})^{-1}\,\,(2\mathbb{N})^{-p\beta}\Big{)}\,$
$\displaystyle\leq
M\,\cdot\|f\|^{2}_{(FS)_{-r,-p}}\,\cdot\,\,\|g\|^{2}_{(FS)_{-r,-p}},$

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

Since the space $(FS)^{\prime}$ is closed under the Wick multiplication, then
its $n$th Wick power is also a well defined element in $(FS)^{\prime}$.
Recall, the $n$th Wick power of $f$ is defined recursively by $f^{\lozenge
n}=f\lozenge f^{\lozenge(n-1)}$, for $n\geq 1$ and $f^{\lozenge 0}=1$.

######  Corollary 1.1.

Let $f\in(FS)_{-r,-p}$, $r\geq 2$, $p\geq 0$. Then its $n$th Wick power
$f^{\lozenge n}$, $n\geq 1$ is an element of $(FS)_{-s,-q}$, for $s\geq r$,
$q>p+1$.

###### Proof.

Applying Theorem 1.1 with $f=g$ we obtain that $f^{\lozenge 2}\in(FS)_{-s,-q}$
for $s\geq r$ and $q=p-1+k$, $k>1$. Now, using the embedding
$f\in(FS)_{-r,-p}\subset(FS)_{-s,-p-1+k}$ we obtain $f,f^{\lozenge
2}\in(FS)_{-s,-p-1+k}$ and hence may apply again Theorem 1.1 to obtain
$f^{\lozenge 3}=f\lozenge f^{\lozenge 2}\in(FS)_{-s_{1},-q_{1}}$ for
$s_{1}\geq s$ and $q_{1}\geq q-1+k_{1}=p-2+k+k_{1}$, where $k_{1}>1$. Without
loss of generality we may let $k_{1}=k$ and write $f^{\lozenge
3}\in(FS)_{-s,-(p-2+2k)}$. By induction one may easily prove that $f^{\lozenge
n}\in(FS)_{-s,-p-(n-1)(k-1)}$. Since $k>1$, we may choose $k=1+\epsilon$,
$\epsilon>0$, small enough so that $(n-1)(k-1)<1$, hence $f^{\lozenge
n}\in(FS)_{-s,-q}$, for $q>p+1$. ∎

####  1.1.3. Stochastic processes

In the same way as it was done in [31] and [32], we extend the definition of
stochastic processes to processes with the chaos expansion form

$u(t,\omega)=\sum_{\alpha\in\mathcal{I}}u_{\alpha}(t){H_{\alpha}}(\omega),$
(4)

where the coefficients $u_{\alpha}$ are elements of the Banach space of
functions $C^{k}[0,T],\;k\in\mathbb{N}$.

######  Definition 1.2.

We say that $u$ given in the chaos expansion form (4) is a _generalized
stochastic process_ in $C^{k}[0,T]\otimes(FS)^{\prime}$ if there exist $r\geq
2$ and $p\geq 0$ such that

$\|u\|_{C^{k}[0,T]\otimes(FS)^{\prime}}^{2}=\sum_{\alpha\in\mathcal{I}}\alpha!(r^{|\alpha|^{3}}!)^{-1}\|u_{\alpha}\|_{C^{k}[0,T]}^{2}(2\mathbb{N})^{-p\alpha}<\infty.$

The differentiation of a stochastic process can be carried out componentwise
in the chaos expansion, i.e. due to the fact that $(FS)^{\prime}$ is a nuclear
space it holds that
$C^{k}([0,T],(FS)^{\prime})=C^{k}[0,T]\hat{\otimes}(FS)^{\prime}$ where
$\hat{\otimes}$ denotes the completion of the tensor product which is the same
for the $\varepsilon-$completion and $\pi-$completion. In the sequel, we will
use the notation $\otimes$ instead of $\hat{\otimes}$. Hence
$C^{k}[0,T]\otimes(FS)_{-r,-p}$ and $C^{k}[0,T]\otimes(FS)_{r,p}$ denote
subspaces of the corresponding completions. This means that a stochastic
process $u(t,\omega)$ is $k$ times continuously differentiable if and only if
all of its coefficients $u_{\alpha}$, $\alpha\in\mathcal{I}$ are in
$C^{k}[0,T]$.

We keep the same notation when $C^{k}[0,T]$ is replaced by another Banach
space. The same holds for Banach space valued stochastic processes, i.e.
elements of $C^{k}([0,T],X)\otimes(FS)^{\prime}$, where $X$ is an arbitrary
Banach space. It holds that

$C^{k}([0,T],X\otimes(FS)^{\prime})=C^{k}([0,T],X)\otimes(FS)^{\prime}=\bigcup_{r\geq
2,p\geq 0}C^{k}([0,T],X)\otimes(FS)_{-r,-p},\,\quad k\in\mathbb{N}.$ (5)

###  1.2. Evolution systems

We fix the notation and recall some known facts about evolution systems (see
[30, Chapter 5]). Let $X$ be a Banach space. Let $\\{A(t)\\}_{t\in[s,T]}$ be a
family of linear operators in $X$ such that $A(t):D(A(t))\subset X\to
X,\;t\in[s,T]$. Further, let $f$ be an $X-$valued function $f:[s,T]\to X.$
Consider the initial value problem

$\displaystyle\frac{d}{dt}u(t)$ $\displaystyle=A(t)u(t)+f(t),\quad 0\leq
s<t\leq T,$ (6) $\displaystyle u(s)$ $\displaystyle=x.$

If $u\in C([s,T],X)\cap C^{1}((s,T],X),$ $u(t)\in D(A(t))$ for all $t\in(s,T]$
and $u$ satisfies (6), then $u$ is a classical solution to (6).

A two parameter family of bounded linear operators $S(t,s),\;0\leq s\leq t\leq
T$ on X is called an evolution system if the following two conditions are
satisfied:

  1. (1)

$S(s,s)=I$ and $S(t,r)S(r,s)=S(t,s),\quad 0\leq s\leq r\leq t\leq T$;

  2. (2)

$(t,s)\mapsto S(t,s)$ is strongly continuous for all $0\leq s\leq t\leq T.$

Clearly, if $S(t,s)$ is an evolution system associated with the homogeneous
evolution problem (6), i.e. if $f\equiv 0,$ then a classical solution to (6)
is given by $u(t)=S(t,s)x,\ t\in[s,T].$

A family $\\{A(t)\\}_{t\in[s,T]}$ of infinitesimal generators of
$C_{0}-$semigroups on $X$ is called stable if there exist constants $m\geq 1$
and $w\in\mathbb{R}$ (stability constants) such that
$(w,\infty)\subseteq\rho(A(t)),\;t\in[s,T]$ and

$\Big{\|}\prod_{j=1}^{k}R(\lambda:A(t_{j}))\Big{\|}\leq\frac{m}{(\lambda-w)^{k}},\quad\lambda>w,$

for every finite sequence $0\leq s\leq t_{1}\leq t_{2}\leq\dots\leq t_{k}\leq
T,\;k=1,2,\dots.$

Let $\\{A(t)\\}_{t\in[s,T]}$ be a stable family of infinitesimal generators
with stability constants $m$ and $w$. Let $B(t),\;t\in[s,T],$ be a family of
bounded linear operators on $X$. If $\|B(t)\|\leq M,\;t\in[s,T],$ then
$\\{A(t)+B(t)\\}_{t\in[s,T]}$ is a stable family of infinitesimal generators
with stability constants $m$ and $w+Mm.$

Let $\\{A(t)\\}_{t\in[s,T]}$ be a stable family of infinitesimal generators of
$C_{0}-$semigroups on $X$ such that the domain $D(A(t))=D$ is independent of
$t$ and for every $x\in D,$ $A(t)x$ is continuously differentiable in $X.$ If
$f\in C^{1}([s,T],X)$ then for every $x\in D$ the evolution problem (6) has a
unique classical solution $u$ given by

$u(t)=S(t,s)x+\int_{s}^{t}S(t,r)f(r)dr,\quad 0\leq s\leq t\leq T.$

From the proof of [30, Theorem 5.3, p. 147] one can obtain

$\displaystyle\frac{d}{dt}u(t)=A(t)S(t,s)x+A(t)\int_{s}^{t}S(t,r)f(r)dr+f(t),\quad
s<t\leq T.$

Since $t\mapsto A(t)$ is continuous in $B(D,X)$ and $(t,s)\mapsto S(t,s)$ is
strongly continuous for all $0\leq s\leq t\leq T,$ we have additionally that
the solution $u$ to (6) exhibits the regularity property $u\in C^{1}([s,T],X)$
and $\frac{d}{dt}u(t)|_{t=s}=A(s)x+f(s).$ Recall that the evolution system
$S(t,s)$ satisfies:

  1. (1)

$\|S(t,s)\|\leq me^{w(t-s)},\ 0\leq s\leq t\leq T;$

  2. (2)

$S(t,s)D\subseteq D;$

  3. (3)

$S(t,s)x$ is continuous in $D$ for all $0\leq s\leq t\leq T$ and $x\in D.$

######  Remark 1.1.

Considering infinitesimal generators depending on $t,$ we follow the standard
approach of Yosida (cf. [37], [21]). We refer to [28] for a method based on an
equivalent operator extension problem (see also references in [28]). The chaos
expansion approach, which is the essence of our paper, requires the existence
results for the propagator system i.e. for the coordinate-wise deterministic
Cauchy problems. For this purpose we demonstrate the applications of the
hyperbolic Cauchy problem given in [30].

###  1.3. Entire functions on Banach algebras

In this subsection we fix some notation and provide some basic properties of
entire functions on Banach spaces that will be used in the sequel (see [5]).

Let $X$ be a Banach algebra with the uniform topology. Continuity of a
function $\Phi,$ defined on a region $Y\subset X$ with values in $X,$ is
defined as usual.

######  Definition 1.3.

A function $\Phi$ has a derivative $\Phi^{\prime}(u_{\mathbf{0}})$ at
$u_{\mathbf{0}}\in X$ if for every $\varepsilon>0$ a $\delta>0$ can be found
such that for all $u\in Y$ with $\|u-u_{\mathbf{0}}\|_{X}<\delta,$

$\left\|\Phi(u)-\Phi(u_{\mathbf{0}})-(u-u_{\mathbf{0}})\Phi^{\prime}(u_{\mathbf{0}})\right\|_{X}<\varepsilon\|u-u_{\mathbf{0}}\|_{X}.$

If $\Phi$ has a derivative throughout a neighbourhood of $u_{\mathbf{0}}$ then
$\Phi$ is said to be analytic at $u_{\mathbf{0}}.$ If $\Phi$ is analytic in
the whole space $X$ then it is said to be entire.

As presented in [5], if $\Phi$ is analytic at $u_{\mathbf{0}}\in Y$ then one
can establish its Taylor expansion, i.e. there exists an infinite sequence
$a_{n}\in X,\;n\in\mathbb{N}$ such that

$\Phi(u)=\sum_{n=0}^{\infty}a_{n}(u-u_{\mathbf{0}})^{n},\quad u\in
Y,\;\|u-u_{\mathbf{0}}\|_{X}<R,$

where $R$ is the ratio of convergence and
$a_{n}=\frac{\Phi^{(n)}(u_{\mathbf{0}})}{n!}.$ The ratio of convergence is
given by $1/R=\lim_{n\to\infty}\sqrt[n]{\|a_{n}\|_{X}}.$ Hence, if $\Phi$ is
an entire function on $X$ then $\lim_{n\to\infty}\sqrt[n]{\|a_{n}\|_{X}}=0$
and one obtains

$\Phi(u)=\sum_{n=0}^{\infty}a_{n}u^{n}\qquad\mbox{and}\qquad\Phi^{(k)}(u)=\sum_{n=0}^{\infty}a_{n+k}u^{n},\quad
u\in X,\;k\in\mathbb{N}.$

In the sequel, we will use the notation $\varphi$ for the function obtained
from the entire function $\Phi$ as follows

$\displaystyle\varphi(\xi):=\sum_{n=0}^{\infty}\|a_{n}\|_{X}\;\xi^{n},\quad\xi\in\mathbb{R}.$

Since $\lim_{n\to\infty}\sqrt[n]{\|a_{n}\|_{X}}=0,$ the function $\varphi$ is
also analytic on the whole $\mathbb{R}.$ Recall, analyticity of the function
$\varphi$ implies that for every compact set $K\subset\mathbb{R}$ there exists
a constant $C>0$ such that for every $k\in\mathbb{N}$ the following bound
holds

$\displaystyle\sup_{\xi\in K}|\varphi^{(k)}(\xi)|\leq C^{k}k!.$ (7)

Note here that if $u_{\mathbf{0}}(t),\;t\in[0,T]$ is an $X-$valued function of
a certain regularity with respect to $t$, e.g. if $u_{\mathbf{0}}\in
C^{1}([0,T],X),$ then by the chain rule theorem the function
$\Phi(u_{\mathbf{0}}(t))=\sum_{n=0}^{\infty}a_{n}u^{n}_{\mathbf{0}}(t),\;t\in[0,T]$
is of the same regularity.

In what follows, we use the notation $\Phi^{\lozenge}$ for a function defined
on the $X-$valued weighted Hida spaces $X\otimes(FS)^{\prime},$ generated from
the entire function $\Phi$ on $X,$ in the following way

$\Phi^{\lozenge}(u)=\sum_{n=0}^{\infty}a_{n}u^{\lozenge n},\quad u\in
X\otimes(FS)^{\prime},$

where $a_{n}\in X,\;n\in\mathbb{N}$ and
$\lim_{n\to\infty}\sqrt[n]{\|a_{n}\|_{X}}=0.$ Since the classical product of
generalized random variables is not well defined in $X\otimes(FS)^{\prime},$
here we interchange the classical product by the Wick product defined in the
previous Subsection 1.1. From Corollary 1.1 one can infer that if $u\in
X\otimes(FS)^{\prime}$ then $\Phi^{\lozenge}(u),$ expressed in this way, is a
well defined element of $X\otimes(FS)^{\prime}.$

##  2\. Stochastic nonlinear evolution equations

We consider the equation (1), i.e. the equation of the form:

$\displaystyle u_{t}(t,\omega)$
$\displaystyle=\mathbf{A}\,u(t,\omega)+\sum_{n=0}^{\infty}a_{n}u^{\lozenge
n}(t,\omega)+f(t,\omega),\quad t\in(0,T],\quad\omega\in\Omega$ (8)
$\displaystyle u(0,\omega)$
$\displaystyle=u^{0}(\omega),\quad\omega\in\Omega.$

We are looking for a solution to (8) as an $X$-valued stochastic process
$u(t)\in X\otimes(FS)^{\prime},\;t\in[0,T]$ represented in the form

$u(t,\omega)=\sum_{\alpha\in\mathcal{I}}u_{\alpha}(t)\,\,H_{\alpha}(\omega),\quad
t\in[0,T],\quad\omega\in\Omega.$ (9)

Let $\mathbf{A}:\mathbb{D}\subset X\otimes(FS)^{\prime}\to
X\otimes(FS)^{\prime}$ be a coordinatewise operator that corresponds to a
family of deterministic operators $A_{\alpha}:\,D_{\alpha}\subset X\to X$,
$\alpha\in\mathcal{I}$

$\mathbf{A}\,u(t,\omega)=\mathbf{A}\left(\sum_{\alpha\in\mathcal{I}}u_{\alpha}(t)\,H_{\alpha}(\omega)\right)=\sum_{\alpha\in\mathcal{I}}A_{\alpha}u_{\alpha}(t)\,\,H_{\alpha}(\omega),\quad
u\in\mathbb{D},$

(see [22, Section 2]). The chaos expansion representation of the Wick-square
is given by

$\displaystyle u^{\lozenge 2}(t,\omega)$
$\displaystyle=\sum_{\alpha\in\mathcal{I}}\Big{(}\sum_{\gamma\leq\alpha}\,u_{\gamma}(t)\,\,u_{\alpha-\gamma}(t)\Big{)}\,H_{\alpha}(\omega)$
$\displaystyle=u^{2}_{\mathbf{0}}(t)\,H_{\mathbf{0}}(\omega)+\sum_{|\alpha|>0}\Big{(}2u_{\mathbf{0}}(t)\,u_{\alpha}(t)+\sum_{0<\gamma<\alpha}\,u_{\gamma}(t)\,\,u_{\alpha-\gamma}(t)\Big{)}\,H_{\alpha}(\omega),$

where $t\in[0,T],\;\omega\in\Omega.$ Let $u^{\lozenge m}_{\alpha}(t)$,
$\alpha\in\mathcal{I}$, $m\in\mathbb{N}$ denote the coefficients of the chaos
expansion of the $m$th Wick power, i.e. $u^{\lozenge
m}(t,\omega)=\sum_{\alpha\in\mathcal{I}}u^{\lozenge
m}_{\alpha}(t)H_{\alpha}(\omega)$, for $m\in\mathbb{N}$. Then, for arbitrary
$n\in\mathbb{N}$, one can infer that the $n$th Wick-power is given by

$\displaystyle u^{\lozenge n}(t,\omega)=u^{\lozenge n-1}(t,\omega)\lozenge
u(t,\omega)=\sum_{\alpha\in\mathcal{I}}\Big{(}\sum_{\gamma\leq\alpha}\,u^{\lozenge
n-1}_{\gamma}(t)\,\,u_{\alpha-\gamma}(t)\Big{)}\,H_{\alpha}(\omega)$ (10)
$\displaystyle=u^{n}_{\mathbf{0}}(t)\,H_{\mathbf{0}}(\omega)+\sum_{|\alpha|>0}\Bigg{(}\binom{n}{1}u_{\mathbf{0}}^{n-1}(t)\,u_{\alpha}(t)+\binom{n}{2}u_{\mathbf{0}}^{n-2}(t)\sum_{0<\gamma_{1}<\alpha}\,u_{\alpha-\gamma_{1}}(t)\,\,u_{\gamma_{1}}(t)$
$\displaystyle+\binom{n}{3}u_{\mathbf{0}}^{n-3}(t)\sum_{0<\gamma_{1}<\alpha}\sum_{0<\gamma_{2}<\gamma_{1}}\,u_{\alpha-\gamma_{1}}(t)\,\,u_{\gamma_{1}-\gamma_{2}}(t)u_{\gamma_{2}}(t)+\dots+$
$\displaystyle+\binom{n}{n}\sum_{0<\gamma_{1}<\alpha}\sum_{0<\gamma_{2}<\gamma_{1}}\dots\sum_{0<\gamma_{n-1}<\gamma_{n-2}}u_{\alpha-\gamma_{1}}(t)\,u_{\gamma_{1}-\gamma_{2}}(t)\dots
u_{\gamma_{n-2}-\gamma_{n-1}}(t)u_{\gamma_{n-1}}(t)\Bigg{)}H_{\alpha}(\omega),$

where $t\in[0,T],\;\omega\in\Omega.$ Moreover, we recall that the Wick power
$u^{\lozenge n}$ of a stochastic process $u\in X\otimes(FS)_{-r,-p}$ is an
element of $X\otimes(FS)_{-r,-q}$, for $q>p+1$, see Corollary 1.1.

Now, the nonlinear part of (8) is of the form

$\displaystyle\sum_{n=0}^{\infty}$ $\displaystyle a_{n}u^{\lozenge
n}(t,\omega)=a_{0}+a_{1}u(t,\omega)+a_{2}u^{\lozenge
2}(t,\omega)+a_{3}u^{\lozenge 3}(t,\omega)+\cdots+a_{n}u^{\lozenge
n}(t,\omega)+\cdots$

which, after using (10), becomes

$\displaystyle=a_{0}+a_{1}\Big{[}u_{\mathbf{0}}(t)\,H_{\mathbf{0}}(\omega)+\sum_{|\alpha|>0}u_{\alpha}(t)\,H_{\alpha}(\omega)\Big{]}$
$\displaystyle+a_{2}\,\Big{[}u^{2}_{\mathbf{0}}(t)\,H_{\mathbf{0}}(\omega)+\sum_{|\alpha|>0}\Big{(}2u_{\mathbf{0}}(t)\,u_{\alpha}(t)+\sum_{0<\gamma<\alpha}\,u_{\gamma}(t)\,u_{\alpha-\gamma}(t)\Big{)}\,H_{\alpha}(\omega)\Big{]}$
$\displaystyle+a_{3}\,\Bigg{[}u^{3}_{\mathbf{0}}(t)\,H_{\mathbf{0}}(\omega)+\sum_{|\alpha|>0}\Bigg{(}3u_{\mathbf{0}}^{2}(t)\,u_{\alpha}(t)+3u_{\mathbf{0}}(t)\sum_{0<\gamma_{1}<\alpha}\,u_{\alpha-\gamma_{1}}(t)\,u_{\gamma_{1}}(t)$
$\displaystyle\hskip
113.81102pt+\sum_{0<\gamma_{1}<\alpha}\sum_{0<\gamma_{2}<\gamma_{1}}\,u_{\alpha-\gamma_{1}}(t)\,u_{\gamma_{1}-\gamma_{2}}(t)u_{\gamma_{2}}(t)\Bigg{)}\,H_{\alpha}(\omega)\Bigg{]}$
$\displaystyle+a_{4}\,\Bigg{[}u^{4}_{\mathbf{0}}(t)\,H_{\mathbf{0}}(\omega)+\sum_{|\alpha|>0}\Bigg{(}4u_{\mathbf{0}}^{3}(t)\,u_{\alpha}(t)+6u_{\mathbf{0}}^{2}(t)\sum_{0<\gamma_{1}<\alpha}\,u_{\alpha-\gamma_{1}}(t)\,u_{\gamma_{1}}(t)$
$\displaystyle\hskip
93.89418pt+4u_{\mathbf{0}}(t)\sum_{0<\gamma_{1}<\alpha}\sum_{0<\gamma_{2}<\gamma_{1}}\,u_{\alpha-\gamma_{1}}(t)\,u_{\gamma_{1}-\gamma_{2}}(t)u_{\gamma_{2}}(t)$
$\displaystyle\hskip
93.89418pt+\sum_{0<\gamma_{1}<\alpha}\sum_{0<\gamma_{2}<\gamma_{1}}\sum_{0<\gamma_{3}<\gamma_{2}}\,u_{\alpha-\gamma_{1}}(t)\,u_{\gamma_{1}-\gamma_{2}}(t)u_{\gamma_{2}-\gamma_{3}}(t)u_{\gamma_{3}}(t)\Bigg{)}\,H_{\alpha}(\omega)\Bigg{]}$
$\displaystyle+\cdots+$
$\displaystyle+a_{n}\,\Bigg{[}u^{n}_{\mathbf{0}}(t)\,H_{\mathbf{0}}(\omega)+\sum_{|\alpha|>0}\Bigg{(}\binom{n}{1}u_{\mathbf{0}}^{n-1}(t)\,u_{\alpha}(t)+\binom{n}{2}u_{\mathbf{0}}^{n-2}(t)\sum_{0<\gamma_{1}<\alpha}\,u_{\alpha-\gamma_{1}}(t)\,u_{\gamma_{1}}(t)$
$\displaystyle\hskip
113.81102pt+\binom{n}{3}u_{\mathbf{0}}^{n-3}(t)\sum_{0<\gamma_{1}<\alpha}\sum_{0<\gamma_{2}<\gamma_{1}}\,u_{\alpha-\gamma_{1}}(t)\,u_{\gamma_{1}-\gamma_{2}}(t)u_{\gamma_{2}}(t)+\dots+$
$\displaystyle\hskip
113.81102pt+\binom{n}{n}\sum_{0<\gamma_{1}<\alpha}\sum_{0<\gamma_{2}<\gamma_{1}}\dots\sum_{0<\gamma_{n-1}<\gamma_{n-2}}\,$
$\displaystyle\hskip
165.02606ptu_{\alpha-\gamma_{1}}(t)\,u_{\gamma_{1}-\gamma_{2}}(t)\dots
u_{\gamma_{n-2}-\gamma_{n-1}}(t)u_{\gamma_{n-1}}(t)\Bigg{)}\,H_{\alpha}(\omega)\Bigg{]}$
$\displaystyle+\cdots$
$\displaystyle=\Phi(u_{\mathbf{0}}(t))\,H_{\mathbf{0}}(\omega)+\sum_{|\alpha|>0}\Bigg{(}\Phi^{\prime}(u_{\mathbf{0}}(t))\,u_{\alpha}(t)+\frac{\Phi^{\prime\prime}(u_{\mathbf{0}}(t))}{2!}\sum_{0<\gamma_{1}<\alpha}\,u_{\alpha-\gamma_{1}}(t)\,u_{\gamma_{1}}(t)$
$\displaystyle\hskip
113.81102pt+\frac{\Phi^{(3)}(u_{\mathbf{0}}(t))}{3!}\sum_{0<\gamma_{1}<\alpha}\sum_{0<\gamma_{2}<\gamma_{1}}\,u_{\alpha-\gamma_{1}}(t)\,u_{\gamma_{1}-\gamma_{2}}(t)u_{\gamma_{2}}(t)+\cdots$
$\displaystyle\hskip
113.81102pt+\frac{\Phi^{(|\alpha|)}(u_{\mathbf{0}}(t))}{|\alpha|!}\sum_{0<\gamma_{1}<\alpha}\sum_{0<\gamma_{2}<\gamma_{1}}\dots\sum_{0<\gamma_{|\alpha|-1}<\gamma_{|\alpha|-2}}\,$
$\displaystyle\hskip
156.49014ptu_{\alpha-\gamma_{1}}(t)u_{\gamma_{1}-\gamma_{2}}(t)\dots
u_{\gamma_{|\alpha|-2}-\gamma_{|\alpha|-1}}(t)u_{\gamma_{|\alpha|-1}}(t)\Bigg{)}H_{\alpha}(\omega).$

Finally,

$\displaystyle\sum_{n=0}^{\infty}$ $\displaystyle a_{n}u^{\lozenge
n}(t,\omega)=\Phi\big{(}u_{\mathbf{0}}(t)\big{)}\,H_{\mathbf{0}}(\omega)+\sum_{|\alpha|>0}\Bigg{(}\Phi^{\prime}\big{(}u_{\mathbf{0}}(t)\big{)}\,u_{\alpha}(t)+r_{\alpha,\Phi}(t)\Bigg{)}\,H_{\alpha}(\omega)$

where $t\in[0,T],\;\omega\in\Omega.$ For later use, let us emphasize here that
for $\alpha\in\mathcal{I},\;|\alpha|=1$ the functions
$r_{\alpha,\Phi}(t)=0,\;t\in[0,T]$ and for
$\alpha\in\mathcal{I},\;|\alpha|\geq 2$ the functions $r_{\alpha,\Phi}$ are of
the form

$\displaystyle
r_{\alpha,\Phi}(t)=\sum_{k=2}^{|\alpha|}\frac{\Phi^{(k)}(u_{\mathbf{0}}(t))}{k!}\sum_{\mathbf{0}<\gamma_{1}<\alpha}\sum_{\mathbf{0}<\gamma_{2}<\gamma_{1}}$
$\displaystyle\dots\sum_{\mathbf{0}<\gamma_{k-1}<\gamma_{k-2}}$ (11)
$\displaystyle u_{\alpha-\gamma_{1}}(t)u_{\gamma_{1}-\gamma_{2}}(t)\dots
u_{\gamma_{k-2}-\gamma_{k-1}}(t)u_{\gamma_{k-1}}(t),$

$t\in[0,T],$ so they contain only the coordinate functions
$u_{\beta},\;\beta<\alpha.$

We rewrite all processes that figure in (8) in their corresponding Wiener-Itô
chaos expansion form and obtain

$\begin{split}\sum_{\alpha\in\mathcal{I}}\,\frac{d}{dt}u_{\alpha}(t)\,H_{\alpha}(\omega)&=\Big{(}A_{\mathbf{0}}u_{\mathbf{0}}(t)+\Phi\big{(}u_{\mathbf{0}}(t)\big{)}+f_{\mathbf{0}}(t)\Big{)}\,H_{\mathbf{0}}(\omega)\\\
&+\sum_{|\alpha|>0}\Bigg{(}A_{\alpha}\,u_{\alpha}(t)+\Phi^{\prime}\big{(}u_{\mathbf{0}}(t)\big{)}\,u_{\alpha}(t)+r_{\alpha,\Phi}(t)+f_{\alpha}(t)\Bigg{)}\,H_{\alpha}(\omega)\\\
\sum_{\alpha\in\mathcal{I}}u_{\alpha}(0)\
H_{\alpha}(\omega)&=\sum_{\alpha\in\mathcal{I}}u^{0}_{\alpha}\
H_{\alpha}(\omega).\end{split}$

Due to the orthogonality of the basis
$\\{H_{\alpha}\\}_{\alpha\in\mathcal{I}}$ this reduces to the system of
infinitely many deterministic Cauchy problems:

  1. $1^{\circ}$

for $\alpha=\mathbf{0}$

$\frac{d}{dt}u_{\mathbf{0}}(t)=A_{\mathbf{0}}u_{\mathbf{0}}(t)+\Phi\big{(}u_{\mathbf{0}}(t)\big{)}+f_{\mathbf{0}}(t),\quad
u_{\mathbf{0}}(0)=u_{\mathbf{0}}^{0},\qquad\text{and}$ (12)

  2. $2^{\circ}$

for $\alpha>\mathbf{0}$

$\frac{d}{dt}u_{\alpha}(t)=\big{(}A_{\alpha}+\Phi^{\prime}\big{(}u_{\mathbf{0}}(t)\big{)}\,Id\big{)}\,u_{\alpha}(t)+r_{\alpha,\Phi}(t)+f_{\alpha}(t),\quad
u_{\alpha}(0)=u_{\alpha}^{0},$ (13)

with $t\in(0,T]$ and $\omega\in\Omega$.

Let

$B_{\alpha,\Phi}(t)=A_{\alpha}+\Phi^{\prime}\big{(}u_{\mathbf{0}}(t)\big{)}\,Id\qquad\text{and}\qquad
g_{\alpha,\Phi}(t)=r_{\alpha,\Phi}(t)+f_{\alpha}(t),\quad t\in(0,T]$

for all $\alpha>\mathbf{0}$. Then, the system (13) can be written in the form

$\frac{d}{dt}u_{\alpha}(t)=B_{\alpha,\Phi}(t)\,u_{\alpha}(t)+g_{\alpha,\Phi}(t),\quad
t\in(0,T];\qquad u_{\alpha}(0)=u_{\alpha}^{0},\quad\alpha>\mathbf{0}.$ (14)

Note that the inhomogeneous part $g_{\alpha,\Phi}$ in (14) does not contain
any of the functions $u_{\beta},\;\beta<\alpha$ for $|\alpha|=1$, while for
$|\alpha|>1$ it involves also $u_{\beta},\;\beta<\alpha$. Hence, we
distinguish these two cases.

  1. (a)

Let $|\alpha|=1$, i.e. $\alpha=\varepsilon_{k}$, $k\in\mathbb{N}.$ Then
$g_{\varepsilon_{k},\Phi}=f_{\varepsilon_{k}}$, $k\in\mathbb{N}$ and thus (14)
transforms to

$\frac{d}{dt}u_{\varepsilon_{k}}(t)=B_{\varepsilon_{k},\Phi}(t)\,u_{\varepsilon_{k}}(t)+f_{\varepsilon_{k}}(t),\quad
t\in(0,T];\qquad u_{\varepsilon_{k}}(0)=u_{\varepsilon_{k}}^{0}.$ (15)

  2. (b)

Let $|\alpha|>1.$ Then

$\frac{d}{dt}u_{\alpha}(t)=B_{\alpha,\Phi}(t)\,u_{\alpha}(t)+g_{\alpha,\Phi}(t),\quad
t\in(0,T];\qquad u_{\alpha}(0)=u_{\alpha}^{0}.$

Each solution $u$ to (8) can be represented in the form (9) and hence its
coefficients $u_{\mathbf{0}}$ and $u_{\alpha}$ for $\alpha>\mathbf{0}$ must
satisfy (12) and (14) respectively. Vice versa, if the coefficients
$u_{\mathbf{0}}$ and $u_{\alpha}$ for $\alpha>\mathbf{0}$ solve (12) and (14)
respectively, and if for every $t\in[0,T]$ the series in (9) represented by
these coefficients exists in $X\otimes(FS)^{\prime}$, then it defines a
solution to (8).

######  Definition 2.1.

An $X-$valued generalized stochastic process
$u(t)=\sum_{\alpha\in\mathcal{I}}u_{\alpha}(t)H_{\alpha}\in
X\otimes(FS)^{\prime},\;t\in[0,T]$ is a coordinatewise classical solution to
(8) if $u_{\mathbf{0}}$ is a classical solution to (12) and for every
$\alpha\in\mathcal{I}\setminus\\{\mathbf{0}\\},$ the coefficient $u_{\alpha}$
is a classical solution to (14). The coordinatewise solution $u(t)\in
X\otimes(FS)^{\prime},\;t\in[0,T]$ is an almost classical solution to (8) if
$u\in C([0,T],X\otimes(FS)^{\prime}).$ An almost classical solution is a
classical solution if $u\in C([0,T],X\otimes(FS)^{\prime})\cap
C^{1}((0,T],X\otimes(FS)^{\prime})$.

We assume that the following hold:

  1. (A1)

The operators $A_{\alpha},\;\alpha\in\mathcal{I}$ are infinitesimal generators
of $C_{0}-$semigroups $\\{T_{\alpha}(s)\\}_{s\geq 0}$ with a common domain
$D_{\alpha}=D,\;\alpha\in\mathcal{I}$ dense in $X$. We assume that there exist
constants $m\geq 1$ and $w\in\mathbb{R}$ such that

$\|T_{\alpha}(s)\|\leq me^{ws},\;s\geq 0\quad\mbox{for
all}\quad\alpha\in\mathcal{I}.$

The action of $\mathbf{A}$ is given by

$\mathbf{A}(u)=\sum_{\alpha\in\mathcal{I}}A_{\alpha}(u_{\alpha})H_{\alpha},$

for $u\in\mathbb{D}\subseteq D\otimes(FS)^{\prime}$ of the form (9), where

$\displaystyle\mathbb{D}=\Big{\\{}u=\sum_{\alpha\in\mathcal{I}}u_{\alpha}\,\,H_{\alpha}$
$\displaystyle\in D\otimes(FS)^{\prime}:$ $\displaystyle\exists r\geq
2,\;\exists p\geq
0,\;\sum_{\alpha\in\mathcal{I}}\|A_{\alpha}(u_{\alpha})\|^{2}_{X}\alpha!(r^{|\alpha|^{3}}!)^{-1}(2\mathbb{N})^{-p\alpha}<\infty\Big{\\}}.$

  2. (A2)

The initial value
$u^{0}=\sum_{\alpha\in\mathcal{I}}u^{0}_{\alpha}H_{\alpha}\in\mathbb{D}$, i.e.
$u_{\alpha}^{0}\in D$ for every $\alpha\in\mathcal{I}$ and there exist $r\geq
2$ and $p\geq 0$ such that

$\displaystyle\|u^{0}\|^{2}_{X\otimes(FS)_{-r,-p}}$
$\displaystyle=\sum_{\alpha\in\mathcal{I}}\|u_{\alpha}^{0}\|_{X}^{2}\alpha!(r^{|\alpha|^{3}}!)^{-1}(2\mathbb{N})^{-p\alpha}<\infty,$
$\displaystyle\|\mathbf{A}u^{0}\|^{2}_{X\otimes(FS)_{-r,-p}}$
$\displaystyle=\sum_{\alpha\in\mathcal{I}}\|A_{\alpha}(u_{\alpha}^{0})\|_{X}^{2}\alpha!(r^{|\alpha|^{3}}!)^{-1}(2\mathbb{N})^{-p\alpha}<\infty.$

  3. (A3)

The inhomogeneous part
$f(t,\omega)=\sum_{\alpha\in\mathcal{I}}f_{\alpha}(t)H_{\alpha}(\omega),\;t\in[0,T],\;\omega\in\Omega$
belongs to $C^{1}([0,T],X)\otimes(FS)^{\prime};$ hence $t\mapsto
f_{\alpha}(t)\in C^{1}([0,T],X),\;\alpha\in\mathcal{I}$ and there exist $r\geq
2$ and $p\geq 0$ such that

$\displaystyle\|f\|$
${}^{2}_{C^{1}([0,T],X)\otimes(FS)_{-r,-p}}=\sum_{\alpha\in\mathcal{I}}\|f_{\alpha}\|^{2}_{C^{1}([0,T],X)}\alpha!(r^{|\alpha|^{3}}!)^{-1}(2\mathbb{N})^{-p\alpha}$
$\displaystyle=\sum_{\alpha\in\mathcal{I}}\Big{(}\sup_{t\in[0,T]}\|f_{\alpha}(t)\|_{X}+\sup_{t\in[0,T]}\|f^{\prime}_{\alpha}(t)\|_{X}\Big{)}^{2}\alpha!(r^{|\alpha|^{3}}!)^{-1}(2\mathbb{N})^{-p\alpha}<\infty.$

  4. (A4)

The Cauchy problem

$\frac{d}{dt}u_{\mathbf{0}}(t)=A_{\mathbf{0}}u_{\mathbf{0}}(t)+\Phi\big{(}u_{\mathbf{0}}(t)\big{)}+f_{\mathbf{0}}(t),\quad
t\in(0,T];\quad u_{\mathbf{0}}(0)=u_{\mathbf{0}}^{0},$

has a unique classical solution $u_{\mathbf{0}}\in C^{1}([0,T],X).$

Now we focus on solving (14) for $\alpha>\mathbf{0}$.

######  Lemma 2.1.

Let the assumptions (A1)-(A4) be fulfilled. Then for every $\alpha>\mathbf{0}$
the evolution system (14) has a unique classical solution $u_{\alpha}\in
C^{1}([0,T],X).$

###### Proof.

First, for every $\alpha>\mathbf{0},$ we consider the family of operators
$B_{\alpha,\Phi}(t)=A_{\alpha}+\Phi^{\prime}\big{(}u_{\mathbf{0}}(t)\big{)}Id,\;t\in[0,T].$
According to assumption (A1), the constant family
$\\{A_{\alpha}(t)\\}_{t\in[0,T]}=\\{A_{\alpha}\\}_{t\in[0,T]}$ is a stable
family of infinitesimal generators of a $C_{0}-$semigroup
$\\{T_{\alpha}(s)\\}_{s\geq 0}$ on $X$ satisfying $\|T_{\alpha}(s)\|\leq
me^{ws}$ with stability constants $m\geq 1$ and $w\in\mathbb{R}.$ Let

$M_{\Phi}=\sup_{t\in[0,T]}\|u_{\mathbf{0}}(t)\|_{X}.$ (16)

The perturbation $\Phi^{\prime}\big{(}u_{\mathbf{0}}(t)\big{)}\,Id:X\to
X,\;t\in[0,T]$ is a family of uniformly bounded linear operators such that

$\displaystyle\|\Phi^{\prime}\big{(}u_{\mathbf{0}}(t)\big{)}x\|_{X}$
$\displaystyle=\|\Phi^{\prime}\big{(}u_{\mathbf{0}}(t)\big{)}\|_{X}\|x\|_{X}=\left\|\sum_{n=0}^{\infty}a_{n+1}u^{n}_{\mathbf{0}}(t)\right\|_{X}\|x\|_{X}$
$\displaystyle\leq\left(\sum_{n=0}^{\infty}\|a_{n+1}\|_{X}\|u_{\mathbf{0}}(t)\|^{n}_{X}\right)\|x\|_{X}\leq\left(\sum_{n=0}^{\infty}\|a_{n+1}\|_{X}M^{n}_{\Phi}\right)\|x\|_{X}$
$\displaystyle\leq\varphi^{\prime}(M_{\Phi})\|x\|_{X},\quad x\in X,\quad
t\in[0,T],$

i.e.
$\|\Phi^{\prime}\big{(}u_{\mathbf{0}}(t)\big{)}Id\|\leq\varphi^{\prime}(M_{\Phi}),\;t\in[0,T].$
Here we have used the notation introduced in Subsection 1.3. Thus, for every
$\alpha>\mathbf{0},$ the family
$\\{A_{\alpha}+\Phi^{\prime}\big{(}u_{\mathbf{0}}(t)\big{)}Id\\}_{t\in[0,T]}$
is a stable family of infinitesimal generators with stability constants $m$
and $w+\varphi^{\prime}(M_{\Phi})m.$ By assumption (A4) the function
$u_{\mathbf{0}}\in C^{1}([0,T],X)$ so we obtain continuous differentiability
of $(A_{\alpha}+\Phi^{\prime}\big{(}u_{\mathbf{0}}(t)\big{)}Id)x,\;t\in[0,T]$
for every $x\in D$ and for every $\alpha>\mathbf{0}.$ Additionally, the domain
of the operators $\Phi^{\prime}\big{(}u_{\mathbf{0}}(t)\big{)}Id$ is the
entire space $X$ which implies that all of the operators
$B_{\alpha,\Phi}(t),\;t\in[0,T]$ have a common domain
$D(B_{\alpha,\Phi}(t))=D(A_{\alpha})=D$ not depending on $t.$ Notice here that
assumption (A1) additionally provides the same domain $D$ of the family
$\\{B_{\alpha,\Phi}(t)\\}_{t\in[0,T]}$ for all $\alpha>\mathbf{0}.$

Finally, one can associate the unique evolution system $S_{\alpha,\Phi}(t,s)$,
for $0\leq s\leq t\leq T$ for all $\alpha>\mathbf{0}$ to the system (14) such
that

$\|S_{\alpha,\Phi}(t,s)\|\leq me^{w_{\Phi}\,(t-s)}\leq
me^{w_{\Phi}(T-s)},\quad 0\leq s\leq t\leq T,\quad\alpha>\mathbf{0},$ (17)

where $w_{\Phi}=w+\varphi^{\prime}(M_{\Phi})m$, see [30, Thm 4.8., p. 145].
Without loss of generality we may assume that $w>0$ and thus will be
$w_{\Phi}>0$.

Now one can solve the infinite system of the Cauchy problems (14) by induction
on the length of the multiindex $\alpha$. Let $|\alpha|=1.$ Since
$f_{\varepsilon_{k}}\in C^{1}([0,T],X),$ we obtain the unique classical
solution $u_{\varepsilon_{k}}\in C^{1}([0,T],X)$ to (15) given by

$u_{\varepsilon_{k}}(t)=S_{\varepsilon_{k},\Phi}(t,0)\,u_{\varepsilon_{k}}^{0}+\int_{0}^{t}\,S_{\varepsilon_{k},\Phi}(t,s)\,f_{\varepsilon_{k}}(s)\,ds,\quad
t\in[0,T].$ (18)

Now let for every $\beta\in\mathcal{I}$ such that $\mathbf{0}<\beta<\alpha$
the unique classical solution to (14) satisfies $u_{\beta}\in C^{1}([0,T],X).$
Then for fixed $|\alpha|>1$ the inhomogeneous part $g_{\alpha,\Phi}\in
C^{1}([0,T],X)$ and the solution to (14) is of the form

$u_{\alpha}(t)=S_{\alpha,\Phi}(t,0)\,u_{\alpha}^{0}+\int_{0}^{t}\,S_{\alpha,\Phi}(t,s)\,g_{\alpha,\Phi}(s)\,ds,\quad
t\in[0,T],$ (19)

where $u_{\alpha}\in C^{1}([0,T],X).$ For more details see [30, Thm 5.3., p.
147]. ∎

We proceed with one technical lemma.

######  Lemma 2.2.

Let $\alpha\in\mathcal{I}\setminus\\{\mathbf{0}\\}$ be given and
$k\in\mathbb{N},\;k\leq|\alpha|.$ Denote by $N(\alpha,k)$ the number of all
possible combinations in which a multiindex $\alpha$ can be written as a sum
of $k$ strictly smaller and nonzero multiindices. Then

$\displaystyle N(\alpha,k)\leq
2^{k|\alpha|},\qquad\alpha\in\mathcal{I}\setminus\\{\mathbf{0}\\},\quad
k\in\mathbb{N},\;k\leq|\alpha|.$

###### Proof.

Here we will use a known combinatorial result about the number of partitions
of a positive integer $n$ into $k$ parts. Namely, a positive integer
$n\in\mathbb{N}$ can be written as a sum of $k$ integers, i.e. in the form
$n=x_{1}+\dots+x_{k},\;x_{i}\in\\{0,1,\dots,n\\},$ in
$\binom{n+k-1}{n}=\binom{n+k-1}{k-1}$ different ways.

Let $\alpha\in\mathcal{I}\setminus\\{\mathbf{0}\\}$ be written as a sum of $k$
multiindices
$\gamma^{1},\dots,\gamma^{k}\in\mathcal{I}\setminus\\{\mathbf{0}\\}.$ Then

$(\alpha_{1},\alpha_{2},\dots,\alpha_{s},0,0,\dots)=(\gamma^{1}_{1},\gamma^{1}_{2},\dots,\gamma^{1}_{s},0,0,\dots)+\dots+(\gamma^{k}_{1},\gamma^{k}_{2},\dots,\gamma^{k}_{s},0,0,\dots),$

i.e. each component of $\alpha$ can be decomposed

$\displaystyle\alpha_{1}=\gamma^{1}_{1}+\dots+\gamma^{k}_{1},\quad\gamma^{i}_{1}\in\\{0,\dots,\alpha_{1}\\},\;i\in\\{1,\dots,k\\}\quad\mbox{in}\quad\binom{\alpha_{1}+k-1}{k-1}\quad\mbox{different
ways;}$
$\displaystyle\alpha_{2}=\gamma^{1}_{2}+\dots+\gamma^{k}_{2},\quad\gamma^{i}_{2}\in\\{0,\dots,\alpha_{2}\\},\;i\in\\{1,\dots,k\\}\quad\mbox{in}\quad\binom{\alpha_{2}+k-1}{k-1}\quad\mbox{different
ways;}$ $\displaystyle...$
$\displaystyle\alpha_{s}=\gamma^{1}_{s}+\dots+\gamma^{k}_{s},\quad\gamma^{i}_{s}\in\\{0,\dots,\alpha_{s}\\},\;i\in\\{1,\dots,k\\}\quad\mbox{in}\quad\binom{\alpha_{s}+k-1}{k-1}\quad\mbox{different
ways.}$

Hence, $N(\alpha,k)\leq\prod_{j=1}^{s}\binom{\alpha_{j}+k-1}{k-1},$ since by
this product we also counted for partitions that include the $\mathbf{0}$
multiindex. Finally, we obtain

$\displaystyle
N(\alpha,k)\leq\prod_{j=1}^{s}\binom{\alpha_{j}+k-1}{k-1}\leq\prod_{j=1}^{s}2^{\alpha_{j}+k-1}=2^{s(k-1)}\prod_{j=1}^{s}2^{\alpha_{j}}=2^{s(k-1)}2^{|\alpha|}\leq
2^{k|\alpha|},$

because $s\leq|\alpha|,$ since $s$ is actually the number of nonzero
coordinates of $\alpha\in\mathcal{I}\setminus\\{\mathbf{0}\\}.$ ∎

###  2.1. Proof of the main theorem

The statement of the main theorem is as follows.

######  Theorem 2.1.

Let the assumptions $(A1)-(A4)$ be fulfilled. Then there exists a unique
almost classical solution $u\in C([0,T],X\otimes(FS)^{\prime})$ to (8).

###### Proof.

According to Lemma 2.1 for every $\alpha>\mathbf{0}$ the evolution equation
(14) has a unique classical solution $u_{\alpha}\in C^{1}([0,T],X).$ Thus, the
formal sum
$u(t,\omega)=\sum_{\alpha\in\mathcal{I}}u_{\alpha}(t)H_{\alpha}(\omega),$
$t\in[0,T],$ $\omega\in\Omega$ has coefficients that are all unique classical
solutions to the corresponding deterministic equation (14). We have to prove
that this formal sum is convergent in the sense of Definition 1.2. Note here
that this solution will be unique due to the uniqueness of the coordinatewise
(classical) solutions $u_{\alpha}$, $\alpha\in\mathcal{I}$ given by (18) and
(19) and due to the uniqueness in the Wiener–Itô chaos expansion.
Additionally, once we show that this formal sum $u$ is convergent, its
continuity follows immediately since all the coefficients $u_{\alpha}$,
$\alpha\in\mathcal{I}$ are continuous, see (5).

Now we prove that $u\in C([0,T],X\otimes(FS)^{\prime})$. Let $u^{0}\in
X\otimes(FS)^{\prime}$ be an initial condition satisfying assumption (A2)
which states that there exist $\tilde{r}\geq 2,$ $\tilde{p}\geq 0$ and
$\tilde{K}>0$ such that
$\sum_{\alpha\in\mathcal{I}}\|u^{0}_{\alpha}\|_{X}^{2}\alpha!(\tilde{r}^{|\alpha|^{3}}!)^{-1}(2\mathbb{N})^{-\tilde{p}\alpha}=\tilde{K}.$
Then there also exist $r\geq 2,$ $p\geq 0$ and $K>0$ such that
$\sum_{\alpha\in\mathcal{I}}\|u^{0}_{\alpha}\|_{X}^{2}\alpha!^{2}(r^{|\alpha|^{3}}!)^{-2}(2\mathbb{N})^{-2p\alpha}=K^{2}$,
or equivalently

$(\exists r\geq 2)\;(\exists p\geq 0)\;(\exists
K>0)\;(\forall\alpha\in\mathcal{I})\quad\|u_{\alpha}^{0}\|_{X}\leq
K\alpha!r^{|\alpha|^{3}}!(2\mathbb{N})^{p\alpha}.$ (20)

The inhomogeneous part $f\in C^{1}([0,T],X)\otimes(FS)^{\prime}$ satisfies
assumption (A3) which states that there exist $\tilde{r}\geq 2$ and
$\tilde{p}\geq 0$ such that
$\sum_{\alpha\in\mathcal{I}}\sup_{t\in[0,T]}\|f_{\alpha}(t)\|_{X}^{2}\alpha!(\tilde{r}^{|\alpha|^{3}}!)^{-1}(2\mathbb{N})^{-\tilde{p}\alpha}<\infty.$
Then there exist $r\geq 2,\;p\geq 0$ and $K>0$ such that

$\sup_{t\in[0,T]}\|f_{\alpha}(t)\|_{X}\leq
K\alpha!r^{|\alpha|^{3}}!(2\mathbb{N})^{p\alpha},\quad\alpha\in\mathcal{I}.$
(21)

The coefficients $u_{\alpha},\;\alpha\in\mathcal{I},\;\alpha>\mathbf{0}$ of
the solution $u$ are given by (18) and (19). Denote by

$\displaystyle
L_{\alpha}:=\sup_{t\in[0,T]}\|u_{\alpha}(t)\|_{X},\quad\alpha\in\mathcal{I}.$

The proof will be divided into three steps. In the first step we will derive a
recurrent formula for $L_{\alpha},\;\alpha\in\mathcal{I}.$ Then, in the second
step, an upper bound for $L_{\alpha},\;\alpha\in\mathcal{I}$ will be provided
and finally, the third step will be devoted to proving that $u\in
C([0,T],X)\otimes(FS)^{\prime}.$

Step 1. For $\alpha=\mathbf{0}$ using (16) one obtains

$L_{\mathbf{0}}=\sup_{t\in[0,T]}\|u_{\mathbf{0}}(t)\|_{X}=M_{\Phi},$

since the solution to (12) satisfies assumption (A4).

Let $|\alpha|=1.$ Then $\alpha=\varepsilon_{k},\;k\in\mathbb{N}$ and using
(18) we have that

$\|u_{\varepsilon_{k}}(t)\|_{X}\leq\|S_{\varepsilon_{k},\Phi}(t,0)\|\|u_{\varepsilon_{k}}^{0}\|_{X}+\int_{0}^{t}\|S_{\varepsilon_{k},\Phi}(t,s)\|\|f_{\varepsilon_{k}}(s)\|_{X}ds,\quad
t\in[0,T].$

From (17) we obtain that

$\displaystyle\int_{0}^{t}\|S_{\alpha,\Phi}(t,s)\|ds\leq\int_{0}^{t}me^{w_{\Phi}(t-s)}ds=m\frac{e^{w_{\Phi}t}-1}{w_{\Phi}}\leq\frac{m}{w_{\Phi}}e^{w_{\Phi}T},\quad
t\in[0,T],\quad\alpha>\mathbf{0}$ (22)

and now (17), (20) and (21) imply that

$\displaystyle L_{\varepsilon_{k}}$
$\displaystyle=\sup_{t\in[0,T]}\|u_{\varepsilon_{k}}(t)\|_{X}\leq\sup_{t\in[0,T]}\Big{\\{}\|S_{\varepsilon_{k},\Phi}(t,0)\|\|u_{\varepsilon_{k}}^{0}\|_{X}+\sup_{s\in[0,t]}\|f_{\varepsilon_{k}}(s)\|_{X}\int_{0}^{t}\|S_{\alpha,\Phi}(t,s)\|ds\Big{\\}}$
$\displaystyle\leq
me^{w_{\Phi}T}K\varepsilon_{k}!r^{|\varepsilon_{k}|^{3}}!(2\mathbb{N})^{p\varepsilon_{k}}+\frac{m}{w_{\Phi}}e^{w_{\Phi}T}K\varepsilon_{k}!r^{|\varepsilon_{k}|^{3}}!(2\mathbb{N})^{p\varepsilon_{k}}=\widetilde{m}K\varepsilon_{k}!r^{|\varepsilon_{k}|^{3}}!(2\mathbb{N})^{p\varepsilon_{k}},$
(23)

for all $k\in\mathbb{N},$ where
$\widetilde{m}=(m+\frac{m}{w_{\Phi}})e^{w_{\Phi}T}.$

Now let $\alpha\in\mathcal{I}$ be such that $|\alpha|\geq 2.$ Then from (19)
one obtains

$u_{\alpha}(t)=S_{\alpha,\Phi}(t,0)u^{0}_{\alpha}+\int_{0}^{t}S_{\alpha,\Phi}(t,s)\big{[}r_{\alpha,\Phi}(s)+f_{\alpha}(s)\big{]}ds,\quad
t\in[0,T].$

From this we have

$\displaystyle L_{\alpha}$
$\displaystyle=\sup_{t\in[0,T]}\|u_{\alpha}(t)\|_{X}$
$\displaystyle\leq\sup_{t\in[0,T]}\Bigg{\\{}\|S_{\alpha,\Phi}(t,0)\|\|u^{0}_{\alpha}\|_{X}+\int_{0}^{t}\|S_{\alpha,\Phi}(t,s)\|\|r_{\alpha,\Phi}(s)\|_{X}ds$
$\displaystyle\hskip
199.16928pt+\int_{0}^{t}\|S_{\alpha,\Phi}(t,s)\|\|f_{\alpha}(s)\|_{X}ds\Bigg{\\}}$
$\displaystyle\leq\sup_{t\in[0,T]}\Bigg{\\{}me^{w_{\Phi}t}\|u^{0}_{\alpha}\|_{X}+\sup_{s\in[0,t]}\|r_{\alpha,\Phi}(s)\|_{X}\int_{0}^{t}\|S_{\alpha,\Phi}(t,s)\|ds$
$\displaystyle\hskip
199.16928pt+\sup_{s\in[0,t]}\|f_{\alpha}(s)\|_{X}\int_{0}^{t}\|S_{\alpha,\Phi}(t,s)\|ds\Bigg{\\}}.$

Using (22) we obtain

$\displaystyle L_{\alpha}$
$\displaystyle=\sup_{t\in[0,T]}\|u_{\alpha}(t)\|_{X}$ $\displaystyle\leq
me^{w_{\Phi}T}\|u^{0}_{\alpha}\|_{X}+\frac{m}{w_{\Phi}}e^{w_{\Phi}T}\sup_{t\in[0,T]}\|r_{\alpha,\Phi}(t)\|_{X}+\frac{m}{w_{\Phi}}e^{w_{\Phi}T}\sup_{t\in[0,T]}\|f_{\alpha}(t)\|_{X}$
$\displaystyle\leq\widetilde{m}K\alpha!r^{|\alpha|^{3}}!(2\mathbb{N})^{p\alpha}+\frac{m}{w_{\Phi}}e^{w_{\Phi}T}\sup_{t\in[0,T]}\|r_{\alpha,\Phi}(t)\|_{X},$

where again $\widetilde{m}=(m+\frac{m}{w_{\Phi}})e^{w_{\Phi}T}.$ Since
$\widetilde{m}\geq\frac{m}{w_{\Phi}}e^{w_{\Phi}T},$ one easily obtains

$\displaystyle
L_{\alpha}\leq\widetilde{m}\Big{(}K\alpha!r^{|\alpha|^{3}}!(2\mathbb{N})^{p\alpha}+\sup_{t\in[0,T]}\|r_{\alpha,\Phi}(t)\|_{X}\Big{)}.$

In the following estimate we will use that

$\displaystyle\sup_{t\in[0,T]}\|\Phi^{(k)}\big{(}u_{\mathbf{0}}(t)\big{)}\|_{X}$
$\displaystyle=\sup_{t\in[0,T]}\left\|\sum_{n=0}^{\infty}a_{n+k}u^{n}_{\mathbf{0}}(t)\right\|_{X}\leq\sup_{t\in[0,T]}\left(\sum_{n=0}^{\infty}\|a_{n+k}\|_{X}\|u_{\mathbf{0}}(t)\|^{n}_{X}\right)$
$\displaystyle\leq\sum_{n=0}^{\infty}\|a_{n+k}\|_{X}\left(\sup_{t\in[0,T]}\|u_{\mathbf{0}}(t)\|_{X}\right)^{n}\leq\sum_{n=0}^{\infty}\|a_{n+k}\|_{X}M^{n}_{\Phi}\leq\varphi^{(k)}(M_{\Phi}),$

see Section 1.3. Now, using (11) one obtains

$\displaystyle\sup_{t\in[0,T]}\|r_{\alpha,\Phi}(t)\|_{X}\leq$
$\displaystyle\leq\sum_{k=2}^{|\alpha|}\frac{\varphi^{(k)}(M_{\Phi})}{k!}\sum_{\mathbf{0}<\gamma_{1}<\alpha}\sum_{\mathbf{0}<\gamma_{2}<\gamma_{1}}\dots\sum_{\mathbf{0}<\gamma_{k-1}<\gamma_{k-2}}L_{\alpha-\gamma_{1}}L_{\gamma_{1}-\gamma_{2}}\dots
L_{\gamma_{k-2}-\gamma_{k-1}}L_{\gamma_{k-1}}$
$\displaystyle\leq\sum_{k=2}^{|\alpha|}\frac{\varphi^{(k)}(M_{\Phi})}{k!}\sum_{\mathbf{0}<\gamma_{1}<\alpha}\sum_{\mathbf{0}<\gamma_{2}<\gamma_{1}}\dots\sum_{\mathbf{0}<\gamma_{k-1}<\gamma_{k-2}}L_{\alpha-\gamma_{1}}L_{\gamma_{1}-\gamma_{2}}\dots
L_{\gamma_{k-2}-\gamma_{k-1}}L_{\gamma_{k-1}}.$

Finally for $|\alpha|\geq 2,\;\alpha\in\mathcal{I}$

$\displaystyle L_{\alpha}\leq\widetilde{m}$
$\displaystyle\Big{(}K\alpha!r^{|\alpha|^{3}}!(2\mathbb{N})^{p\alpha}$
$\displaystyle+\sum_{k=2}^{|\alpha|}\frac{\varphi^{(k)}(M_{\Phi})}{k!}\sum_{\mathbf{0}<\gamma_{1}<\alpha}\sum_{\mathbf{0}<\gamma_{2}<\gamma_{1}}\dots\sum_{\mathbf{0}<\gamma_{k-1}<\gamma_{k-2}}L_{\alpha-\gamma_{1}}L_{\gamma_{1}-\gamma_{2}}\dots
L_{\gamma_{k-2}-\gamma_{k-1}}L_{\gamma_{k-1}}\Big{)}.$ (24)

In the sequel, for the sake of clarity, we will provide an explicit form of
the formula (2.1) for the cases $|\alpha|=2$ and $|\alpha|=3.$ For
$\alpha\in\mathcal{I}$ such that $|\alpha|=2$ the following holds:

$\displaystyle
L_{\alpha}\leq\widetilde{m}\Big{(}K\alpha!r^{|\alpha|^{3}}!(2\mathbb{N})^{p\alpha}+\frac{\varphi^{\prime\prime}(M_{\Phi})}{2!}\sum_{\mathbf{0}<\gamma_{1}<\alpha}L_{\alpha-\gamma_{1}}L_{\gamma_{1}}\Big{)},$

which is for $\alpha=2\varepsilon_{k},\;k\in\mathbb{N}$

$\displaystyle
L_{2\varepsilon_{k}}\leq\widetilde{m}\Big{(}K\;2!\;r^{|2\varepsilon_{k}|^{3}}!(2\mathbb{N})^{p2\varepsilon_{k}}+\frac{\varphi^{\prime\prime}(M_{\Phi})}{2!}L_{\varepsilon_{k}}^{2}\Big{)}$

and for $\alpha=\varepsilon_{k}+\varepsilon_{l},\;k,l\in\mathbb{N},\;k\neq l$

$\displaystyle
L_{\varepsilon_{k}+\varepsilon_{l}}\leq\widetilde{m}\Big{(}K\;1!1!\;r^{|\varepsilon_{k}+\varepsilon_{l}|^{3}}!(2\mathbb{N})^{p(\varepsilon_{k}+\varepsilon_{l})}+\frac{\varphi^{\prime\prime}(M_{\Phi})}{2!}2L_{\varepsilon_{k}}L_{\varepsilon_{l}}\Big{)}.$

Further, for $|\alpha|=3$ one obtains

$\displaystyle L_{\alpha}\leq\widetilde{m}$
$\displaystyle\Big{(}K\alpha!r^{|\alpha|^{3}}!(2\mathbb{N})^{p\alpha}$
$\displaystyle+\frac{\varphi^{\prime\prime}(M_{\Phi})}{2!}\sum_{\mathbf{0}<\gamma_{1}<\alpha}L_{\alpha-\gamma_{1}}L_{\gamma_{1}}+\frac{\varphi^{(3)}(M_{\Phi})}{3!}\sum_{\mathbf{0}<\gamma_{1}<\alpha}\sum_{\mathbf{0}<\gamma_{2}<\gamma_{1}}L_{\alpha-\gamma_{1}}L_{\gamma_{1}-\gamma_{2}}L_{\gamma_{2}}\Big{)},$

which is for $\alpha=3\varepsilon_{k},\;k\in\mathbb{N}$

$\displaystyle
L_{3\varepsilon_{k}}\leq\widetilde{m}\Big{(}K\;3!\;r^{3^{3}}!(2\mathbb{N})^{p3\varepsilon_{k}}+\frac{\varphi^{\prime\prime}(M_{\Phi})}{2!}2L_{2\varepsilon_{k}}L_{\varepsilon_{k}}+\frac{\varphi^{(3)}(M_{\Phi})}{3!}L_{\varepsilon_{k}}^{3}\Big{)},$

for $\alpha=2\varepsilon_{k}+\varepsilon_{l},\;k,l\in\mathbb{N},\;k\neq l$

$\displaystyle L_{2\varepsilon_{k}+\varepsilon_{l}}\leq\widetilde{m}$
$\displaystyle\Big{(}K\;2!1!\;r^{3^{3}}!(2\mathbb{N})^{p(2\varepsilon_{k}+\varepsilon_{l})}$
$\displaystyle+\frac{\varphi^{\prime\prime}(M_{\Phi})}{2!}\left(2L_{2\varepsilon_{k}}L_{\varepsilon_{l}}+2L_{\varepsilon_{k}+\varepsilon_{l}}L_{\varepsilon_{k}}\right)+\frac{\varphi^{(3)}(M_{\Phi})}{3!}3L_{\varepsilon_{k}}^{2}L_{\varepsilon_{l}}\Big{)}$

and for
$\alpha=\varepsilon_{k}+\varepsilon_{l}+\varepsilon_{s},\;k,l,s\in\mathbb{N},\;k\neq
l,\;k\neq s,\;l\neq s$

$\displaystyle L_{\varepsilon_{k}+\varepsilon_{l}+\varepsilon_{s}}\leq$
$\displaystyle\widetilde{m}\Big{(}K\;1!1!1!\;r^{3^{3}}!(2\mathbb{N})^{p(\varepsilon_{k}+\varepsilon_{l}+\varepsilon_{s})}$
$\displaystyle+\frac{\varphi^{\prime\prime}(M_{\Phi})}{2!}\left(2L_{\varepsilon_{k}+\varepsilon_{l}}L_{\varepsilon_{s}}+2L_{\varepsilon_{k}+\varepsilon_{s}}L_{\varepsilon_{l}}+2L_{\varepsilon_{l}+\varepsilon_{s}}L_{\varepsilon_{k}}\right)+\frac{\varphi^{(3)}(M_{\Phi})}{3!}6L_{\varepsilon_{k}}L_{\varepsilon_{l}}L_{\varepsilon_{s}}\Big{)}.$

Step 2. This part of the proof is devoted to showing that there exist
$r_{0}\geq 2$ and $p_{0}\geq 0$ such that

$\displaystyle
L_{\alpha}\leq\alpha!(2\mathbb{N})^{p_{0}\alpha}r_{0}^{|\alpha|^{3}}!\quad\mbox{for
all}\quad\alpha\in\mathcal{I},\;|\alpha|\geq 1,$ (25)

where $p_{0}>p$ and
$r_{0}>2(r+\widetilde{m}K+\widetilde{m}\widetilde{C}),\;\widetilde{C}=(2C)^{2}$
for $C$ given in (7). The proof is derived by mathematical induction with
respect to the length of the multiindex $\alpha\in\mathcal{I},$ i.e. with
respect to $n=|\alpha|,\;n\in\mathbb{N}.$

First, for $|\alpha|=1$ from (2.1) one directly obtains

$\displaystyle L_{\alpha}$
$\displaystyle\leq\alpha!\widetilde{m}Kr^{|\alpha|^{3}}!(2\mathbb{N})^{p\alpha}\leq\alpha!(2\mathbb{N})^{p_{0}\alpha}r_{0}^{|\alpha|^{3}}!=\alpha!(2\mathbb{N})^{p_{0}\alpha}r_{0}!,$
(26)

since $p_{0}>p$ and $r_{0}>r+\widetilde{m}K.$

Next, for $|\alpha|=2,$ let $J_{2}$ be given as follows

$\displaystyle
J_{2}=\sum_{\mathbf{0}<\gamma_{1}<\alpha}L_{\alpha-\gamma_{1}}L_{\gamma_{1}}.$

The sum over $\gamma_{1}\in\mathcal{I},$ where $\mathbf{0}<\gamma_{1}<\alpha,$
has $N(\alpha,2)\leq 2^{2|\alpha|}=2^{4}$ terms, as given in Lemma 2.2. Using
this and the estimate (26) for both terms $L_{\alpha-\gamma_{1}}$ and
$L_{\gamma_{1}},$ since $|\alpha-\gamma_{1}|=|\gamma_{1}|=1,$ one obtains

$\displaystyle J_{2}$ $\displaystyle\leq
N(\alpha,2)(\alpha-\gamma_{1})!(2\mathbb{N})^{p_{0}(\alpha-\gamma_{1})}r_{0}^{|\alpha-\gamma_{1}|^{3}}!(\gamma_{1})!(2\mathbb{N})^{p_{0}\gamma_{1}}r_{0}^{|\gamma_{1}|^{3}}!\leq
2^{4}\alpha!(2\mathbb{N})^{p_{0}\alpha}r_{0}!r_{0}!$ $\displaystyle\leq
2^{4}\alpha!(2\mathbb{N})^{p_{0}\alpha}r_{0}^{2}!.$

Recall, the estimate (7) says that since $\varphi$ is analytic and
$\\{u_{0}(t),t\in[0,T]\\}$ is a compact set, then for all $k\in\mathbb{N}$

$\displaystyle\sup_{t\in[0,T]}\frac{\varphi^{(k)}(u_{0}(t))}{k!}\leq
C^{k},\quad\mbox{for some}\;C>0.$

Hence, for $|\alpha|=2$ the following holds

$\displaystyle L_{\alpha}\leq$
$\displaystyle\widetilde{m}\left(K\alpha!r^{|\alpha|^{3}}!(2\mathbb{N})^{p\alpha}+C^{2}J_{2}\right)\leq\widetilde{m}\left(K\alpha!r^{|\alpha|^{3}}!(2\mathbb{N})^{p\alpha}+C^{2}2^{4}\alpha!(2\mathbb{N})^{p_{0}\alpha}r_{0}^{2}!\right)$
$\displaystyle\leq\alpha!(2\mathbb{N})^{p_{0}\alpha}r_{0}^{|\alpha|^{3}}!\left(\frac{\widetilde{m}Kr^{|\alpha|^{3}}!}{r_{0}^{|\alpha|^{3}}!}+\frac{C^{2}2^{4}r_{0}^{2}!}{r_{0}^{|\alpha|^{3}}!}\right)\leq\alpha!(2\mathbb{N})^{p_{0}\alpha}r_{0}^{8}!\left(\frac{\widetilde{m}Kr^{8}!}{r_{0}^{8}!}+\frac{C^{2}2^{4}r_{0}^{2}!}{r_{0}^{8}!}\right),$

since $p_{0}>p.$ Finally, using that $r_{0}^{8}!=(r_{0}^{2}r_{0}^{6})!\geq
r_{0}^{2}!r_{0}^{6}!$ and
$r_{0}>2(r+\widetilde{m}K+\widetilde{m}\widetilde{C}),$ where
$\widetilde{C}=(2C)^{2},$ one obtains

$\displaystyle L_{\alpha}$
$\displaystyle\leq\alpha!(2\mathbb{N})^{p_{0}\alpha}r_{0}^{8}!\left(\frac{\widetilde{m}K}{r_{0}^{8}}+\frac{C^{2}2^{4}}{r_{0}^{6}!}\right)\leq\alpha!(2\mathbb{N})^{p_{0}\alpha}r_{0}^{8}!\left(\frac{\widetilde{m}K}{r_{0}}+\frac{C^{2}2^{4}}{r_{0}^{2}}\right)$
$\displaystyle\leq\alpha!(2\mathbb{N})^{p_{0}\alpha}r_{0}^{8}!\left(\frac{1}{2}+\frac{1}{2}\right)=\alpha!(2\mathbb{N})^{p_{0}\alpha}r_{0}^{8}!.$

Now, let us assume that the inductive hypothesis (25) is true for all
$\alpha\in\mathcal{I}\setminus\\{\mathbf{0}\\}$ such that $|\alpha|\leq n.$ It
remains to prove that then it also holds for $|\alpha|=n+1.$

We start with (2.1) for $|\alpha|\geq 2,\;\alpha\in\mathcal{I}$:

$\displaystyle L_{\alpha}\leq\widetilde{m}$
$\displaystyle\Big{(}K\alpha!r^{|\alpha|^{3}}!(2\mathbb{N})^{p\alpha}$
$\displaystyle+\sum_{k=2}^{|\alpha|}\frac{\varphi^{(k)}(M_{\Phi})}{k!}\sum_{\mathbf{0}<\gamma_{1}<\alpha}\sum_{\mathbf{0}<\gamma_{2}<\gamma_{1}}\dots\sum_{\mathbf{0}<\gamma_{k-1}<\gamma_{k-2}}L_{\alpha-\gamma_{1}}L_{\gamma_{1}-\gamma_{2}}\dots
L_{\gamma_{k-2}-\gamma_{k-1}}L_{\gamma_{k-1}}\Big{)}.$

Let

$J_{k}=\sum_{\mathbf{0}<\gamma_{1}<\alpha}\sum_{\mathbf{0}<\gamma_{2}<\gamma_{1}}\dots\sum_{\mathbf{0}<\gamma_{k-1}<\gamma_{k-2}}L_{\alpha-\gamma_{1}}L_{\gamma_{1}-\gamma_{2}}\dots
L_{\gamma_{k-2}-\gamma_{k-1}}L_{\gamma_{k-1}},\qquad k=2,\dots,|\alpha|.$

This sum has $N(\alpha,k)\leq 2^{k|\alpha|}$ terms. Note that
$1\leq|\alpha-\gamma_{1}|,|\gamma_{1}-\gamma_{2}|,\dots,|\gamma_{k-2}-\gamma_{k-1}|,|\gamma_{k-1}|\leq
n.$ From (25) and the fact that
$(\alpha-\gamma_{1})!(\gamma_{1}-\gamma_{2})!\cdots(\gamma_{k-2}-\gamma_{k-1})!(\gamma_{k-1})!\leq\alpha!$
this implies

$\displaystyle J_{k}$ $\displaystyle\leq
2^{k|\alpha|}\alpha!(2\mathbb{N})^{p_{0}(\alpha-\gamma_{1})}r_{0}^{|\alpha-\gamma_{1}|^{3}}!(2\mathbb{N})^{p_{0}(\gamma_{1}-\gamma_{2})}r_{0}^{|\gamma_{1}-\gamma_{2}|^{3}}!\times\cdots$
$\displaystyle\hskip
128.0374pt\cdots\times(2\mathbb{N})^{p_{0}(\gamma_{k-2}-\gamma_{k-1})}r_{0}^{|\gamma_{k-2}-\gamma_{k-1}|^{3}}!(2\mathbb{N})^{p_{0}(\gamma_{k-1})}r_{0}^{|\gamma_{k-1}|^{3}}!$
$\displaystyle\leq
2^{k|\alpha|}\alpha!(2\mathbb{N})^{p_{0}\alpha}r_{0}^{|\alpha-\gamma_{1}|^{3}}!r_{0}^{|\gamma_{1}-\gamma_{2}|^{3}}!\cdots
r_{0}^{|\gamma_{k-2}-\gamma_{k-1}|^{3}}!r_{0}^{|\gamma_{k-1}|^{3}}!.$

Note that for $r_{0}\geq 2$ and
$|\alpha-\gamma_{1}|,|\gamma_{1}-\gamma_{2}|>0$ the inequality (3) implies

$r_{0}^{|\alpha-\gamma_{1}|^{3}}!r_{0}^{|\gamma_{1}-\gamma_{2}|^{3}}!\leq
r_{0}^{|\alpha-\gamma_{2}|^{3}}!.$

Also for $\beta\in\mathcal{I}$ it holds

$(|\alpha|-|\beta|)^{3}+|\beta|^{3}\leq(|\alpha|-1)^{3}+1,\qquad\beta<\alpha,\;|\beta|\geq
1.$

The first inequality together with (3) implies

$r_{0}^{|\alpha-\gamma_{1}|^{3}}!r_{0}^{|\gamma_{1}-\gamma_{2}|^{3}}!\cdots
r_{0}^{|\gamma_{k-2}-\gamma_{k-1}|^{3}}!r_{0}^{|\gamma_{k-1}|^{3}}!\leq
r_{0}^{|\alpha-\gamma_{k-1}|^{3}+|\gamma_{k-1}|^{3}}!$

and now applying the second inequality, one obtains

$r_{0}^{|\alpha-\gamma_{1}|^{3}}!r_{0}^{|\gamma_{1}-\gamma_{2}|^{3}}!\cdots
r_{0}^{|\gamma_{k-2}-\gamma_{k-1}|^{3}}!r_{0}^{|\gamma_{k-1}|^{3}}!\leq
r_{0}^{(|\alpha|-1)^{3}+1}!.$

All these estimates imply

$\displaystyle J_{k}\leq
2^{k|\alpha|}\alpha!(2\mathbb{N})^{p_{0}\alpha}r_{0}^{(|\alpha|-1)^{3}+1}!,\qquad
k=2,\dots,|\alpha|.$

Again using the estimate for the analytic function $\varphi$ one obtains

$\displaystyle L_{\alpha}$
$\displaystyle\leq\widetilde{m}\left(K\alpha!r^{|\alpha|^{3}}!(2\mathbb{N})^{p\alpha}+\sum_{k=2}^{|\alpha|}C^{k}J_{k}\right)$
$\displaystyle\leq\widetilde{m}\;\alpha!\left(Kr^{|\alpha|^{3}}!(2\mathbb{N})^{p\alpha}+\sum_{k=2}^{|\alpha|}C^{k}2^{k|\alpha|}(2\mathbb{N})^{p_{0}\alpha}r_{0}^{(|\alpha|-1)^{3}+1}!\right)$
$\displaystyle\leq\alpha!(2\mathbb{N})^{p_{0}\alpha}\left(\widetilde{m}Kr^{|\alpha|^{3}}!+\widetilde{m}r_{0}^{(|\alpha|-1)^{3}+1}!\sum_{k=2}^{|\alpha|}C^{k}2^{k|\alpha|}\right),$

provided by $p_{0}>p.$ Also, we need

$\displaystyle\sum_{k=2}^{|\alpha|}C^{k}2^{k|\alpha|}=\sum_{k=2}^{|\alpha|}(C2^{|\alpha|})^{k}\leq(C2^{|\alpha|})^{|\alpha|+1}=(2C)^{|\alpha|^{2}+|\alpha|}\leq\widetilde{C}^{|\alpha|^{2}},\qquad|\alpha|\geq
2,$

where $\widetilde{C}=(2C)^{2}.$ Continuing the series of estimates for
$L_{\alpha},\;|\alpha|=n+1$ one obtains

$\displaystyle L_{\alpha}$
$\displaystyle\leq\alpha!(2\mathbb{N})^{p_{0}\alpha}\left(\widetilde{m}Kr^{|\alpha|^{3}}!+\widetilde{m}r_{0}^{(|\alpha|-1)^{3}+1}!\widetilde{C}^{|\alpha|^{2}}\right)$
$\displaystyle\leq\alpha!(2\mathbb{N})^{p_{0}\alpha}r_{0}^{|\alpha|^{3}}!\left(\frac{\widetilde{m}Kr^{|\alpha|^{3}}!}{r_{0}^{|\alpha|^{3}}!}+\frac{\widetilde{m}\widetilde{C}^{|\alpha|^{2}}r_{0}^{(|\alpha|-1)^{3}+1}!}{r_{0}^{|\alpha|^{3}}!}\right)$
$\displaystyle\leq\alpha!(2\mathbb{N})^{p_{0}\alpha}r_{0}^{|\alpha|^{3}}!\left(\frac{\widetilde{m}K}{r_{0}}+\frac{\widetilde{m}\widetilde{C}^{(n+1)^{2}}r_{0}^{n^{3}+1}!}{r_{0}^{(n+1)^{3}}!}\right).$

Finally, using $a^{b+c}\geq a^{b}+a^{c},$ for $a\geq 2$, we have

$\frac{r_{0}^{n^{3}+1}!}{r_{0}^{n^{3}+3n^{2}+3n+1}!}\leq\frac{1}{r_{0}^{3n^{2}+3n}!}\,.$

Thus, we obtain

$\displaystyle L_{\alpha}$
$\displaystyle\leq\alpha!(2\mathbb{N})^{p_{0}\alpha}r_{0}^{|\alpha|^{3}}!\left(\frac{\widetilde{m}K}{r_{0}}+\frac{\widetilde{m}\widetilde{C}^{(n+1)^{2}}}{r_{0}^{3n^{2}+3n}!}\right)\leq\alpha!(2\mathbb{N})^{p_{0}\alpha}r_{0}^{|\alpha|^{3}}!\left(\frac{\widetilde{m}K}{r_{0}}+\frac{(\widetilde{m}\widetilde{C})^{(n+1)^{2}}}{r_{0}^{(n+1)^{2}}}\right)$
$\displaystyle\leq\alpha!(2\mathbb{N})^{p_{0}\alpha}r_{0}^{|\alpha|^{3}}!,$

since $r_{0}>2(r+\widetilde{m}K+\widetilde{m}\widetilde{C}).$

Step 3. In this last step it is left to show that $u\in
C([0,T],X)\otimes(FS)^{\prime},$ i.e. that there exist $s\geq 2$ and $q\geq 0$
such that $u\in C([0,T],X)\otimes(FS)_{-s,-q}.$ Namely, it will be shown that
for $u(t,\omega)=\sum_{\alpha\in\mathcal{I}}u_{\alpha}(t)H_{\alpha}(\omega)$
there exist $s\geq 2$ and $q\in\mathbb{N}$ such that

$\displaystyle\sum_{\alpha\in\mathcal{I}}\sup_{t\in[0,T]}\|u_{\alpha}(t)\|^{2}_{X}\alpha!(s^{|\alpha|^{3}}!)^{-1}(2\mathbb{N})^{-q\alpha}=\sum_{\alpha\in\mathcal{I}}L_{\alpha}^{2}\alpha!(s^{|\alpha|^{3}}!)^{-1}(2\mathbb{N})^{-q\alpha}<\infty.$
(27)

Note that, for all $\alpha\in\mathcal{I},\;|\alpha|\geq 1$ by (25) and (3) it
holds that

$\displaystyle L_{\alpha}^{2}$
$\displaystyle\leq\alpha!^{2}(2\mathbb{N})^{2p_{0}\alpha}(r_{0}^{|\alpha|^{3}}!)^{2}\leq\alpha!^{2}(2\mathbb{N})^{2p_{0}\alpha}r_{0}^{2|\alpha|^{3}}!.$

Moreover, since $\alpha!\leq r_{0}^{|\alpha|^{3}}$ then for all
$\alpha\in\mathcal{I},\;|\alpha|\geq 1$ it holds that

$\displaystyle L_{\alpha}^{2}$
$\displaystyle\leq(2\mathbb{N})^{2p_{0}\alpha}r_{0}^{4|\alpha|^{3}}!.$

Finally, the series (27) is convergent since for $s=r_{0}^{4}$ and
$q>2p_{0}+1$ the following holds

$\displaystyle\sum_{\alpha\in\mathcal{I}}L_{\alpha}^{2}\alpha!(s^{|\alpha|^{3}}!)^{-1}(2\mathbb{N})^{-q\alpha}$
$\displaystyle\leq
M_{\Phi}^{2}+\sum_{\alpha\in\mathcal{I}\setminus\\{\mathbf{0}\\}}(2\mathbb{N})^{2p_{0}\alpha}r_{0}^{4|\alpha|^{3}}!(r_{0}^{4|\alpha|^{3}}!)^{-1}(2\mathbb{N})^{-q\alpha}$
$\displaystyle<M_{\Phi}^{2}+\sum_{\alpha\in\mathcal{I}\setminus\\{\mathbf{0}\\}}(2\mathbb{N})^{(2p_{0}-q)\alpha}<\infty,$

where we used that $2p_{0}-q<-1$ and $M_{\Phi}$ is given in (16). ∎

######  Remark 2.1.

Note that if assumption $(A4)$ is violated in so far that the governing
equation (12) has more than one solution (as many nonlinear PDEs have non-
unique solutions), then each of its solutions $u_{\mathbf{0}}(t),\;t\in[0,T]$
will generate a corresponding family of operators
$B_{\alpha,\Phi}(t)=A_{\alpha}+\Phi^{\prime}(u_{\mathbf{0}}(t))\,Id$,
$t\in[0,T],$ $\alpha\in\mathcal{I}\setminus\\{\mathbf{0}\\}$, which further
generates a corresponding semigroup $S_{\alpha,\Phi}(t,s),$ $0\leq s\leq t\leq
T,$ $\alpha\in\mathcal{I}\setminus\\{\mathbf{0}\\}$ that provides the
coordinatewise solutions $u_{\alpha}(t)$, $t\in[0,T],$
$\alpha\in\mathcal{I}\setminus\\{\mathbf{0}\\}$, via (14). Hence each solution
$u_{\mathbf{0}}(t),$ $t\in[0,T]$ of the PDE (12) will provide its own family
of recursively generated coefficients $u_{\alpha}(t)$, $t\in[0,T],$
$\alpha\in\mathcal{I}\setminus\\{\mathbf{0}\\}$, and therefore also a
stochastic solution $u(t),$ $t\in[0,T]$ to the initial SPDE (8). The nonlinear
stochastic equation (8) will therefore have exactly as many solutions as the
equation (12) in $(A4)$.

##  3\. Applications

In this section we present some examples of stochastic nonlinear equations
that are of the form (1) and on which our proposed method and hence also
Theorem 2.1 can be applied. We fully work out the first example (the heat
equation with an exponential nonlinearity) as an illustration and note that
all subsequent examples can be solved by a similar pattern following the
formula derived in the proof of Theorem 2.1.

###  3.1. Stochastic heat equations with exponential nonlinearity

The stochastic reaction-diffusion equation with Wick-exponential type
nonlinearity of the form

$\displaystyle u_{t}$ $\displaystyle=\triangle u+a+be^{\lozenge u}+f,\quad
a,b\not=0,\,ab<0,$ $\displaystyle u(0)$ $\displaystyle=u^{0}$

can be solved by applying Theorem 2.1. This equation is also called the
stochastic Fujita–Gelfand equation in combustion. Here the nonlinearity
$\Phi^{\lozenge}(u)=e^{\lozenge u}=\sum_{n=1}^{\infty}\frac{1}{n!}u^{\lozenge
n}$ is a convex function that appears in explosion dynamics. The corresponding
deterministic problem is used to model convection in a heat transfer process
when exothermic chemical reactions occur in the fluid body [19].

We analyze the one-dimensional case when $f$ is an $\mathbb{R}-$valued process
with zero expectation. For technical simplicity let $a=2,b=-2$ and we consider
an initial condition of the form
$u^{0}(x,\omega)=u_{\mathbf{0}}^{0}(x)+N(\omega)=u_{\mathbf{0}}^{0}(x)+\sum_{\alpha\in\mathcal{I}}H_{\alpha}(\omega)$,
where $N$ denotes a spatial random noise with expectation of the form
$u_{\mathbf{0}}^{0}(x)=-2\ln(1+e^{-x})$ and constant noise part given by the
coefficients $u_{\alpha}^{0}(x)=1$,
$\alpha\in\mathcal{I}\setminus\\{\mathbf{0}\\}$. Equation (12) hence reduces
to

$\frac{d}{dt}u_{\mathbf{0}}(t,x)=\frac{d^{2}}{dx^{2}}u_{\mathbf{0}}(t,x)+2-2e^{u_{\mathbf{0}}(t,x)},\quad
u_{\mathbf{0}}(0,x)=-2\ln(1+e^{-x}).$

By [33] this equation has a solution to the form

$u_{\mathbf{0}}(t,x)=-2\ln(1+e^{-x-t})$ (28)

that is indeed of $C^{1}([0,T],\mathbb{R})$ class, hence $(A4)$ is satisfied.
For the nonlinearity $\Phi(u)=2-2e^{u}$ we have $\Phi^{\prime}(u)=-2e^{u}$,
hence by (13), the corresponding family of operators is

$B_{\alpha,\Phi}(t)=\triangle+\Phi^{\prime}(u_{\mathbf{0}}(t))=\frac{d^{2}}{dx^{2}}-2(1+e^{-x-t})^{-2},$

actually not depending on $\alpha$. The associated semigroup to this operator
will be of the form $S_{\alpha,\Phi}(t,s)=e^{-2t(1+e^{-x-s})^{-2}}Z_{t},$
where $Z_{t}$ is the semigroup associated to the Laplace operator. Hence,
$S_{\alpha,\Phi}$ also does not depend on $\alpha$ and reads as

$S_{\alpha,\Phi}(t,s)\,g(x)=e^{-2t(1+e^{-x-s})^{-2}}\frac{1}{\sqrt{4\pi
t}}\int_{\mathbb{R}}g(x-y)e^{-\frac{y^{2}}{4t}}dy,\quad\mbox{ for }\;g\in
L^{p}(\mathbb{R}).$

Its action onto the initial condition will be

$S_{\alpha,\Phi}(t,0)\,u_{\alpha}^{0}(x)=e^{-2t(1+e^{-x})^{-2}}\frac{1}{\sqrt{4\pi
t}}\int_{\mathbb{R}}e^{-\frac{y^{2}}{4t}}dy=e^{-2t(1+e^{-x})^{-2}}.$

By (11) we have

$\displaystyle
r_{\alpha,\Phi}(t,x)=-2(1+e^{-x-t})^{-2}\sum_{k=2}^{|\alpha|}\frac{1}{k!}$
$\displaystyle\sum_{\mathbf{0}<\gamma_{1}<\alpha}\sum_{\mathbf{0}<\gamma_{2}<\gamma_{1}}\dots\sum_{\mathbf{0}<\gamma_{k-1}<\gamma_{k-2}}$
$\displaystyle u_{\alpha-\gamma_{1}}(t,x)u_{\gamma_{1}-\gamma_{2}}(t,x)\dots
u_{\gamma_{k-2}-\gamma_{k-1}}(t,x)u_{\gamma_{k-1}}(t,x),$

and $g_{\alpha,\Phi}(t,x)=r_{\alpha,\Phi}(t,x)+f_{\alpha}(t,x)$. Finally, by
(19) we obtain the coefficients $u_{\alpha}$ of the solution recursively as:

$\displaystyle
u_{\alpha}(t,x)=e^{-2t(1+e^{-x})^{-2}}+\int_{0}^{t}e^{-2t(1+e^{-x-s})^{-2}}\Big{(}\frac{1}{\sqrt{4\pi
t}}\int_{\mathbb{R}}\Big{[}f_{\alpha}(s,x-y)-2(1+e^{-x-y-s})^{-2}\times$
$\displaystyle\times\sum_{k=2}^{|\alpha|}\frac{1}{k!}\sum_{\mathbf{0}<\gamma_{1}<\alpha}\sum_{\mathbf{0}<\gamma_{2}<\gamma_{1}}\dots\sum_{\mathbf{0}<\gamma_{k-1}<\gamma_{k-2}}u_{\alpha-\gamma_{1}}(s,x-y)u_{\gamma_{1}-\gamma_{2}}(s,x-y)\dots
u_{\gamma_{k-1}}(s,x-y)\Big{]}$ $\displaystyle\times
e^{-\frac{y^{2}}{4t}}dy\Big{)}ds,$ (29)

for $\alpha>\mathbf{0}$. The final solution is thus fully determined by (28)
and (3.1) and it is given by the chaos expansion
$u(t,x,\omega)=\sum_{\alpha\in\mathcal{I}}u_{\alpha}(t,x)H_{\alpha}(\omega)$.

###  3.2. Stochastic heat equations with polynomial nonlinearities

Stochastic nonlinear heat equations with Wick-polynomial nonlinearities of the
form

$\displaystyle u_{t}$ $\displaystyle=\triangle u+au+bu^{\lozenge n}+f,\quad
a,b\not=0,\,n>1$ $\displaystyle u(0)$ $\displaystyle=u^{0}$

where $f\in C([0,T],X)\otimes(FS)^{\prime}$ is a generalized stochastic
process and $u^{0}\in X\otimes(FS)^{\prime}$ is a generalized random variable,
belong to the class of problems (1). These problems arise in quantum
mechanics, plasma physics and mathematical biology.

Particularly, the Fisher–KPP equation is one of the most important models in
mathematical biology and ecology. It describes a population that evolves under
the combined effects of spatial diffusion and local logistic growth and
saturation, for example in migration and population behavior in biological
invasion. In one spatial dimension the stochastic Fisher–KPP equation is given
by

$\displaystyle u_{t}$ $\displaystyle=\triangle u+u-u^{\lozenge 2}$
$\displaystyle u(0)$ $\displaystyle=u^{0}.$

The stochastic Allen–-Cahn equation is of the form

$\displaystyle u_{t}$ $\displaystyle=\triangle u+u-u^{\lozenge 3}+f,$
$\displaystyle u(0)$ $\displaystyle=u^{0}.$

It is used to describe phase separation and the evolution of interfaces
between the phases for systems without mass conservation taking into account
the effects due to the thermal fluctuations of the system.

Amplitude equations are used as models for different stripes patterns such as
ripples in sand, stripes of seashells arising in a variety of natural
phenomena. One of the most well known amplitude equation is the
Newell–Whitehead–Segel equation which describes the appearance of the stripe
pattern in two dimensional systems. The Newell–Whitehead–Segel equation models
the interaction of the effect of the diffusion term with the nonlinear effect
of the reaction term. Moreover, this type of equations describes the dynamical
behavior near the bifurcation point of the Rayleigh–Benard convection of
binary fluid mixtures.

The stochastic Newell–Whitehead–Segel equation is an reaction-diffusion
equation of the form

$\displaystyle u_{t}$ $\displaystyle=\triangle u+au-bu^{\lozenge n}+f,\quad
a,b>0,\quad n\geq 2$ $\displaystyle u(0)$ $\displaystyle=u^{0}.$

###  3.3. Stochastic heat equations with logarithmic nonlinearities

Stochastic heat equations with logarithmic nonlinearities of the form

$\displaystyle u_{t}$ $\displaystyle=\triangle u+\log^{\lozenge}(1+u)+f$
$\displaystyle u(0)$ $\displaystyle=u^{0},$

with
$\log^{\lozenge}(1+u)=\sum_{n=1}^{\infty}(-1)^{n+1}\frac{1}{n}\;u^{\lozenge
n}$, as well as of the form

$\displaystyle u_{t}$ $\displaystyle=\triangle
u+u\lozenge\log^{\lozenge}|u|+f$ $\displaystyle u(0)$ $\displaystyle=u^{0}$

belong to the class (1) and can be solved by Theorem 2.1.

###  3.4. Other types of nonlinearities

Other types of nonlinearities $\Phi$ which are analytic functions such as
trigonometric functions or hyperbolic functions can be considered in the form
(1). For example, a nonlinear stochastic equation of the form

$\displaystyle u_{t}$ $\displaystyle=\triangle
u+\cos^{\lozenge}(u)+\,\cosh^{\lozenge}(u)+f,\quad a\not=0$ $\displaystyle
u(0)$ $\displaystyle=u^{0}$

with

$\displaystyle\Phi^{\lozenge}(u)$
$\displaystyle=\cos^{\lozenge}u+\cosh^{\lozenge}u$
$\displaystyle=\sum_{n=0}^{\infty}(-1)^{n}\frac{1}{(2n)!}u^{\lozenge(2n)}+\sum_{n=0}^{\infty}\frac{1+(-1)^{n}}{2(n)!}u^{\lozenge
n}=\sum_{n=0}^{\infty}\frac{1}{(4n)!}\;u^{\lozenge(4n)}$

satisfies the conditions of Theorem 2.1 and has a unique almost classical
solution in $C([0,T],X)\otimes(FS)^{\prime}$.

###  3.5. Stochastic reaction-diffusion equation with multiplicative noise

In [7] global existence and uniqueness is proved for a stochastic reaction-
diffusion equation with polynomial nonlinearity (2) and multiplicative noise
in a bounded domain. These equations can be generalized to problems of the
form

$\displaystyle u_{t}$ $\displaystyle=\triangle u+\Phi^{\lozenge}(u)+u\lozenge
W_{t}$ (30) $\displaystyle u(0)$ $\displaystyle=u^{0},$

where $W_{t}$ denotes the white noise process. Our proposed method can be
applied also to the problem (30). Equations of the form (30) with
nonlinearities $\Phi$ which are not necessarily polynomial, but rather belong
to another class of analytic functions (exponential, logarithmic,
trigonometric or hypergeometric) can be solved with our method. Additionally,
instead of a multiplicative white noise, we can consider a general type of
noise terms, i.e. stochastic problems of the form

$\displaystyle u_{t}$ $\displaystyle={\textbf{A}}u+{\textbf{B}}\lozenge
u+\Phi^{\lozenge}(u)+f$ $\displaystyle u(0)$ $\displaystyle=u^{0},$

following the same procedure as we did in [23].

## Funding

This work was supported by the Ministry of Education, Science and
Technological Development of the Republic of Serbia under Grant No.
451-03-9/2021-14/200125

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