# Nonlinear random perturbations of PDEs and quasi-linear equations in Hilbert
spaces depending on a small parameter

Sandra Cerrai  Department of Mathematics
University of Maryland
4176 Campus Drive, College Park, MD 20742, United States<EMAIL_ADDRESS>,
Giuseppina Guatteri  Dipartimento di Matematica F. Brioschi
Politecnico di Milano
via Bonardi 9, 20133 Milano, Italy<EMAIL_ADDRESS>and
Gianmario Tessitore  Dipartimento di Matematica e Applicazioni
Università di Milano Bicocca
Via Roberto Cozzi 55, 20125 Milano, Italy<EMAIL_ADDRESS>
###### Abstract.

We study a class of quasi-linear parabolic equations defined on a separable
Hilbert space, depending on a small parameter in front of the second order
term. Through the nonlinear semigroup associated with such equation, we
introduce the corresponding SPDE and we study the asymptotic behavior of its
solutions, depending on the small parameter. We show that a large deviations
principle holds and we give an explicit description of the action functional.

##  1\. Introduction

Consider the partial differential equation

$\frac{dX^{x}}{dt}(t)=AX^{x}(t)+b(X^{x}(t)),\ \ \ \ \ X^{x}(0)=x\in\,H,$ (1.1)

defined on a separable Hilbert space $H$, endowed with the scalar product
$\langle\cdot,\cdot\rangle_{H}$ and the corresponding norm $\|\cdot\|_{H}$.
Here $A:D(A)\subset H\to H$ is the generator of a strongly continuous
semigroup and $b:D(b)\subseteq H\to H$ is some non-linear mapping. Next,
consider the following stochastic perturbation of (1.1)

$dX^{x}_{\epsilon}(t)=\left[AX^{x}_{\epsilon}(t)+b(X^{x}_{\epsilon}(t))\right]\,dt+\sqrt{\epsilon}\,\sigma(X^{x}_{\epsilon}(t))\,dW_{t},\
\ \ \ \ X^{x}_{\epsilon}(0)=x\in\,H,$ (1.2)

where $\epsilon>0$ is a small parameter, $W_{t}$, $t\geq 0$, is a cylindrical
Wiener process and $\sigma$ is a mapping, defined on $H$ and taking values in
some space of bounded linear operators defined on the reproducing kernel of
the noise into $H$. We assume that the differential operator $A$, the
coefficients $b$ and $\sigma$ and the noise $W_{t}$ are such that both (1.1)
and (1.2) are well-posed.

If the parameter $\epsilon$ is small, the trajectories of the perturbed system
(1.2) remain close to those of the unperturbed system (1.1) on any bounded
time interval. In particular, if there exist a domain $G\subset H$ and a point
$x_{0}\in\,G$ such that any trajectory of (1.1) starting in $G$ remains in $G$
and converges to $x_{0}$, as time goes to infinity, then with overwhelming
probability the trajectories of (1.2) starting from any $x\in G$ enter any
neighborhood of $x_{0}$, before eventually leaving the domain $G$ because of
the effect of the noise. As know, this is a consequence of the large
deviations of $X_{\epsilon}(t)$ from $X(t)$ which are described by the action
functional

$I^{x}_{T}(f)=\frac{1}{2}\inf\left\\{\int_{0}^{T}\|\varphi(t)\|_{H}^{2}\,dt\,:\,f=X^{x,\varphi}\right\\},$

where we have denoted by $X^{x,\varphi}$ the solution of the controlled
version of (1.1)

$\frac{dX^{x,\varphi}}{dt}(t)=AX^{x,\varphi}(t)+b(X^{x,\varphi}(t))+\sigma(X^{x,\varphi}(t))\varphi(t),\
\ \ \ \ X^{x,\varphi}(0)=x,$

and by the quasi-potential

$V(x_{0},x)=\inf\left\\{I_{T}(f)\,:\,f\in\,C([0,T];H),\ f(0)=x_{0},\ f(T)=x,\
T>0\right\\}.$

It is known that the stochastic PDE (1.2) is related to the linear Kolmogorov
equation on the Hilbert space $H$

$\left\\{\begin{array}[]{l}\displaystyle{D_{t}u_{\epsilon}(t,x)=\frac{\epsilon}{2}\text{Tr}\left[\sigma\sigma^{\star}(x)D^{2}_{x}u_{\epsilon}(t,x)\right]+\langle
Ax+b(x),Du_{\epsilon}(t,x)\rangle_{H},\ \ \ x\in\,H,\ \ \ t>0,}\\\\[10.0pt]
\displaystyle{u_{\epsilon}(0,x)=g(x),\ \ \ x\in\,H.}\end{array}\right.$ (1.3)

Actually, under suitable conditions on the operator $A$, the coefficients $b$
and $\sigma$ and the initial condition $g$, equation (1.3) admits a unique
classical solution $u_{\epsilon}$, which can be written in terms of the linear
transition semigroup $P^{\epsilon}_{t}$ associated with (1.2). Namely

$u_{\epsilon}(t,x)=P^{\epsilon}_{t}g(x)=\mathbb{E}g(X_{\epsilon}(t,x)),\ \ \ \
t\geq 0,\ \ x\in\,H.$

In particular, the description of the small noise asymptotics of the solutions
of equation (1.2) provided by the theory of large deviations allows to give a
detailed description of the long-time behavior of the solutions of infinite
dimensional PDE (1.3).

In [8], Freidlin and Koralov have considered more general stochastic
perturbations of the dynamical system (1.1), when $H=\mathbb{R}^{d}$, $A=0$
and $b:\mathbb{R}^{d}\to\mathbb{R}^{d}$ is a Lipschitz-continuous mapping.
They have introduced the following quasi-linear parabolic problem

$\left\\{\begin{array}[]{l}\displaystyle{\partial_{t}u_{\epsilon}(t,x)=\frac{\epsilon}{2}\sum_{i,j=1}^{d}a_{i,j}(x,u_{\epsilon}(t,x))\,\partial_{ij}u_{\epsilon}(t,x)+\sum_{i=1}^{d}b_{i}(x)\,\partial_{i}u_{\epsilon}(t,x),\
\ \ x\in\,\mathbb{R}^{d},\ \ \ t>0,}\\\\[10.0pt]
\displaystyle{u_{\epsilon}(0,x)=g(x),\ \ \
x\in\,\mathbb{R}^{d},}\end{array}\right.$ (1.4)

where $a_{ij}(x,r)=(\sigma\sigma^{\star})_{ij}(x,r)$, and by invoking the
classical theory of quasi-linear PDEs, they have shown that, under reasonable
assumptions on the coefficients $f$ and $\sigma$, equation (1.4) admits a
unique classical solution $u_{\epsilon}$. Next, for every $t>0$ and
$x\in\,\mathbb{R}^{d}$, they have introduced the following randomly perturbed
system

$\left\\{\begin{array}[]{l}\displaystyle{dX_{\epsilon}^{t,x}(s)=b(X_{\epsilon}^{t,x}(s))\,ds+\sqrt{\epsilon}\,\sigma(X_{\epsilon}^{t,x}(s),u_{\epsilon}(t-s,X_{\epsilon}^{t,x}(s)))\,dB_{s},}\\\\[10.0pt]
\displaystyle{X_{\epsilon}^{t,x}(0)=x,}\end{array}\right.$ (1.5)

where $B_{t}$, $t\geq 0$, is a $d$-dimensional Brownian motion. As in the
linear case, the PDE (1.4) and the SDE (1.5) are related by the following
relation

$u_{\epsilon}(t,x)=\mathbb{E}g(X^{t,x}_{\epsilon}(t))=:T^{\epsilon}_{t}g(x),$
(1.6)

but now $T^{\epsilon}_{t}$ is a non-linear semigroup. This is in fact reason
why equation (1.5) can be seen as a non-linear perturbation of the
deterministic system.

The study of the large deviation principle and of the quasi-potential for
(1.5), has allowed Freidlin and Koralov to study the long-time behavior of the
solutions to equation (1.4), restricted to the domain $G$ (that now is a
bounded domain in $\mathbb{R}^{d}$) and endowed with the boundary condition
$u_{\epsilon}(t,x)=g(x)$, for every $x\in\,\partial G$. In this case

$u_{\epsilon}(t,x)=\mathbb{E}g(X^{t,x}_{\epsilon}(t\wedge\tau^{x}_{\epsilon})),$

where $\tau^{x}_{\epsilon}$ is the first exit time of $X^{t,x}_{\epsilon}$
from the domain $G$. In particular, the asymptotic description of
$\tau^{x}_{\epsilon}$ in terms of the quasi-potential has made possible to
study the asymptotic behavior of $u_{\epsilon}$ on exponential time scales
$t(\epsilon)\sim\exp(\lambda/\epsilon)$. Freidlin and Koralov’s idea is to
introduce a family of linear equations obtained from (1.4) by freezing the
second variable in $\sigma\sigma^{\star}$ and putting it equal to a constant
$c$. This allows them to describe the asymptotics of
$u_{\epsilon}(\exp(\lambda/\epsilon),x)$, for different values of
$\lambda\in\,(0,\infty)$, in terms of some function $c(\lambda)$ obtained from
$V_{G}(c)$, the minimum of the quasi-potential in $G$ for the linear problem
corresponding to $c$, and from $g(x^{\star}(c))$, where $x^{\star}(c)$ is the
point of $\partial G$ where the quasi-potential attains its minimum, for
different values of $c$.

The present paper represents the beginning of a longer term project where we
aim to develop an analogous theory for infinite dimensional dynamical systems
described by PDEs. As in the finite dimensional case studied in [8], also
here, as a first and fundamental step, we need to be able to study the well-
posedness of the following quasi-linear equations

$\left\\{\begin{array}[]{l}\displaystyle{D_{t}u_{\epsilon}(t,x)=\frac{\epsilon}{2}\text{Tr}\left[\sigma\sigma^{\star}(x,u_{\epsilon}(t,x))D^{2}_{x}u_{\epsilon}(t,x)\right]+\langle
Ax+b(x),Du_{\epsilon}(t,x)\rangle_{H},\ \ \ x\in\,H,\ \ \ t>0,}\\\\[10.0pt]
\displaystyle{u_{\epsilon}(0,x)=g(x),\ \ \ x\in\,H,}\end{array}\right.$ (1.7)

However, unlike in finite dimension, where a well-established theory of
deterministic quasi-linear PDEs is available, it seems that the current
literature does not provide any Hilbert space counterpart to such classical
theory, and everything has to be done.

In our analysis we will proceed in several steps and here we are considering
the case when $\sigma:H\times\mathbb{R}\to\mathscr{L}(H)$ is Lipschitz
continuous and there exist a bounded and non-negative symmetric operator $Q$,
a continuous mapping $f$ defined on $H\times\mathbb{R}$ with values in the
space of trace-class operators and a constant $\delta>0$ such that

$\sigma^{\star}\sigma(x,r)=Q+\delta\,f(x,r),\ \ \ \ x\in\,H,\ \
r\in\,\mathbb{R}.$

This allows to rewrite equation (1.7) as

$\left\\{\begin{array}[]{l}\displaystyle{D_{t}u_{\epsilon}(t,x)=\mathscr{L}_{\epsilon}u_{\epsilon}(t,x)+\frac{\epsilon}{2}\text{Tr}\left[\delta\,f(x,u_{\epsilon})(t,x)D^{2}_{x}u_{\epsilon}(t,x)\right]+\langle
b(x),Du_{\epsilon}(t,x)\rangle_{H},}\\\\[10.0pt]
\displaystyle{u_{\epsilon}(0,x)=g(x),\ \ \ x\in\,H,}\end{array}\right.$

where

$\mathscr{L}_{\epsilon}\varphi(x)=\frac{\epsilon}{2}\text{Tr}\left[QD_{x}^{2}\varphi(x)\right]+\langle
Ax,D_{x}\varphi(x)\rangle_{H}.$

In particular, if we denote by $R^{\epsilon}_{t}$ the Ornstein-Uhlenbeck
semigroup associated with the operator $\mathscr{L}_{\epsilon}$, we can
rewrite equation (1.7) in mild form as

$u_{\epsilon}(t,x)=R^{\epsilon}_{t}g(x)+\int_{0}^{t}R^{\epsilon}_{t-s}\left(\frac{\epsilon}{2}\text{\mbox{Tr}}\left[\delta
F(u_{\epsilon}(s,\cdot))D^{2}_{x}u_{\epsilon}(s,\cdot)\right]+\langle
b(\cdot),Du_{\epsilon}(s,\cdot)\rangle_{H}\right)(x)\,ds.$ (1.8)

By assuming suitable conditions on $A$ and $Q$ (see Hypothesis 2), the
semigroup $R^{\epsilon}_{t}$ has a smoothing effect for every fixed
$\epsilon>0$, and this allows to prove by a fixed point argument that if
$\delta$ is sufficiently small there exists a local solution in a suitable
space of smooth Hölder continuous functions. Moreover, we show that any mild
solution $u_{\epsilon}$ defined on a given interval $[0,T]$ is in fact a
classical solution. In particular, $u_{\epsilon}(t,\cdot)$ is twice
continuously differentiable in $H$ for every $t>0$ and
$QD^{2}_{x}u_{\epsilon}(t,x)$ is a trace-class operator,
$u_{\epsilon}(\cdot,x)$ is differentiable in $(0,+\infty)$ for every
$x\in\,D(A)$ and equation (1.4) holds. Finally, in order to prove that for
every $T>0$ and $\epsilon>0$ the solution is defined on the whole interval
$[0,T]$ and is unique, we prove suitable a-priori bounds.

Once we have proven the existence and uniqueness of a classical solution for
equation (1.4), we can introduce the stochastic PDE

$\left\\{\begin{array}[]{l}\displaystyle{dX_{\epsilon}^{t,x}(s)=\left[AX_{\epsilon}^{t,x}(s)+b(X_{\epsilon}^{t,x}(s))\right]\,ds+\sqrt{\epsilon}\,\sigma(X_{\epsilon}^{t,x}(s),u_{\epsilon}(t-s,X_{\epsilon}^{t,x}(s)))\,dW_{s},}\\\\[10.0pt]
\displaystyle{X_{\epsilon}^{t,x}(0)=x,}\end{array}\right.$ (1.9)

where $W_{t}$ is a cylindrical Wiener process in $H$, defined on some
stochastic basis $(\Omega,\mathscr{F},\\{\mathscr{F}_{t}\\}_{t\geq
0},\mathbb{P})$. Due to the regularity of the coefficients and of the function
$u_{\epsilon}$, we can show that there exists $\bar{\delta}>0$ such that, for
every $\delta\leq\bar{\delta}$ and for every $t>0$ and $x\in\,H$, equation
(1.9) admits a unique mild solution in $L^{2}(\Omega;C([0,T];H))$. Moreover,
we show that, as in the finite dimensional case, the quasi-linear equation
(1.4) and the stochastic PDE are related through formula (1.6) and, in
particular, a maximum principle holds for equation (1.7).

It is worth noticing that as a consequence of the Markov property, the
following relation holds

$u_{\epsilon}(t-s,X_{\epsilon}^{t,x}(s)))=\mathbb{E}(g(X_{\epsilon}^{t-s,y}(t-s))\big{|}_{y=X_{\epsilon}^{t,x}(s)}=\mathbb{E}(g(X_{\epsilon}^{t,x}(t))|\mathscr{F}_{s}),$

for every $s\in\,[0,t]$ and $x\in\,H$, so that equation (1.5) reads as

$\left\\{\begin{array}[]{l}\displaystyle{dX_{\epsilon}^{t,x}(s)=\left[AX_{\epsilon}^{t,x}(s)+b(X_{\epsilon}^{t,x}(s))\right]\,ds+\sqrt{\epsilon}\,\sigma(X_{\epsilon}^{t,x}(s),\mathbb{E}(g(X_{\epsilon}^{t,x}(t))|\mathscr{F}_{s}))\,dW_{s},}\\\\[10.0pt]
\displaystyle{X_{\epsilon}^{t,x}(0)=x.}\end{array}\right.$ (1.10)

Setting $Y^{t,x}(s):=\mathbb{E}(g(X_{\epsilon}^{t,x}(t))|\mathscr{F}_{s}))$,
the equation above can be further rewritten as a coupled forward backward
infinite dimensional stochastic system

$\left\\{\begin{array}[]{l}\displaystyle{dX_{\epsilon}^{t,x}(s)=\left[AX_{\epsilon}^{t,x}(s)+b(X_{\epsilon}^{t,x}(s))\right]\,ds+\sqrt{\epsilon}\,\sigma(X_{\epsilon}^{t,x}(s),Y^{t,x}(s)))\,dW_{s},\qquad
0\leq s\leq t}\\\\[10.0pt]
\displaystyle{-d_{s}Y^{t,x}(s)=-Z^{t,x}(s)dW_{s},\qquad 0\leq s\leq
t}\\\\[10.0pt] \displaystyle{Y^{t,x}(t)=g(X_{\epsilon}^{t,x}(t))}\\\\[10.0pt]
\displaystyle{X_{\epsilon}^{t,x}(0)=x.}\end{array}\right.$ (1.11)

Coupled forward-backward systems of stochastic equations of the general form

$\left\\{\begin{array}[]{l}\displaystyle{dX(s)=b(X(s),Y(s),Z(s))ds+\sigma(X(s),Y(s))\,dW_{s},\qquad
0\leq s\leq t}\\\\[10.0pt]
\displaystyle{-d_{s}Y(s)=\psi((X(s),Y(s),Z(s))\,ds-Z(s)dW_{s},\qquad 0\leq
s\leq t}\\\\[10.0pt] \displaystyle{Y(t)=g(X(t))}\\\\[10.0pt]
\displaystyle{X_{\epsilon}(0)=x,}\end{array}\right.$ (1.12)

have been extensively studied in the finite dimensional case, see [14] where
several results are collected. Since [1], it has been clear that arbitrary
forward-backward stochastic systems do not always admit a solution. Different
techniques have been developed to prove existence and uniqueness both locally
in time and in arbitrarily long time intervals. In particular the classical
theory of PDEs, applied to the corresponding nonlinear Kolmogorov equations,
offers a wide range of results stating well posedness of system (1.12) (see,
for instance [14] [10] or [11]) that include existence and uniqueness of a
global solution to the finite dimensional analogue of system (1.11) when
$\sigma$ is not degenerate. In the infinite dimensional case, in which large
part of the analytic techniques are not available any more, very few results
on existence and uniqueness of a solution to system (1.11) in arbitrary time
interval are at hand (for local existence and uniqueness see [9]). It seems
that the techniques more likely to be extended in infinite dimensions are the
ones introduced in [15] where quantitative conditions on dissipativity of $b$
and bounds on the Lipschitz norm of $\sigma$ and $g$ are required. Such
restrictions go in the same direction as the condition on $\delta$ that we
have to impose here, see above.

We finally notice that, if we show that system (1.11) is well posed, then we
can define a candidate solution to the PDE (1.7) by setting

$\hat{u}_{\epsilon}(t-s,\xi)=\mathbb{E}(Y^{t,x}(s)|X^{t,x}(s)=\xi)$

but, unless we have a satisfactory analytic theory for equation (1.11), the
proof that $\hat{u}_{\epsilon}$ is the unique solution of (1.11) (in which
formulation?) is still to be done and does not seem obvious at all. Once such
relation would be understood, it could also be possible to study the large
deviations principle, see below, for more general nonlinear perturbations of
(1.1) defined through systems like (1.12) (see [5] for a similar approach in
the finite dimensional case where the connection between $\hat{u}_{\epsilon}$
and equation (1.11) is a straight-forward consequence of existence and
uniqueness of a regular solution to (1.11) and Ito rule).

As we mentioned at the beginning of this introduction, we are interested in
applying our results to the study of the asymptotic behavior of (1.9) and
(1.7), as $\epsilon\downarrow 0$. This is a multi-step project and here we are
addressing the problem of the validity of a large deviation principle for the
trajectories of the solutions of equation (1.9). Thus, in the last section of
our paper we prove that the family
$\\{X_{\epsilon}^{t,x}\\}_{\epsilon\in\,(0,1)}$ satisfies a large deviation
principle in the space $C([0,t];H)$, which is governed by the action
functional

$I_{t,x}(X)=\frac{1}{2}\,\inf\left\\{\int_{0}^{t}\|\varphi(s)\|_{H}^{2}\,ds\,:\,X(s)=X^{t,x}_{\varphi}(s),\
s\in\,[0,t]\right\\},$

where $X^{t,x}_{\varphi}$ is the unique mild solution of problem

$X^{\prime}(s)=AX(s)+b(X(s))+\sigma(X(s),g(Z^{X(s)}(t-s)))\varphi(s),\ \ \ \
X(0)=x,$

and for every $y\in\,H$

$Z^{y}(s)=e^{sA}y+\int_{0}^{s}e^{(s-r)A}b(Z^{y}(r))\,dr.$

##  2\. Notations and preliminaries

Throughout this paper, $H$ is a separable Hilbert space, endowed with the
scalar product $\langle\cdot,\cdot\rangle_{H}$ and the corresponding norm
$\|\cdot\|_{H}$. In what follows we shall introduce some notations and
preliminary results (we refer to [2], [6] and [12] for all details).

###  2.1. Operator spaces

We denote by $\mathscr{L}(H)$ the Banach space of all bounded linear operators
$A:H\to H$, endowed with the sup-norm

$\|A\|_{\mathscr{L}(H)}=\sup_{\|x\|_{H}\leq 1}\|Ax\|_{H}.$

An operator $A\in\,\mathscr{L}(H)$ is symmetric if it coincides with its
adjoint $A^{\star}$, that is if $\langle Ax,y\rangle_{H}=\langle
x,Ay\rangle_{H}$, for all $x,y\in\,H$. Moreover, it is non-negative if
$\langle Ax,x\rangle_{H}\geq 0$, for all $x\in\,H$. We shall denote by
$\mathscr{L}^{+}(H)$ the subspace of all non-negative and symmetric operators
in $\mathscr{L}(H)$.

An operator $A\in\,\mathscr{L}(H)$ is called an Hilbert-Schmidt operator if
there exists an orthonormal basis $\\{e_{i}\\}_{i\in\,\mathbb{N}}$ of $H$ such
that

$\sum_{i=1}^{\infty}\|Ae_{i}\|_{H}^{2}<\infty.$

The subspace of Hilbert-Schmidt operators, denoted by $\mathscr{L}_{2}(H)$, is
a Hilbert space, endowed with the scalar product

$\langle A,B\rangle_{\mathscr{L}_{2}(H)}=\sum_{i=1}^{\infty}\langle
Ae_{i},Be_{i}\rangle_{H}.$

As know, for every $B\in\,\mathscr{L}^{+}(H)$ there exists a unique
$C\in\,\mathscr{L}(H)$, denoted by $\sqrt{B}$ such that $C^{2}=B$. Thus, for
any $A\in\,\mathscr{L}(H)$ we can define

$|A|:=\sqrt{A^{\star}A}.$

We recall that an operator $A\in\,\mathscr{L}(H)$ is compact if and only if
$|A|$ is compact. Moreover, if $A$ is a symmetric compact operator, then there
exists an orthonormal basis $\\{e_{i}\\}_{i\in\,\mathbb{N}}$ of $H$ and a
sequence $\\{\alpha_{i}\\}_{i\in\,\mathbb{N}}$ converging to zero such that
$Ae_{i}=\alpha_{i}e_{i}$, for all $i\in\,\mathbb{N}$. With these notations, we
say that a compact operator $A\in\,\mathscr{L}(H)$ is nuclear or trace-class
if there exists an orthonormal basis of $H$ consisting of eigenvectors of
$|A|$ corresponding to the eigenvalues $\\{\alpha_{i}\\}_{i\in\,\mathbb{N}}$,
such that

$\sum_{i=1}^{\infty}\alpha_{i}<\infty.$

In particular, if the operator $A$ is symmetric, it is nuclear if and only if
there exists an orthonormal basis of $H$ consisting of eigenvectors of $A$
corresponding to the eigenvalues $\\{\alpha_{i}\\}_{i\in\,\mathbb{N}}$, such
that

$\sum_{i=1}^{\infty}|\alpha_{i}|<\infty.$

We denote by $\mathscr{L}_{1}(H)$ the set of nuclear operators.

It is possible to prove that for every $A\in\,\mathscr{L}_{1}(H)$ the series

$\text{Tr}A:=\sum_{i=1}^{\infty}\langle Ae_{i},e_{i}\rangle_{H}$

does not depend on the choice of the orthonormal basis
$\\{e_{i}\\}_{i\in\,\mathbb{N}}$. Moreover, a symmetric operator $A$ belongs
to $\mathscr{L}_{1}(H)$ if and only if the series above converges absolutely
for every orthonormal basis $\\{e_{i}\\}_{i\in\,\mathbb{N}}$. The space
$\mathscr{L}_{1}(H)$ is a Banach space, endowed with the norm

$\|A\|_{\mathscr{L}_{1}(H)}=\text{Tr}\,|A|,$

and

$|\text{Tr}A|\leq\|A\|_{\mathscr{L}_{1}(H)}.$ (2.1)

It is possible to prove that
$\mathscr{L}_{1}(H)\subset\mathscr{L}_{2}(H)\subset\mathscr{L}(H)$ with

$\|A\|_{\mathscr{L}(H)}\leq\|A\|_{\mathscr{L}_{2}(H)}\leq\|A\|_{\mathscr{L}_{1}(H)},$

and for $j=1,2$ it holds

$\|AB\|_{\mathscr{L}_{j}(H)}\leq\|A\|_{\mathscr{L}_{j}(H)}\|B\|_{\mathscr{L}(H)},\
\ \ \
\|AB\|_{\mathscr{L}_{j}(H)}\leq\|B\|_{\mathscr{L}_{j}(H)}\|A\|_{\mathscr{L}(H)}.$

Moreover, if $A,B\in\,\mathscr{L}_{2}(H)$, then $AB\in\,\mathscr{L}_{1}(H)$,
with

$\|AB\|_{\mathscr{L}_{1}(H)}\leq\|A\|_{\mathscr{L}_{2}(H)}\|B\|_{\mathscr{L}_{2}(H)}.$

###  2.2. Functional spaces

If $E$ is an arbitrary Banach space, endowed with the norm $\|\cdot\|_{E}$, we
denote by $B_{b}(H;E)$ the space of Borel and bounded functions $\varphi:H\to
E$. $B_{b}(H;E)$ is a Banach space, endowed with the sup-norm

$\|\varphi\|_{0}=\sup_{x\in\,H}\|\varphi(x)\|_{E}.$

Moreover, we denote by $C_{b}(H;E)$ the closed subspace of uniformly
continuous and bounded functions.

For every integer $n\geq 1$, we denote by $C^{n}_{b}(H;E)$ the space of all
functions $\varphi\in\,C_{b}(H;E)$ which are $n$-times Fréchet differentiable,
with uniformly continuous and bounded Fréchet derivatives
$D^{l}\varphi:H\to\mathscr{L}^{h}(H;E)$ (1) (1) (1)We denote by
$\mathscr{L}^{l}(H;E)$ the space of $l$-linear bounded operators $A:H^{l}\to
E$. When $l=1$, we identify $\mathscr{L}^{1}(H;\mathbb{R})$ with $H$ and when
$l=2$ we identify $\mathscr{L}^{2}(H;\mathbb{R})$ with $\mathscr{L}(H)$., for
all $l\leq n$. We have that $C^{k}_{b}(H;E)$ is a Banach space, endowed with
the norm

$\|\varphi\|_{n}=\|\varphi\|_{0}+\sum_{l=1}^{n}\|D^{l}\varphi\|_{0}.$

Next, for every $\vartheta\in\,(0,1)$ we denote by $C^{\vartheta}_{b}(H;E)$
the space of all functions $\varphi\in\,C_{b}(H;E)$ such that

$\displaystyle{[\varphi]_{\vartheta}:=\sup_{\begin{subarray}{c}x,y\in\,H\\\
x\neq
y\end{subarray}}\frac{\|\varphi(x)-\varphi(y)\|_{E}}{\|x-y\|_{H}^{\vartheta}}<\infty.}$

$C^{\vartheta}_{b}(H;E)$ is a Banach space, endowed with the norm

$\|\varphi\|_{\vartheta}=\|\varphi\|_{0}+[\varphi]_{\vartheta}.$

Finally, for every integer $n\in\,\mathbb{N}$ and $\vartheta\in\,(0,1)$, we
denote by $C^{n+\vartheta}_{b}(H;E)$ the space of all functions
$\varphi\in\,C^{n}_{b}(H;E)$ such that

$[D^{n}\varphi]_{\vartheta}:=\sup_{\begin{subarray}{c}x,y\in\,H\\\ x\neq
y\end{subarray}}\frac{\|D^{n}\varphi(x)-D^{n}\varphi(y)\|_{\mathscr{L}^{n}(H;E)}}{\|x-y\|_{H}^{\vartheta}}<\infty.$

$C^{n+\vartheta}_{b}(H;E)$ is a Banach space, endowed with the norm

$\|\varphi\|_{n+\vartheta}=\|\varphi\|_{0}+\sum_{l=1}^{n}\|D^{l}\varphi\|_{0}+[D^{n}\varphi]_{\vartheta}=\|\varphi\|_{n}+[D^{n}\varphi]_{\vartheta}.$

Notice that in case $E=\mathbb{R}$, we simply write $B_{b}(H)$ instead of
$B_{b}(H;E)$, and for every $\alpha\geq 0$ we write $C^{\alpha}_{b}(H)$
instead of $C^{\alpha}_{b}(H;\mathbb{R})$.

Now, we want to see how classical interpolatory estimates for functions
defined on $\mathbb{R}^{n}$ are still valid for functions defined on the
infinite dimensional Hilbert space $H$. To this purpose, we recall that, as
shown in [6, Theorem 2.3.5], for every $0\leq\alpha<\beta<\gamma$ there exists
a constant $c=c(\alpha,\beta,\gamma)>0$ such that for every
$\varphi\in\,C^{\gamma}_{b}(H)$

$\|\varphi\|_{\beta}\leq
c\,\|\varphi\|_{\alpha}^{\frac{\gamma-\beta}{\gamma-\alpha}}\|\varphi\|_{\gamma}^{\frac{\beta-\alpha}{\gamma-\alpha}}.$
(2.2)

However, in what follows we will need the following additional interpolatory
estimates.

######  Lemma 2.1.

Let us fix $\vartheta\in\,(0,1)$. Then, for every $\varphi\in\,C^{1}_{b}(H)$
we have

$[\varphi]_{\vartheta}\leq
c_{1,\vartheta}\,\|\varphi\|_{0}^{1-\vartheta}\,\|D\varphi\|_{0}^{\vartheta}.$
(2.3)

Moreover, for every $\varphi\in\,C^{2+\vartheta}_{b}(H)$ we have

$\|D^{2}\varphi\|_{0}\leq
c_{2,\vartheta}\,\|D\varphi\|_{0}^{\frac{\vartheta}{1+\vartheta}}\,[D^{2}\varphi]_{\vartheta}^{\frac{1}{1+\vartheta}},$
(2.4)

and

$\|D\varphi\|_{0}\leq
c_{3,\vartheta}\,\|\varphi\|_{0}^{\frac{1+\vartheta}{2+\vartheta}}\,[D^{2}\varphi]_{\vartheta}^{\frac{1}{2+\vartheta}}.$
(2.5)

###### Proof.

Let us fix $\varphi\in\,C^{1}_{b}(H)$ and $x,y\in\,H$. Then, for every
$\vartheta\in\,(0,1)$ we have

$|\varphi(x+y)-\varphi(x)|\leq
2\,\|\varphi\|_{0}^{1-\vartheta}\,\left|\int_{0}^{1}\langle D\varphi(x+\lambda
y),y\rangle_{H}\,d\lambda\right|^{\vartheta}\leq
2\,\|\varphi\|_{0}^{1-\vartheta}\,\|D\varphi\|_{0}^{\vartheta}\|y\|_{H}^{\vartheta},$

so that (2.3) follows.

Now, if we fix $\varphi\in\,C^{2+\vartheta}_{b}(H)$, for every $\mu>0$ and
$x,z\in\,H$, with $\|z\|_{H}=1$, we have

$\begin{array}[]{ll}\displaystyle{\varphi(x+\mu
z)=}&\displaystyle{\varphi(x)+\mu\,\langle
D\varphi(x),z\rangle_{H}+\frac{\mu^{2}}{2}\langle
D^{2}\varphi(x)z,z\rangle_{H}}\\\\[10.0pt]
&\displaystyle{+\mu^{2}\int_{0}^{1}(1-r)\langle[D^{2}\varphi(x+r\mu
z)-D^{2}\varphi(x)]z,z\rangle_{H}\,dr.}\end{array}$ (2.6)

By proceeding as in [6, proof of Theorem 2.3.5], we use (2.6) to prove (2.4).
Actually, thanks to (2.6) we have

$\begin{array}[]{ll}\displaystyle{\frac{\mu^{2}}{2}\left|\langle
D^{2}\varphi(x)z,z\rangle_{H}\right|}&\displaystyle{\leq\left|\varphi(x+\mu
z)-\varphi(x)-\mu\,\langle
D\varphi(x),z\rangle_{H}\right|+\mu^{2+\vartheta}[D^{2}\varphi]_{\vartheta}\int_{0}^{1}(1-r)r^{\vartheta}\,dr,}\end{array}$

so that

$\|D^{2}\varphi\|_{0}\leq\frac{2}{\mu}\,\|D\varphi\|_{0}+c_{\vartheta}\,\mu^{\vartheta}\,[D^{2}\varphi]_{\vartheta},\
\ \ \ \mu>0.$

If we take the minimum over $\mu>0$, we get (2.4).

Finally, by using again (2.6), we have

$\mu\,\left|\langle D\varphi(x),z\rangle_{H}\right|\leq\left|\varphi(x+\mu
z)-\varphi(x)\right|+\frac{\mu^{2}}{2}\left|\langle
D^{2}\varphi(x)z,z\rangle_{H}\right|+\mu^{2+\vartheta}[D^{2}\varphi]_{\vartheta}\int_{0}^{1}(1-r)r^{\vartheta}\,dr,$

so that, in view of (2.4), we get

$\begin{array}[]{ll}\displaystyle{\|D\varphi\|_{0}}&\displaystyle{\leq\frac{2}{\mu}\,\|\varphi\|_{0}+\frac{c_{2,\vartheta}\mu}{2}\,\|D\varphi\|_{0}^{\frac{\vartheta}{1+\vartheta}}\,[D^{2}\varphi]_{\vartheta}^{\frac{1}{1+\vartheta}}+c_{\vartheta}\mu^{1+\vartheta}[D^{2}\varphi]_{\vartheta}}\\\\[10.0pt]
&\displaystyle{\leq\frac{2}{\mu}\,\|\varphi\|_{0}+\frac{1}{2}\|D\varphi\|_{0}+c_{\vartheta}\mu^{1+\vartheta}[D^{2}\varphi]_{\vartheta}.}\end{array}$

This implies that

$\|D\varphi\|_{0}\leq\frac{4}{\mu}\,\|\varphi\|_{0}+c_{\vartheta}\mu^{1+\vartheta}[D^{2}\varphi]_{\vartheta},\
\ \ \ \ \mu>0,$

and if we minimize once again with respect to $\mu>0$ we obtain (2.5). ∎

######  Remark 2.2.

As a consequence of (2.3), (2.4) and (2.5), we have that for every
$\vartheta\in\,(0,1)$ there exists some $c_{\vartheta}>0$ such that for every
$\varphi\in\,C^{2+\vartheta}_{b}(H)$

$[\varphi]_{\vartheta}\,\|D^{2}\varphi\|_{0}\leq
c_{\vartheta}\,\|\varphi\|_{0}\,[D^{2}\varphi]_{\vartheta}.$ (2.7)

Actually, from (2.4) and (2.5), we have

$\|D^{2}\varphi\|_{0}\leq
c_{2,\vartheta}\left(c_{3,\vartheta}\,\|\varphi\|_{0}^{\frac{1+\vartheta}{2+\vartheta}}\,[D^{2}\varphi]_{\vartheta}^{\frac{1}{2+\vartheta}}\right)^{\frac{\vartheta}{1+\vartheta}}\,[D^{2}\varphi]_{\vartheta}^{\frac{1}{1+\vartheta}}\leq
c_{2,\vartheta}\,c_{3,\vartheta}^{\frac{1}{1+\vartheta}}\,\|\varphi\|_{0}^{\frac{\vartheta}{2+\vartheta}}\,[D^{2}\varphi]_{\vartheta}^{\frac{2}{2+\vartheta}}.$
(2.8)

Moreover, thanks to (2.3) and (2.5) we have

$[\varphi]_{\vartheta}\leq
c_{1,\vartheta}\,\|\varphi\|_{0}^{1-\vartheta}\,\left(c_{3,\vartheta}\,\|\varphi\|_{0}^{\frac{1+\vartheta}{2+\vartheta}}\,[D^{2}\varphi]_{\vartheta}^{\frac{1}{2+\vartheta}}\right)^{\vartheta}=c_{1,\vartheta}\,c_{3,\vartheta}^{\vartheta}\,\|\varphi\|_{0}^{\frac{2}{2+\vartheta}}\,[D^{2}\varphi]_{\vartheta}^{\frac{\vartheta}{2+\vartheta}}.$

Therefore, if we combine together this last inequality with (2.8), we obtain
(2.7).

###  2.3. The Ornstein-Uhlenbeck semigroup

By following [6, Chapter 6], we recall here some results about the Ornstein-
Uhlenbeck semigroup and the associated Kolmogorov equation.

Let $A:D(A)\subset H\to H$ be the generator of a $C_{0}$-semigroup $e^{tA}$
and let $Q$ be an operator in $\mathscr{L}^{+}(H)$. For every $t\geq 0$ we
define

$Q_{t}:=\int_{0}^{t}e^{sA}Qe^{sA^{\star}}\,ds,$

and we assume that $Q_{t}\in\,\mathscr{L}_{1}(H)$, for every $t\geq 0$. Thus,
we can introduce the centered Gaussian measure $\mathscr{N}_{Q_{t}}$ defined
on $H$ with covariance $Q_{t}$, and we can define

$R_{t}\varphi(x):=\int_{H}\varphi(e^{tA}x+y)\,\mathscr{N}_{Q_{t}}(dy),\ \ \ \
\ x\in\,H,\ \ \ t\geq 0,$ (2.9)

for every $\varphi$ in $B_{b}(H)$. $R_{t}$ is the Ornstein-Uhlenbeck semigroup
associated with $A$ and $Q$. In what follows, we assume that

$e^{tA}(H)\subset Q_{t}^{1/2}(H),\ \ \ \ t>0.$ (2.10)

and we define

$\Lambda_{t}:=Q_{t}^{-1/2}e^{tA},\ \ \ \ t>0.$

As shown e.g. in [6, Theorem 6.2.2], as a consequence of assumption (2.10) we
have that $R_{t}\varphi\in\,C^{\infty}_{b}(H)$, for every
$\varphi\in\,B_{b}(H)$ and $t>0$, and for every $n\in\,\mathbb{N}\cup\\{0\\}$
there exists some $c_{n}>0$ such that

$\|D^{n}R_{t}\varphi\|_{0}\leq
c_{n}\,\|\Lambda_{t}\|_{\mathscr{L}(H)}^{n}\,\|\varphi\|_{0}.$ (2.11)

Moreover, if we fix $\alpha\in\,(0,1)$ and assume
$\varphi\in\,C^{\alpha}_{b}(H)$ we have

$[D^{n}R_{t}\varphi]_{\alpha}\leq
c_{n}\,\|\Lambda_{t}\|_{\mathscr{L}(H)}^{n}\,\|e^{tA}\|_{\mathscr{L}(H)}^{\alpha}\,[\varphi]_{\alpha},$
(2.12)

so that we conclude that for all $\alpha\in\,[0,1)$ and
$\varphi\in\,C^{\alpha}_{b}(H)$

$\|D^{n}R_{t}\varphi\|_{\alpha}\leq
c_{n}\,\|\Lambda_{t}\|_{\mathscr{L}(H)}^{n}\,\|\varphi\|_{t,\alpha},\ \ \ \ \
t>0,$ (2.13)

where, for every $t\geq 0$ and $\alpha\in\,(0,1)$,

$\|\varphi\|_{t,\alpha}:=\left(\|\varphi\|_{0}+\|e^{tA}\|_{\mathscr{L}(H)}^{\alpha}\,[\varphi]_{\alpha}\right),$
(2.14)

By the interpolation inequality (2.2), for every $n\in\,\mathbb{N}\cup\\{0\\}$
and $0\leq\alpha\leq\beta<1$ and for every $\varphi\in\,C^{\beta}_{b}(H)$ and
$t>0$, we have

$\begin{array}[]{ll}\displaystyle{\|D^{n}R_{t}\varphi\|_{1+\alpha}}&\displaystyle{\leq\|D^{n}R_{t}\varphi\|_{\beta}^{\beta-\alpha}\,\|D^{n}R_{t}\varphi\|_{1+\beta}^{1-(\beta-\alpha)}}\\\\[10.0pt]
&\displaystyle{=\|D^{n}R_{t}\varphi\|_{\beta}^{\beta-\alpha}\left(\|D^{n}R_{t}\varphi\|_{0}+\|D^{n+1}R_{t}\varphi\|_{\beta}\right)^{1-(\beta-\alpha)}.}\end{array}$

Hence, thanks to (2.13), we get

$\begin{array}[]{ll}\displaystyle{\|D^{n}R_{t}\varphi\|_{1+\alpha}}&\displaystyle{\leq
c_{n}^{\,\beta-\alpha}\|\Lambda_{t}\|_{\mathscr{L}(H)}^{n(\beta-\alpha)}\,\|\varphi\|_{t,\beta}^{\beta-\alpha}\left(c_{n}\,\|\Lambda_{t}\|_{\mathscr{L}(H)}^{n}\,\|\varphi\|_{0}+c_{n+1}\,\|\Lambda_{t}\|_{\mathscr{L}(H)}^{n+1}\|\varphi\|_{t,\beta}\right)^{1-(\beta-\alpha)}}\\\\[10.0pt]
&\displaystyle{\leq
c_{\alpha,\beta,n}\,\|\varphi\|_{t,\beta}\left(\|\Lambda_{t}\|_{\mathscr{L}(H)}^{n}+\|\Lambda_{t}\|_{\mathscr{L}(H)}^{n+1-(\beta-\alpha)}\right).}\end{array}$

In particular, this allows to conclude that for every $n\in\,\mathbb{N}$ and
$0\leq\alpha\leq\beta\leq 1$ and for every $\varphi\in\,C_{b}^{\beta}(H)$

$\|D^{n}R_{t}\varphi\|_{\alpha}\leq
c_{\alpha,\beta,n}\,\left(\|\Lambda_{t}\|_{\mathscr{L}(H)}^{n-1}+\|\Lambda_{t}\|_{\mathscr{L}(H)}^{n-(\beta-\alpha)}\right)\,\|\varphi\|_{t,\beta}.$
(2.15)

Next, we recall that in [6, Proposition 6.2.9] it is shown that for every
$\varphi\in\,C^{1}_{b}(H)$ and $x\in\,H$

$\text{Tr}\,[QD^{2}R_{t}\varphi(x)]=\int_{H}\langle
Q_{t}^{-1/2}y,\Lambda_{t}Qe^{tA^{\star}}D\varphi(e^{tA}x+y)\rangle_{H}\,\mathscr{N}_{Q_{t}}(dy),$

so that, if we assume that

$\Lambda_{t}Qe^{tA^{\star}}\in\,\mathscr{L}_{2}(H),$ (2.16)

we have

$\sup_{x\in\,H}\|QD^{2}R_{t}\varphi(x)\|_{\mathscr{L}_{1}(H)}\leq\|\Lambda_{t}Qe^{tA^{\star}}\|_{\mathscr{L}_{2}(H)}\,\|D\varphi\|_{0},\
\ \ \ \ t>0.$

Therefore, thanks to the semigroup law and (2.15) we conclude that for every
$\varphi\in\,C^{\beta}_{b}(H)$

$\sup_{x\in\,H}\|QD^{2}R_{t}\varphi(x)\|_{\mathscr{L}_{1}(H)}\leq
c\,\|\Lambda_{t/2}Qe^{tA^{\star}/2}\|_{\mathscr{L}_{2}(H)}\|\Lambda_{t/2}\|_{\mathscr{L}(H)}^{1-\beta}\|\varphi\|_{\beta},\
\ \ \ \ t>0.$ (2.17)

Moreover, we recall that in [6, Proposition 6.2.5] it is shown that if the
operator $\Lambda_{t}A$ has a continuous extension $\overline{\Lambda_{t}A}$
to $H$, for every $t>0$, then for every $\varphi\in\,B_{b}(H)$ and $x\in\,H$

$DR_{t}\varphi(x)\in\,D(A^{\star}),\ \ \ \
\|A^{\star}DR_{t}\varphi\|_{0}\leq\|\overline{\Lambda_{t}A}\|_{\mathscr{L}(H)}\,\|\varphi\|_{0},\
\ \ \ t>0.$ (2.18)

Now, we introduce the parabolic equation in $H$

$D_{t}u(t,x)=\frac{1}{2}\text{Tr}\left[QD_{x}^{2}u(t,x)\right]+\langle
x,A^{\star}D_{x}u(t,x)\rangle_{H},\ \ \ \ \ u(0,x)=\varphi(x).$ (2.19)

######  Definition 1.

A function $u:[0,+\infty)\times H\to\mathbb{R}$ is a classical solution of
problem (2.19) if

  1. 1.

$u$ is continuous in $[0,+\infty)\times H$ and $u(0,\cdot)=\varphi$.

  2. 2.

$u(t,\cdot)\in\,C^{2}_{b}(H)$, for all $t>0$, and
$QD^{2}_{x}u(t,x)\in\,\mathscr{L}_{1}(H)$, for all $t>0$ and $x\in\,H$.

  3. 3.

$D_{x}u(t,x)\in\,D(A^{\star})$, for all $t>0$ and $x\in\,H$.

  4. 4.

$u(\cdot,x)$ is differentiable in $(0,+\infty)$ for every $x\in\,H$ and $u$
satisfies equation (2.19).

In [6, Theorem 6.2.4] it is shown that if we assume conditions (2.10) and
(2.16) and we assume that the operator $\Lambda_{t}A$ has a continuous
extension to $H$, then for every $\varphi\in\,B_{b}(H)$ the function

$u(t,x)=R_{t}\varphi(x)$

is the unique classical solution of equation (2.19).

##  3\. Assumptions and main results

###  3.1. Assumptions

In what follows, we shall make the following hypotheses.

######  Hypothesis 1.

  1. (1)

The mapping $\sigma:H\times\mathbb{R}\to\mathscr{L}(H)$ is Lipschitz
continuous and there exist an operator $Q\in\,\mathscr{L}^{+}(H)$, a
continuous mapping $f:H\times\mathbb{R}\to\mathscr{L}_{1}(H)$ and a constant
$\delta>0$ such that

$\sigma^{\star}\sigma(x,r)=Q+\delta\,f(x,r),\ \ \ \ x\in\,H,\ \
r\in\,\mathbb{R}.$ (3.1)

  2. (2)

For every fixed $x\in\,H$, the function
$f(x,\cdot):\mathbb{R}\to\mathscr{L}_{1}(H)$ is differentiable. Both $f$ and
$\partial_{r}f$ are Lipschitz continuous in both variables, uniformly with
respect to the other. Moreover

$\sup_{x\in\,H}\|f(x,r)\|_{\mathscr{L}_{1}(H)}\leq c\left(1+|r|\right),\ \ \ \
\ \ r\in\,\mathbb{R}.$ (3.2)

######  Remark 3.1.

Let $H=L^{2}(\mathscr{O})$, for some smooth and bounded domain
$\mathscr{O}\subset\mathbb{R}^{d}$, with $d\geq 1$. Let
$\\{e_{i}\\}_{i\in\,\mathbb{N}}$ be an orthonormal basis of $H$ and let
$\\{\lambda_{i}\\}_{i\in\,\mathbb{N}}$ be a sequence of non-negative real
numbers. We assume that $e_{i}\in\,L^{\infty}(\mathscr{O})$, for every
$i\in\,\mathbb{N}$, and

$\sum_{i=1}^{\infty}\lambda_{i}\,\|e_{i}\|_{L^{\infty}(\mathscr{O})}<\infty.$
(3.3)

For every $x,y\in\,H$ and $r\in\,\mathbb{R}$, we define

$[f(x,r)y](\xi)=\sum_{i=1}^{\infty}\mathfrak{f}_{i}(x(\xi),r)\lambda_{i}\langle
y,e_{i}\rangle_{H}e_{i}(\xi),\ \ \ \ \ \xi\in\,\mathscr{O},$

for some continuous functions
$\mathfrak{f}_{i}:\mathbb{R}\times\mathbb{R}\to\mathbb{R}$ such that
$\mathfrak{f}_{i}(s,\cdot):\mathbb{R}\to\mathbb{R}$ is differentiable, for
every $s\in\,\mathbb{R}$ and $i\in\,\mathbb{N}$. We assume that both
$\mathfrak{f}_{i}$ and $\partial_{r}\mathfrak{f}_{i}$ are Lipschitz continuous
in both variables, uniformly with respect to the other variable, and uniformly
with respect to $i\in\,\mathbb{N}$. Moreover, we assume that

$\sup_{i\in\,\mathbb{N}}\,\sup_{s\in\,\mathbb{R}}|\mathfrak{f}_{i}(s,r)|\leq
c\left(1+|r|\right),\ \ \ \ \ \ r\in\,\mathbb{R}.$ (3.4)

With this choice of $H$ and $f$, we have that condition 2 in Hypothesis 1
holds.

Actually, since $\mathfrak{f}_{i}(\cdot,r):\mathbb{R}\to\mathbb{R}$ is
Lipschitz continuous, uniformly with respect to $r\in\,\mathbb{R}$ and
$i\in\,\mathbb{N}$, for every $x,y\in\,H$ and $r\in\,\mathbb{R}$ we have

$\begin{array}[]{l}\displaystyle{\sum_{i=1}^{\infty}|\langle\left[f(x,r)-f(y,r)\right]e_{i},e_{i}\rangle_{H}|=\sum_{i=1}^{\infty}\lambda_{i}\,|\langle\left[\mathfrak{f}_{i}(x(\cdot),r)-\mathfrak{f}_{i}(y(\cdot),r)\right]e_{i},e_{i}\rangle_{H}|}\\\\[18.0pt]
\displaystyle{\leq\sum_{i=1}^{\infty}\|\mathfrak{f}_{i}(x(\cdot),r)-\mathfrak{f}_{i}(y(\cdot),r)\|_{H}\|e_{i}\|_{L^{\infty}(\mathscr{O})}\lambda_{i}\leq
c\,\|x-y\|_{H}\,\,\sum_{i=1}^{\infty}\lambda_{i}\|e_{i}\|_{L^{\infty}(\mathscr{O})}.}\end{array}$

In particular, thanks to (3.3), we can conclude that
$f(\cdot,r):H\to\mathscr{L}_{1}(H)$ is Lipschitz continuous, uniformly with
respect to $r\in\,\mathbb{R}$. In view of our assumptions, the same is true
for $\partial_{r}f$.

The Lipschitz continuity of $f(x,\cdot),\
\partial_{r}f(x,\cdot):\mathbb{R}\to\mathscr{L}_{1}(H)$, uniform with respect
to $x\in\,H$, is proved in a similar way. However, in this case (3.3) is not
required and we only need the weaker condition

$\sum_{i=1}^{\infty}\lambda_{i}<\infty.$

Finally, (3.2) is an immediate consequence of (3.4).

Now, we see some consequences of Hypothesis 1.

######  Lemma 3.2.

For any function $\varphi:H\to\mathbb{R}$ we define

$F(\varphi)(x)=f(x,\varphi(x)),\ \ \ \ x\in\,H.$ (3.5)

Then, under Hypothesis 1 we have that $F$ maps $C^{\vartheta}_{b}(H)$ into
$C^{\vartheta}_{b}(H;\mathscr{L}_{1}(H))$ and for every
$\varphi\in\,C^{\vartheta}_{b}(H)$

$\|F(\varphi)\|_{\vartheta}\leq c\left(1+\|\varphi\|_{\vartheta}\right).$
(3.6)

Moreover for every $\varphi_{1},\ \varphi_{2}\in\,C^{\vartheta}_{b}(H)$ it
holds

$\|F(\varphi_{1})-F(\varphi_{2})\|_{\vartheta}\leq
c\left(1+\|\varphi_{1}\|_{\vartheta}+\|\varphi_{2}\|_{\vartheta}\right)\|\varphi_{1}-\varphi_{2}\|_{\vartheta}.$
(3.7)

###### Proof.

Due to (3.2), if $\varphi\in\,C^{\vartheta}_{b}(H)$ we have

$\|F(\varphi)\|_{0}\leq\sup_{x\in\,H}\|f(x,\varphi(x))\|_{\mathscr{L}_{1}(H)}\leq
c\left(1+\|\varphi\|_{0}\right).$ (3.8)

Moreover, for every $x,y\in\,H$

$\begin{array}[]{l}\displaystyle{\|f(x,\varphi(x))-f(y,\varphi(y))\|_{\mathscr{L}_{1}(H)}}\\\\[10.0pt]
\displaystyle{\leq\|f(x,\varphi(x))-f(y,\varphi(x))\|_{\mathscr{L}_{1}(H)}^{\vartheta}\,\|f(x,\varphi(x))-f(y,\varphi(x))\|_{\mathscr{L}_{1}(H)}^{1-\vartheta}}\\\\[10.0pt]
\displaystyle{+\|f(y,\varphi(x))-f(y,\varphi(y))\|_{\mathscr{L}_{1}(H)}\leq
c\,\|x-y\|_{H}^{\vartheta}\,\left(1+\|\varphi\|_{0}^{1-\vartheta}\right)+c\,|\varphi(x)-\varphi(y)|}\\\\[10.0pt]
\displaystyle{\leq
c\,\|x-y\|_{H}^{\vartheta}\,\left(1+\|\varphi\|_{0}^{1-\vartheta}+[\varphi]_{\vartheta}\right)\leq
c\,\|x-y\|_{H}^{\vartheta}\,\left(1+\|\varphi\|_{0}+[\varphi]_{\vartheta}\right),}\end{array}$

so that

$[F(\varphi)]_{\vartheta}\leq
c\,\left(1+\|\varphi\|_{0}+[\varphi]_{\vartheta}\right).$ (3.9)

This, together with (3.8) allows to conclude that
$F(\varphi)\in\,C^{\vartheta}_{b}(H)$ and (3.6) holds.

Concerning (3.7), for every $\varphi_{1},\varphi_{2}\in\,C_{b}^{\vartheta}(H)$
we have

$\begin{array}[]{ll}\displaystyle{\|F(\varphi_{1})-F(\varphi_{2})\|_{0}}&\displaystyle{\leq
c\sup_{x\in\,H}\|f(x,\varphi_{1}(x))-f(x,\varphi_{2}(x))\|_{\mathscr{L}_{1}(H)}\,\leq
c\,\|\varphi_{1}-\varphi_{2}\|_{0}.}\end{array}$ (3.10)

Moreover, for every $x,y\in\,H$ we have

$\begin{array}[]{l}\displaystyle{(f(x,\varphi_{1}(x))-f(x,\varphi_{2}(x)))-(f(y,\varphi_{1}(y))-f(y,\varphi_{2}(y)))}\\\\[10.0pt]
\displaystyle{=\int_{0}^{1}\left[\gamma(s\varphi_{1}+(1-s)\varphi_{2})(x)(\varphi_{1}-\varphi_{2})(x)-\gamma(s\varphi_{1}+(1-s)\varphi_{2})(y)(\varphi_{1}-\varphi_{2})(y)\right]\,ds,}\end{array}$

where we have defined

$\gamma(\varphi)(x)=\partial_{r}f(x,\varphi(x)),\ \ \ \ \ x\in\,H.$

This implies that

$\begin{array}[]{ll}\displaystyle{[f(\cdot,\varphi_{1})-f(\cdot,\varphi_{2})]_{\vartheta}\leq\int_{0}^{1}\|\gamma(s\varphi_{1}+(1-s)\varphi_{2})\|_{\vartheta}\,ds\,\|\varphi_{1}-\varphi_{2}\|_{\vartheta}.}\end{array}$

Since, we are assuming that $\partial_{r}f$, like $f$, is Lipschitz continuous
with respect to each variable, uniformly with respect to the other, and is
clearly uniformly bounded, by using the same arguments we have used to prove
(3.8) and (3.9), we have

$\|\gamma(s\varphi_{1}+(1-s)\varphi_{2})\|_{\vartheta}\leq
c\left(1+s\,\|\varphi_{1}\|_{\vartheta}+(1-s)\,\|\varphi_{2}\|_{\vartheta}\right),$

and hence

$[F(\varphi_{1})-F(\varphi_{2})]_{\vartheta}\leq
c\left(1+\|\varphi_{1}\|_{\vartheta}+\|\varphi_{2}\|_{\vartheta}\right)\|\varphi_{1}-\varphi_{2}\|_{\vartheta}.$

This, together with (3.10), implies (3.7).

∎

######  Hypothesis 2.

  1. (1)

The operator $A:D(A)\subset H\to H$ generates a $C_{0}$-semigroup $e^{tA}$ and
there exist $M,\omega>0$ such that

$\|e^{tA}\|_{\mathscr{L}(H)}\leq Me^{-\omega t}.$ (3.11)

  2. (2)

If $Q$ is the operator introduced in Hypothesis 1 and if we define

$Q_{t}:=\int_{0}^{t}e^{sA}Qe^{sA^{\star}}\,ds,\ \ \ \ \ t\geq 0,$

we have that $Q_{t}\in\,\mathscr{L}^{+}_{1}(H)$, for every $t\geq 0$.

  3. (3)

For every $t>0$, we have

$e^{tA}(H)\subset Q_{t}^{1/2}(H).$ (3.12)

  4. (4)

If we define

$\Lambda_{t}:=Q_{t}^{-1/2}e^{tA},\ \ \ \ t>0,$

there exists some $\lambda>0$ such that

$\|\Lambda_{t}\|_{\mathscr{L}(H)}\leq c\,(t\wedge 1)^{-1/2}e^{-\lambda t},\ \
\ \ \ t>0.$ (3.13)

  5. (5)

For every $t>0$ we have that
$\Lambda_{t}Qe^{tA^{\star}}\in\,\mathscr{L}_{2}(H)$ and for every
$\vartheta\in\,(0,1)$ there exist $\beta_{\vartheta}<1$ and
$\alpha_{\vartheta}>0$ such that

$\kappa_{\vartheta}(t):=\|\Lambda_{t}Qe^{tA^{\star}}\|_{\mathscr{L}_{2}(H)}\|\Lambda_{t}\|_{\mathscr{L}(H)}^{1-\vartheta}\leq
c\,(t\wedge 1)^{-\beta_{\vartheta}}e^{-\alpha_{\vartheta}t},\ \ \ \ t>0.$
(3.14)

######  Hypothesis 3.

For every $(x,r)\in\,H\times\mathbb{R}$ and $t>0$ we have

$e^{tA}\sigma(x,r)\in\,\mathscr{L}_{2}(H).$

Moreover,

$\|e^{tA}\sigma(x,r)\|_{\mathscr{L}_{2}(H)}\leq c\,(t\wedge
1)^{-\frac{1}{4}}\,\left(1+\|x\|_{H}+|r|\right),\ \ \ \ t>0,$

and for every $(x,r),(y,s)\in\,H\times\mathbb{R}$

$\|e^{tA}\sigma(x,r)-e^{tA}\sigma(y,r)\|_{\mathscr{L}_{2}(H)}\leq c\,(t\wedge
1)^{-\frac{1}{4}}\,\left(\|x-y\|_{H}+|r-s|\right),\ \ \ \ t>0.$

######  Remark 3.3.

Let $\\{e_{i}\\}_{i\in\,\mathbb{N}}$ be an orthonormal basis in $H$ and assume
that $Ae_{i}=-\alpha_{i}e_{i}$ and $Qe_{i}=\gamma_{i}\,e_{i}$, for every
$i\in\,\mathbb{N}$, with $\alpha_{i},\ \gamma_{i}>0$, and
$\alpha_{i}\uparrow+\infty$, as $i\to\infty$. By proceeding as in [6, Example
6.2.11], we have that

$Q_{t}e_{i}=\frac{\gamma_{i}}{2\alpha_{i}}\left(1-e^{-2\alpha_{i}t}\right)e_{i},\
\ \ \ i\in\,\mathbb{N},$

so that $Q_{t}\in\,\mathscr{L}_{1}(H)$ if and only if

$\sum_{i=1}^{\infty}\frac{\gamma_{i}}{\alpha_{i}}<\infty.$ (3.15)

Moreover,

$\Lambda_{t}e_{i}=\left(\frac{2\,\alpha_{i}t\,e^{-\alpha_{i}t}}{\gamma_{i}\,(1-e^{-2\alpha_{i}t})}\right)^{1/2}t^{-1/2}e^{-\frac{\alpha_{i}}{2}t}e_{i},\
\ \ \ i\in\,\mathbb{N}.$

In particular, if $\gamma_{i}\geq\gamma_{0}>0$, we have

$\|\Lambda_{t}\|_{\mathscr{L}(H)}\leq c\,t^{-1/2}e^{-\frac{\alpha_{1}}{2}t},\
\ \ \ \ t>0,$

so that (3.13) holds. Furthermore,

$\|\Lambda_{t}Qe^{tA^{\star}}\|^{2}_{\mathscr{L}_{2}(H)}=2\sum_{i=1}^{\infty}\frac{\alpha_{i}\,\gamma_{i}\,e^{-2\alpha_{i}t}}{e^{2\alpha_{i}t}-1}\leq
c\,t^{-1}e^{-2\alpha_{1}t},$ (3.16)

When $A$ is the realization of the Laplace operator in an interval, endowed
with Dirichlet boundary conditions, we have that $\alpha_{i}\sim i^{2}$ and
(3.15) is satisfied, for every choice of $Q\in\,\mathscr{L}(H)$. If we assume
that $Q=I$, we have that (3.13) holds. Moreover, thanks to (3.16) we have

$\|\Lambda_{t}Qe^{tA^{\star}}\|_{\mathscr{L}_{2}(H)}\|\Lambda_{t}\|_{\mathscr{L}(H)}^{1-\vartheta}\leq
c\,t^{-1/2}\,e^{-\alpha_{1}t}\,t^{-(1-\vartheta)/2}e^{-\frac{\alpha_{1}(1-\vartheta)t}{2}}=c\,t^{-(1-\vartheta/2)}e^{-\frac{\alpha_{1}(3-\vartheta)t}{2}},$

and Condition (5) in Hypothesis 2 holds for every $\vartheta\in\,(0,1)$.
Notice also that in this case Hypothesis 3 is satisfied.

######  Hypothesis 4.

The mapping $b:H\to H$ is Lipschitz continuous and bounded.

###  3.2. Main results

As we have done in Section 2 for the linear Kolmogorov equation (2.19), we
introduce here the notion of classical solution for the quasi-linear problem

$\left\\{\begin{array}[]{l}\displaystyle{D_{t}u_{\epsilon}(t,x)=\frac{\epsilon}{2}\text{Tr}\left[\sigma^{\star}\sigma(x,u_{\epsilon}(t,x))D^{2}_{x}u_{\epsilon}(t,x)\right]+\langle
Ax+b(x),Du_{\epsilon}(t,x)\rangle_{H},\ \ \ x\in\,H,\ \ \ t>0,}\\\\[10.0pt]
\displaystyle{u_{\epsilon}(0,x)=g(x),\ \ \ x\in\,H.}\end{array}\right.$ (3.17)

######  Definition 2.

A function $u_{\epsilon}:[0,+\infty)\times H\to\mathbb{R}$ is a classical
solution of problem (3.17) if the following conditions are satisfied.

  1. 1.

It is continuous in $[0,+\infty)\times H$ and $u_{\epsilon}(0,\cdot)=g$.

  2. 2.

$u_{\epsilon}(t,\cdot)\in\,C^{2}_{b}(H)$, for all $t>0$, and
$QD^{2}_{x}u_{\epsilon}(t,x)\in\,\mathscr{L}_{1}(H)$, for all
$(t,x)\in\,(0,+\infty)\times H$.

  3. 3.

$u_{\epsilon}(\cdot,x)$ is differentiable in $(0,+\infty)$, for every
$x\in\,D(A)$.

  4. 4.

It satisfies equation (3.17), for every $(t,x)\in\,(0,+\infty)\times D(A)$.

In what follows, for every $\epsilon\in\,(0,1)$, $0<\vartheta<\eta<1$,
$\varrho\in\,(0,1/2)$ and $T>0$, we denote by
$C_{\epsilon,\varrho,\eta}((0,T];C^{2+\vartheta}_{b}(H))$ the space of all
functions $u\in\,C([0,T];C^{\eta}_{b}(H))\cap C((0,T];C^{2+\vartheta}_{b}(H))$
such that

$\|u\|_{\epsilon,\varrho,\eta,\vartheta,T}:=\sup_{t\in\,(0,T]}\left(\|u(t,\cdot)\|_{\eta}+\epsilon^{\varrho}(t\wedge
1)^{\varrho}\|D_{x}u(t,\cdot)\|_{\vartheta}+\epsilon^{\varrho+\frac{1}{2}}(t\wedge
1)^{\varrho+\frac{1}{2}}\|D^{2}_{x}u(t,\cdot)\|_{\vartheta}\right)<\infty.$

######  Theorem 3.4.

Assume Hypotheses 1 to 4, and fix an arbitrary $g\in\,C^{\eta}_{b}(H)$, for
some $\eta>1/2$. Then there exists $\bar{\delta}>0$ such that for every
$\delta\leq\bar{\delta}$ and $\epsilon\in\,(0,1)$ there exists a unique
classical solution $u_{\epsilon}$ for equation (3.17). Moreover, if we fix
$\vartheta\in\,(0,(\eta-1/2)\wedge 1)$ and we define

$\varrho=\frac{1-(\eta-\vartheta)}{2},$

we have that
$u_{\epsilon}\in\,C_{\epsilon,\varrho,\eta}((0,T];C^{2+\vartheta}_{b}(H))$,
for every $T>0$ and $\epsilon\in\,(0,1)$, and

$\|u_{\epsilon}\|_{\epsilon,\varrho,\eta,\vartheta,T}\leq
c_{\epsilon}\,\|g\|_{\eta},\ \ \ \ \ \epsilon\in\,(0,1),$ (3.18)

for some constant $c_{\epsilon}>0$ independent of $T>0$.

Once proved the existence and uniqueness of a classical solution
$u_{\epsilon}$ for problem (3.17), for every $\epsilon>0$, we fix arbitrary
$t>0$ and $x\in\,H$ and we introduce the following stochastic PDE

$\left\\{\begin{array}[]{rl}\displaystyle{dX(s)=}&\displaystyle{\left[AX(s)+b(X(s))+\sigma(X(s),u_{\epsilon}(t-s,X(s)))\,\varphi(s)\right]\,ds}\\\\[14.0pt]
&\displaystyle{+\sqrt{\epsilon}\,\sigma(X(s),u_{\epsilon}(t-s,X(s)))\,dW_{s},}\\\\[10.0pt]
\displaystyle{X(0)=}&\displaystyle{x.}\end{array}\right.$ (3.19)

Here $W_{t}$, $t\geq 0$, is a cylindrical Wiener process on $H$, defined on
the filtered probability space
$(\Omega,\mathscr{F},\\{\mathscr{F}_{t}\\}_{t\geq 0},\mathbb{P})$, such that
for every $h,k\in\,H$ and $t,s\geq 0$

$\mathbb{E}\,\langle W_{t},h\rangle_{H}\langle W_{s},k\rangle_{H}=(t\wedge
s)\,\langle h,k\rangle_{H},$

and $\varphi$ is a predictable process in $L^{2}(\Omega;L^{2}(0,t;H))$.

######  Definition 3.

An adapted process $X^{t,x}_{\varphi,\epsilon}\in\,L^{2}(\Omega;C([0,t];H))$
is a mild solution for equation (3.19) if for every $s\in\,[0,t]$

$\begin{array}[]{ll}\displaystyle{X_{\varphi,\epsilon}^{t,x}(s)=}&\displaystyle{e^{sA}x+\int_{0}^{s}e^{(s-r)A}b(X_{\varphi,\epsilon}^{t,x}(r))\,dr+\int_{0}^{s}e^{(s-r)A}\sigma(X_{\varphi,\epsilon}^{t,x}(r),u_{\epsilon}(t-r,X_{\varphi,\epsilon}^{t,x}(r)))\,\varphi(r)\,dr}\\\\[18.0pt]
&\displaystyle{+\sqrt{\epsilon}\,\int_{0}^{s}e^{(s-r)A}\sigma(X_{\varphi,\epsilon}^{t,x}(r),u_{\epsilon}(t-r,X_{\varphi,\epsilon}^{t,x}(r)))\,dW_{r}.}\end{array}$
(3.20)

######  Theorem 3.5.

Suppose that Hypotheses 1 to 4 hold, and fix any $g\in\,C^{\eta}_{b}(H)$, with
$\eta>1/2$, $\epsilon\in\,(0,1)$ and $\delta\in\,[0,\bar{\delta})$, where
$\bar{\delta}$ is the constant introduced in Theorem 3.4. Moreover, fix an
arbitrary predictable process in $L^{2}(\Omega;C([0,t];H))$ such that

$\int_{0}^{t}\|\varphi(s)\|_{H}^{2}\,ds\leq M,\ \ \ \ \
\mathbb{P}-\text{a.s.}$ (3.21)

for some $M>0$. Then, if we assume that $b:H\to H$ is Lipschitz-continuous,
equation (3.19) admits a unique mild solution
$X^{t,x}_{\varphi,\epsilon}\in\,L^{2}(\Omega;C([0,t];H))$, for every $x\in\,H$
and $t>0$.

In what follows, the solution of the uncontrolled version of equation (3.19),
corresponding to $\varphi=0$, will be denoted by $X^{t,x}_{\epsilon}$.

Once proved Theorem 3.5, we are interested in studying the limiting behavior
of $X^{t,x}_{\epsilon}$ as $\epsilon\downarrow 0$. More precisely, we want to
prove that for every fixed $t>0$ and $x\in\,H$ the family
$\\{\mathscr{L}(X^{t,x}_{\epsilon})\\}_{\epsilon\in\,(0,1)}$ satisfies a large
deviation principle in the space $C([0,t];H)$ (with speed $\epsilon$) with
respect to a suitable action functional $I_{t,x}$ that we will describe
explicitly.

In order to state our result, we have to introduce some notations. First, we
introduce the unperturbed problem

$Z^{\prime}(s)=AZ(s)+b(Z(s)),\ \ \ \ Z(0)=y\in\,H.$ (3.22)

Since we are assuming that $b:H\to H$ is Lipschitz continuous, for every $T>0$
and $y\in\,H$ there exists a unique $Z^{y}\in\,C([0,T];H)$ such that

$Z^{y}(s)=e^{sA}y+\int_{0}^{s}e^{(s-r)A}b(Z^{y}(r))\,dr.$

Next, for every $x\in\,H$, $t>0$ and $\varphi\in\,L^{2}(0,t;H)$ we introduce
the controlled problem

$X^{\prime}(s)=AX(s)+b(X(s))+\sigma(X(s),g(Z^{X(s)}(t-s)))\varphi(s),\ \ \ \
X(0)=x.$ (3.23)

In Section 8 we will see that under the same assumptions of Theorem 3.5,
equation (3.23) admits a unique mild solution
$X^{t,x}_{\varphi}\in\,C([0,t];H)$. This will allow to state the last main
result of this paper.

######  Theorem 3.6.

In addition to the conditions assumed in Theorem 3.5, suppose that
$g:H\to\mathbb{R}$ is Lipschitz-continuous. Then, for every fixed $t>0$ and
$x\in\,H$ the family
$\\{\mathscr{L}(X^{t,x}_{\epsilon})\\}_{\epsilon\in\,(0,1)}$ satisfies a large
deviation principle in the space $C([0,t];H)$, with speed $\epsilon$, with
respect to the action functional

$I_{t,x}(X)=\frac{1}{2}\,\inf\left\\{\int_{0}^{t}\|\varphi(s)\|_{H}^{2}\,ds\,:\,X(s)=X^{t,x}_{\varphi}(s),\
s\in\,[0,t]\right\\},$ (3.24)

where $X^{t,x}_{\varphi}$ is the unique mild solution of problem (3.23).

##  4\. The well-posedness of the stochastic PDE (3.19)

In this section we will assume that, for some $T>0$, $\eta>1/2$,
$\vartheta\in\,(0,(\eta-1/2)\wedge 1)$, $\varrho<1/4$, and
$\epsilon\in\,(0,1)$ there exists a mild solution
$u_{\epsilon}\in\,C_{\epsilon,\varrho,\eta}((0,T];C^{2+\vartheta}_{b}(H))$ for
equation (3.17). We will show how this allows to prove Theorem 3.5 for every
$t\in\,(0,T]$.

We fix $t\in\,(0,T]$ and a predictable process
$\varphi\in\,L^{2}(\Omega;L^{2}(0,t;H))$ satisfying (3.21) and we consider the
stochastic equation

$\left\\{\begin{array}[]{rl}\displaystyle{dX(s)=}&\displaystyle{\left[AX(s)+b(X(s))+\sigma(X(s),u_{\epsilon}(t-s,X(s)))\varphi(s)\right]\,ds}\\\\[14.0pt]
&\displaystyle{+\sqrt{\epsilon}\,\sigma(X(s),u_{\epsilon}(t-s,X(s)))\,dW_{s},}\\\\[10.0pt]
\displaystyle{X(0)=}&\displaystyle{x.}\end{array}\right.$ (4.1)

For every $\epsilon\in\,(0,1)$, $s\in\,[0,t]$ and $x\in\,H$, we define

$\Sigma_{t,\epsilon}(s,x):=\sigma(x,u_{\epsilon}(t-s,x)).$ (4.2)

A process $X\in\,L^{2}(\Omega;C([0,t];H))$ is a mild solution of equation
(4.1) if it is a fixed point of the mapping $\Lambda_{t,\epsilon}$ defined by

$\begin{array}[]{ll}\displaystyle{\Lambda_{t,\epsilon}(X)(s):=}&\displaystyle{e^{sA}x+\int_{0}^{s}e^{(s-r)A}b(X(r))\,ds+\int_{0}^{s}e^{(s-r)A}\Sigma_{t,\epsilon}(r,X(r))\,\varphi(r)\,dr}\\\\[14.0pt]
&\displaystyle{+\sqrt{\epsilon}\int_{0}^{s}e^{(s-r)A}\Sigma_{t,\epsilon}(r,X(r))\,dW(r).}\end{array}$

According to Hypothesis 3, for every $\tau>0$, $s\in\,[0,t]$ and $x,y\in\,H$
we have

$\|e^{\tau
A}\left(\Sigma_{t,\epsilon}(s,x)-\Sigma_{t,\epsilon}(s,y)\right)\|_{\mathscr{L}_{2}(H)}\leq
c\,(\tau\wedge
1)^{-\frac{1}{4}}\left(\|x-y\|_{H}+|u_{\epsilon}(t-s,x)-u_{\epsilon}(t-s,y)|\right).$

Since
$u_{\epsilon}\in\,C_{\epsilon,\varrho,\eta}((0,T];C^{2+\vartheta}_{b}(H))$, we
have

$\begin{array}[]{ll}\displaystyle{|u_{\epsilon}(t-s,x)-u_{\epsilon}(t-s,y)|}&\displaystyle{\leq\|D_{x}u_{\epsilon}(t-s,\cdot)\|_{0}\,\|x-y\|_{H}}\\\\[14.0pt]
&\displaystyle{\leq\epsilon^{-\varrho}\,((t-s)\wedge
1)^{-\varrho}\|u_{\epsilon}\|_{\epsilon,\varrho,\eta,\vartheta,T}\|x-y\|_{H},}\end{array}$

so that

$\begin{array}[]{l}\displaystyle{\|e^{\tau
A}\left(\Sigma_{t,\epsilon}(s,x)-\Sigma_{t,\epsilon}(s,y)\right)\|_{\mathscr{L}_{2}(H)}\leq
c\,(\tau\wedge 1)^{-\frac{1}{4}}\left(1+\epsilon^{-\varrho}((t-s)\wedge
1)^{-\varrho}\|u_{\epsilon}\|_{\epsilon,\varrho,\eta,\vartheta,T}\right)\|x-y\|_{H}.}\end{array}$
(4.3)

Now, for every $\beta\geq 0$ we denote by $\mathscr{K}_{\beta,t}(H)$ the
Banach space of all $H$-valued predictable processes $X$ such that

$\|X\|_{\mathscr{K}_{\beta,t}(H)}^{2}:=\sup_{s\in\,[0,t]}e^{-\beta
s}\,\mathbb{E}\,\|X(s)\|^{2}_{H}<\infty.$

If $X_{1},X_{2}\in\,\mathscr{K}_{\beta,t}(H)$, in view of (4.3) we have

$\begin{array}[]{l}\displaystyle{\mathbb{E}\left\|\int_{0}^{s}e^{(s-r)A}\left[\Sigma_{t,\epsilon}(r,X_{1}(r))-\Sigma_{t,\epsilon}(r,X_{2}(r))\right]\,\varphi(r)\,dr\right\|_{H}^{2}}\\\\[18.0pt]
\displaystyle{\leq\mathbb{E}\int_{0}^{s}\left\|e^{(s-r)A}\left[\Sigma_{t,\epsilon}(r,X_{1}(r))-\Sigma_{t,\epsilon}(r,X_{2}(r))\right]\right\|_{\mathscr{L}_{2}(H)}^{2}\,dr\int_{0}^{s}\|\varphi(r)\|_{H}^{2}\,dr}\\\\[18.0pt]
\displaystyle{\leq cM\,\mathbb{E}\int_{0}^{s}((s-r)\wedge
1)^{-\frac{1}{2}}\left(1+\epsilon^{-2\varrho}((t-r)\wedge
1)^{-2\varrho}\|u_{\epsilon}\|^{2}_{\epsilon,\varrho,\eta,\vartheta,T}\right)\,\|X_{1}(r)-X_{2}(r)\|_{H}^{2}\,dr}\\\\[18.0pt]
\displaystyle{\leq
cM\,\|X_{1}-X_{2}\|^{2}_{\mathscr{K}_{\beta,t}(H)}\int_{0}^{s}((s-r)\wedge
1)^{-\frac{1}{2}}\left(1+e^{-2\varrho}((t-r)\wedge
1)^{-2\varrho}\|u_{\epsilon}\|^{2}_{\epsilon,\varrho,\eta,\vartheta,T}\right)e^{\beta
r}\,dr.}\end{array}$

Since we are assuming that $\varrho<1/4$, for every $s\in\,[0,t]$

$\int_{0}^{s}((s-r)\wedge 1)^{-\frac{1}{2}}\left(1+e^{-2\varrho}((t-r)\wedge
1)^{-2\varrho}\|u_{\epsilon}\|^{2}_{\epsilon,\varrho,\eta,\vartheta,T}\right)e^{-\beta(s-r)}\,dr\leq
c_{\epsilon,\beta,t}(s),$

for some continuous increasing function
$c_{\epsilon,\beta,t}:[0,t]\to[0,+\infty)$ such that

$\lim_{\beta\to\infty}\sup_{s\in\,[0,t]}c_{\epsilon,\beta,t}(s)=0.$

Therefore, we pick $\beta_{1}=\beta_{1}(\epsilon,t)>0$ such that

$\sup_{s\in\,[0,t]}c_{\epsilon,\beta_{1},t}(s)\leq\frac{1}{6},$

we have

$\begin{array}[]{l}\displaystyle{\sup_{s\in\,[0,t]}e^{-\beta_{1}s}\,\mathbb{E}\left\|\int_{0}^{s}e^{(s-r)A}\left[\Sigma_{t,\epsilon}(r,X_{1}(r))-\Sigma_{t,\epsilon}(r,X_{2}(r))\right]\,\varphi(r)\,dr\right\|_{H}^{2}}\\\\[18.0pt]
\displaystyle{\leq\frac{1}{6}\,\|X_{1}-X_{2}\|^{2}_{\mathscr{K}_{\beta_{1},t}(H)}.}\end{array}$

Moreover, we have

$\begin{array}[]{l}\displaystyle{\mathbb{E}\left\|\int_{0}^{s}e^{(s-r)A}\left[\Sigma_{t,\epsilon}(r,X_{1}(r))-\Sigma_{t,\epsilon}(r,X_{2}(r))\right]\,dW(r)\right\|_{H}^{2}}\\\\[18.0pt]
\displaystyle{\leq c\int_{0}^{s}((s-r)\wedge
1)^{-\frac{1}{2}}\left(1+\epsilon^{-2\varrho}((t-r)\wedge
1)^{-2\varrho}\|u_{\epsilon}\|^{2}_{\epsilon,\varrho,\eta,\vartheta,T}\right)\,\mathbb{E}\|X_{1}(r)-X_{2}(r)\|_{H}^{2}\,dr}\\\\[18.0pt]
\displaystyle{\leq
c\,\|X_{1}-X_{2}\|^{2}_{\mathscr{K}_{\beta,t}(H)}\int_{0}^{s}((s-r)\wedge
1)^{-\frac{1}{2}}\left(1+\epsilon^{-2\varrho}((t-r)\wedge
1)^{-2\varrho}\|u_{\epsilon}\|^{2}_{\epsilon,\varrho,\eta,\vartheta,T}\right)e^{\beta
r}\,dr.}\end{array}$

Then, by proceeding as above

$\begin{array}[]{l}\displaystyle{\sup_{s\in\,[0,t]}e^{-\beta_{1}s}\,\mathbb{E}\left\|\int_{0}^{s}e^{(s-r)A}\left[\Sigma_{t,\epsilon}(r,X_{1}(r))-\Sigma_{t,\epsilon}(r,X_{2}(r))\right]\,dW(r)\right\|_{H}^{2}}\\\\[18.0pt]
\displaystyle{\leq\frac{1}{6}\,\|X_{1}-X_{2}\|^{2}_{\mathscr{K}_{\beta_{1},t}(H)}.}\end{array}$

Finally, due to the Lipschitz-continuity of $b$, we have that there exists
$\beta_{2}>0$ such that that

$\sup_{s\in\,[0,t]}e^{-\beta_{2}s}\mathbb{E}\left\|\int_{0}^{s}e^{(s-r)A}\left[b(X_{1}(r))-b(X_{2}(r))\right]\,dr\right\|_{H}^{2}\leq\frac{1}{6}\,\|X_{1}-X_{2}\|^{2}_{\mathscr{K}_{\beta_{2},t}(H)}.$

Therefore, is we take $\bar{\beta}:=\beta_{1}\vee\beta_{2}$ , we have that
$\Lambda_{t,\epsilon}$ is a contraction in $\mathscr{K}_{\bar{\beta},t}(H)$
and its fixed point is the unique mild solution $X^{t,x}_{\varphi,\epsilon}$
of equation (4.1).

Finally, by using a stochastic factorization argument, it is possible to prove
that $X_{\varphi}^{t,x}$ belongs to $L^{2}(\Omega;C([0,t];H))$ (for all
details about stochastic factorization see [7, Subsection 5.3.1]).

##  5\. Local existence of mild solutions for the quasi-linear problem

In this section we will prove that the quasi-linear problem (3.17) admits a
local mild solution, for every $\epsilon\in\,(0,1)$.

In view of (3.1) and (3.5), problem (3.17) can be rewritten as

$\left\\{\begin{array}[]{l}\displaystyle{D_{t}u_{\epsilon}(t,x)=\mathscr{L}_{\epsilon}u_{\epsilon}(t,x)+\frac{\epsilon}{2}\text{Tr}\left[\delta\,F(u_{\epsilon})(t,x)D^{2}_{x}u_{\epsilon}(t,x)\right]+\langle
b(x),Du_{\epsilon}(t,x)\rangle_{H},}\\\\[10.0pt]
\displaystyle{u_{\epsilon}(0,x)=g(x),\ \ \ x\in\,H,}\end{array}\right.$ (5.1)

where $\mathscr{L}_{\epsilon}$ is the linear Kolmogorov operator

$\mathscr{L}_{\epsilon}\varphi(x)=\frac{\epsilon}{2}\text{Tr}\left[QD_{x}^{2}\varphi(x)\right]+\langle
Ax,D_{x}\varphi(x)\rangle_{H}.$

As we have recalled in Subsection 2.3, for every $\varphi\in\,B_{b}(H)$ the
unique classical solution of the linear problem

$D_{t}v_{\epsilon}(t,x)=\mathscr{L}_{\epsilon}v_{\epsilon}(t,x),\ \ \ \
v_{\epsilon}(0,x)=\varphi(x),$

is given by the Ornstein-Uhlenbeck semigroup

$v_{\epsilon}(t,x)=R_{t}^{\epsilon}\varphi(x)=\int_{H}\varphi(e^{tA}x+y)\mathscr{N}_{\epsilon
Q_{t}}(dy).$

Before proceeding with the study of equation (5.1), we show how, in view of
Hypothesis 2, the properties of the Ornstein-Uhlenbeck semigroup described in
Subsection 2.3 apply to the semigroup $R^{\epsilon}_{t}$.

Thanks to (3.11) and (3.13), inequality (2.13) gives for every
$n\in\,\mathbb{N}\cup\\{0\\}$ and $\alpha\in\,(0,1)$

$\|D^{n}R^{\epsilon}_{t}\varphi\|_{\alpha}\leq
c_{n,\alpha}\,\epsilon^{-\frac{n}{2}}(t\wedge 1)^{-\frac{n}{2}}e^{-\lambda
nt}\|\varphi\|_{t,\alpha},\ \ \ \ t>0,\ \ \epsilon\in\,(0,1),$ (5.2)

where

$\|\varphi\|_{t,\alpha}:=\left(\|\varphi\|_{0}+e^{-\omega\alpha
t}[\varphi]_{\alpha}\right).$

In the same way, inequality (2.15) gives for every $n\in\,\mathbb{N}$ and
$0\leq\alpha\leq\beta\leq 1$

$\|D^{n}R^{\epsilon}_{t}\varphi\|_{\alpha}\leq
c_{n,\alpha,\beta}\,\epsilon^{-\frac{n-(\beta-\alpha)}{2}}(t\wedge
1)^{-\frac{n-(\beta-\alpha)}{2}}e^{-\lambda nt}\|\varphi\|_{t,\beta},\ \ \ \
t>0,\ \ \epsilon\in\,(0,1).$ (5.3)

Finally, since

$\|R^{\epsilon}_{t}\varphi\|_{0}\leq\|\varphi\|_{0},\ \ \ \
[R^{\epsilon}_{t}\varphi]_{\beta}\leq e^{-\omega\beta t}[\varphi]_{\beta},\ \
\ \ \epsilon>0,$

and

$[R^{\epsilon}_{t}\varphi]_{\alpha}\leq\|DR^{\epsilon}_{t}\varphi\|_{0}^{\frac{\alpha-\beta}{1-\beta}}[R^{\epsilon}_{t}\varphi]_{\beta}^{\frac{1-\alpha}{1-\beta}},$

thanks to (5.2) and (5.3) we get

$[R^{\epsilon}_{t}\varphi]_{\alpha}\leq
c_{\alpha,\beta}\,\epsilon^{-\frac{\alpha-\beta}{2}}(t\wedge
1)^{-\frac{\alpha-\beta}{2}}e^{-\omega_{\alpha,\beta}t}\|\varphi\|_{t,\beta},\
\ \ \ t>0,\ \ \epsilon\in\,(0,1),$ (5.4)

for some $\omega_{\alpha,\beta}>0$, and

$\|R^{\epsilon}_{t}\varphi\|_{\alpha}\leq
c_{\alpha,\beta}\,\epsilon^{-\frac{\alpha-\beta}{2}}(t\wedge
1)^{-\frac{\alpha-\beta}{2}}\|\varphi\|_{t,\beta},\ \ \ \ t>0,\ \
\epsilon\in\,(0,1).$ (5.5)

Now, we introduce the notion of mild solution for equation (3.17).

######  Definition 4.

A function $u_{\epsilon}\in\,C([0,+\infty);H)$ such that
$u_{\epsilon}(t,\cdot)\in\,C^{2}_{b}(H)$, for every $t>0$, is a mild solution
for problem (3.17) if for every $(t,x)\in\,[0,+\infty)\times H$

$u_{\epsilon}(t,x)=R^{\epsilon}_{t}g(x)+\int_{0}^{t}R^{\epsilon}_{t-s}\left(\frac{\epsilon}{2}\text{\em{Tr}}\left[\delta
F(u_{\epsilon}(s,\cdot))D^{2}_{x}u_{\epsilon}(s,\cdot)\right]+\langle
b(\cdot),Du_{\epsilon}(s,\cdot)\rangle_{H}\right)(x)\,ds.$

For every $R>0$ we define

$\mathscr{Y}^{\epsilon,R}_{\varrho,\eta,\vartheta,T}:=\left\\{\,u\in\,C_{\epsilon,\varrho,\eta}((0,T];C^{2+\vartheta}_{b}(H))\
:\ \|u\|_{\epsilon,\varrho,\eta,\vartheta,T}\leq R\right\\},$

and for every $v\in\,\mathscr{Y}^{\epsilon,R}_{\varrho,\eta,\vartheta,T}$ and
$\delta>0$ we define

$\Gamma_{\epsilon,\delta}(v)(t,x):=\int_{0}^{t}R^{\epsilon}_{t-s}\gamma_{\epsilon,\delta}(v,s)(x)\,ds,\
\ \ \ \ t\in\,[0,T],\ \ \ x\in\,H,$

where

$\gamma_{\epsilon,\delta}(v,s)(x):=\frac{\epsilon}{2}\text{Tr}\left[\delta
F(v(s,\cdot))(x)D^{2}_{x}v(s,x)\right]+\langle b(x),Dv(s,x)\rangle_{H}.$

In particular, $u_{\epsilon}$ is a mild solution for problem (3.17) if and
only if

$u_{\epsilon}(t,x)=R^{\epsilon}_{t}g(x)+\Gamma_{\epsilon,\delta}(u_{\epsilon})(t,x).$

First, we investigate the dependence of $\gamma_{\epsilon,\delta}$ on
$v\in\,\mathscr{Y}^{\epsilon,R}_{\varrho,\eta,\vartheta,T}$.

######  Lemma 5.1.

For every $v\in\,\mathscr{Y}^{\epsilon,R}_{\varrho,\eta,\vartheta,T}$ and
$\delta>0$

$\|\gamma_{\epsilon,\delta}(v,s)\|_{\vartheta}\leq
c\,\epsilon^{\frac{1}{2}-\varrho}\,\delta R\left(1+R\right)(s\wedge
1)^{-(\varrho+\frac{1}{2})}+c\,\epsilon^{-\varrho}R(s\wedge 1)^{-\varrho},\ \
\ \ \ s\in\,(0,T].$ (5.6)

Moreover, for every
$v_{1},v_{2}\in\,\mathscr{Y}^{\epsilon,R}_{\varrho,\eta,\vartheta,T}$ and
$\delta>0$

$\begin{array}[]{l}\displaystyle{\|\gamma_{\epsilon,\delta}(v_{1},s)-\gamma_{\epsilon,\delta}(v_{2},s)\|_{\vartheta}\leq
c\,\,\epsilon^{\frac{1}{2}-\varrho}\delta\,R(1+R)\,(s\wedge
1)^{-(\varrho+\frac{1}{2})}\|v_{1}(s,\cdot)-v_{2}(s,\cdot)\|_{\vartheta}}\\\\[18.0pt]
\displaystyle{+c\,\epsilon\,\delta(1+R)\,\|D^{2}_{x}v_{1}(s,\cdot)-D^{2}_{x}v_{2}(s,\cdot)\|_{\vartheta}+c\,\|D_{x}v_{1}(s,\cdot)-D_{x}v_{2}(s,\cdot)\|_{\vartheta}.}\end{array}$
(5.7)

###### Proof.

In view of (3.6) and Hypothesis 3, we have

$\begin{array}[]{ll}\displaystyle{\|\gamma_{\epsilon,\delta}(v,s)\|_{\vartheta}}&\displaystyle{\leq
c\,\epsilon\,\delta\,\|F(v(s,\cdot))\|_{\vartheta}\,\|D_{x}^{2}v(s,\cdot)\|_{\vartheta}+\|b\|_{\vartheta}\,\|D_{x}v(s,\cdot)\|_{\vartheta}}\\\\[10.0pt]
&\displaystyle{\leq
c\,\epsilon\,\delta\left(1+\|v(s,\cdot)\|_{\vartheta}\right)\|D^{2}_{x}v(s,\cdot)\|_{\vartheta}+c\,\|D_{x}v(s,\cdot)\|_{\vartheta},}\end{array}$

and since we are assuming that
$v\in\,\mathscr{Y}^{\epsilon,R}_{\varrho,\eta,\vartheta,T}$, this implies
(5.6).

Next, if $v_{1},v_{2}\in\,\mathscr{Y}^{\epsilon,R}_{\varrho,\eta,\vartheta,T}$
and $\delta>0$ we have

$\begin{array}[]{l}\displaystyle{\|\gamma_{\epsilon,\delta}(v_{1},s)-\gamma_{\epsilon,\delta}(v_{2},s)\|_{\vartheta}\leq
c\,\epsilon\,\delta\,\|F(v_{1}(s,\cdot))-F(v_{1}(s,\cdot))\|_{\vartheta}\,\|D_{x}^{2}v_{1}(s,\cdot)\|_{\vartheta}}\\\\[18.0pt]
\displaystyle{+c\,\epsilon\,\delta\,\|F(v_{2}(s,\cdot))\|_{\vartheta}\,\|D_{x}^{2}v_{1}(s,\cdot)-D_{x}^{2}v_{2}(s,\cdot)\|_{\vartheta}+\|b\|_{\vartheta}\,\|D_{x}v_{1}(s,\cdot)-D_{x}v_{2}(s,\cdot)\|_{\vartheta}.}\end{array}$

Thus, according to (3.6) and (3.7),

$\begin{array}[]{l}\displaystyle{\|\gamma_{\epsilon,\delta}(v_{1},s)-\gamma_{\epsilon,\delta}(v_{2},s)\|_{\vartheta}}\\\\[14.0pt]
\displaystyle{\leq
c\,\epsilon\,\delta\,\|v_{1}(s,\cdot)-v_{2}(s,\cdot)\|_{\vartheta}\left(1+\|v_{1}(s,\cdot)\|_{\vartheta}+\|v_{2}(s,\cdot)\|_{\vartheta}\right)\|D^{2}_{x}v_{1}(s,\cdot)\|_{\vartheta}}\\\\[18.0pt]
\displaystyle{+c\,\epsilon\,\delta\left(1+\|v_{2}(s,\cdot)\|_{\vartheta}\right)\|D^{2}_{x}v_{1}(s,\cdot)-D^{2}_{x}v_{2}(s,\cdot)\|_{\vartheta}+c\,\|D_{x}v_{1}(s,\cdot)-D_{x}v_{2}(s,\cdot)\|_{\vartheta}.}\end{array}$

Recalling that
$v_{1},v_{2}\in\,\mathscr{Y}^{\epsilon,R}_{\varrho,\eta,\vartheta,T}$, this
implies (5.7).

∎

######  Remark 5.2.

If for every fixed $\epsilon,\delta>0$ we define

$\alpha_{\epsilon,\delta}(R,s):=\epsilon^{\frac{1}{2}-\varrho}\delta
R(1+R)(s\wedge 1)^{-(\varrho+\frac{1}{2})}+\epsilon^{-\varrho}R(s\wedge
1)^{-\varrho},\ \ \ \ \ s>0,\ \ \ R>0,$

and

$\mathfrak{a}_{\epsilon,\delta}(R,s):=\epsilon^{\frac{1}{2}-\varrho}\delta(1+R)^{2}(s\wedge
1)^{-(\varrho+\frac{1}{2})}+\epsilon^{-\varrho}(s\wedge 1)^{-\varrho},\ \ \ \
\ s>0,\ \ \ R>0,$

due (5.6) and (5.7) we have that for every
$v,v_{1},v_{2}\in\,\mathscr{Y}^{\epsilon,R}_{\varrho,\eta,\vartheta,T}$ and
$s\in\,(0,T]$

$\|\gamma_{\epsilon,\delta}(v,s)\|_{\vartheta}\leq
c\,\alpha_{\epsilon,\delta}(R,s),$ (5.8)

and

$\|\gamma_{\epsilon,\delta}(v_{1},s)-\gamma_{\epsilon,\delta}(v_{2},s)\|_{\vartheta}\leq
c\,\mathfrak{a}_{\epsilon,\delta}(R,s)\,\|v_{1}-v_{2}\|_{\epsilon,\varrho,\eta,\vartheta,T}.$
(5.9)

Notice that for all $\beta<1$ and $\mu>0$ and for all $t\geq 0$

$\begin{array}[]{l}\displaystyle{\int_{0}^{t}((t-s)\wedge
1)^{-\beta}e^{-\mu(t-s)}\alpha_{\epsilon,\delta}(R,s)\,ds}\\\\[14.0pt]
\displaystyle{\leq\epsilon^{\frac{1}{2}-\varrho}\delta
R(1+R)\int_{0}^{t}((t-s)\wedge 1)^{-\beta}e^{-\mu(t-s)}(s\wedge
1)^{-(\varrho+\frac{1}{2})}\,ds}\\\\[14.0pt]
\displaystyle{+\epsilon^{-\varrho}R\int_{0}^{t}((t-s)\wedge
1)^{-\beta}e^{-\mu(t-s)}(s\wedge 1)^{-\varrho}\,ds,}\end{array}$

and this implies that there exists some constant $c>0$ only dependent on
$\beta$ and $\mu$ such that

$\int_{0}^{t}((t-s)\wedge
1)^{-\beta}e^{-\mu(t-s)}\alpha_{\epsilon,\delta}(R,s)\,ds\leq c\,(t\wedge
1)^{\frac{1}{2}-(\varrho+\beta)}\epsilon^{\frac{1}{2}-\varrho}\lambda_{\epsilon,\delta}(T,t),$
(5.10)

where

$\lambda_{\epsilon,\delta}(R,t):=\delta
R(1+R)\,+\epsilon^{\frac{1}{2}}R\,(t\wedge 1)^{\frac{1}{2}}.$

In an analogous way

$\int_{0}^{t}(t-s)^{-\beta}e^{-\mu(t-s)}\mathfrak{a}_{\epsilon,\delta}(R,s)\,ds\leq
c\,(t\wedge
1)^{\frac{1}{2}-(\varrho+\beta)}\epsilon^{\frac{1}{2}-\varrho}\mathfrak{l}_{\epsilon,\delta}(R,t),$
(5.11)

where

$\mathfrak{l}_{\epsilon,\delta}(R,t):=\delta(1+R)^{2}\,+\epsilon^{\frac{1}{2}}(t\wedge
1)^{\frac{1}{2}}.$

Next we prove the following estimates for $\Gamma_{\epsilon,\delta}$ on
$\mathscr{Y}^{\epsilon,R}_{\varrho,\eta,\vartheta,T}$.

######  Lemma 5.3.

For every $v\in\,\mathscr{Y}^{\epsilon,R}_{\epsilon,\varrho,\eta,\vartheta,T}$
and $\epsilon,\delta\in\,(0,1)$ it holds

$\|\Gamma_{\epsilon,\delta}(v)\|_{\epsilon,\varrho,\eta,\vartheta,T}\leq
c\,\lambda_{\epsilon,\delta}(R,T)\left[\epsilon^{\frac{1-(\eta-\vartheta)}{2}-\varrho}\,(T\wedge
1)^{\frac{1-(\eta-\vartheta)}{2}-\varrho}\,(T\vee 1)+1\right].$ (5.12)

###### Proof.

Step 1. We have

$\|\Gamma_{\epsilon,\delta}(v)(t)\|_{\eta}\leq
c\,\epsilon^{\frac{1-(\eta-\vartheta)}{2}-\varrho}\lambda_{\epsilon,\delta}(R,t)\,(t\wedge
1)^{\frac{1-(\eta-\vartheta)}{2}-\varrho}(t\vee 1),\ \ \ \ \ t\in\,[0,T].$
(5.13)

Proof of Step 1. In view of (5.5), for every $t\in\,[0,T]$

$\begin{array}[]{ll}\displaystyle{\|\Gamma_{\epsilon,\delta}(v)(t)\|_{\eta}}&\displaystyle{\leq\int_{0}^{t}\|R^{\epsilon}_{t-s}\gamma_{\epsilon,\delta}(v,s)\|_{\eta}\,ds\leq
c\,\epsilon^{-\frac{\eta-\vartheta}{2}}\int_{0}^{t}((t-s)\wedge
1)^{-\frac{\eta-\vartheta}{2}}\|\gamma_{\epsilon,\delta}(v,s)\|_{\vartheta}\,ds}\\\\[14.0pt]
&\displaystyle{\leq
c\,\epsilon^{-\frac{\eta-\vartheta}{2}}\int_{0}^{t}((t-s)\wedge
1)^{-\frac{\eta-\vartheta}{2}}\alpha_{\epsilon,\delta}(R,s)\,ds.}\end{array}$

Then, by adapting (5.10) to the case $\mu=0$, we get

$\|\Gamma_{\epsilon,\delta}(v)(t)\|_{\eta}\leq
c\,\epsilon^{-\frac{\eta-\vartheta}{2}}\epsilon^{\frac{1}{2}-\varrho}\lambda_{\epsilon,\delta}(R,t)(t\wedge
1)^{\frac{1-(\eta-\vartheta)}{2}-\varrho}\,(t\vee 1),$

and (5.13) follows.

Step 2. We have

$(t\wedge 1)^{\varrho}\,\|D\Gamma_{\epsilon,\delta}(v)(t)\|_{\vartheta}\leq
c\,\epsilon^{-\varrho}\left[\delta\,R(1+R)+\epsilon^{-\frac{1}{2}}R(t\wedge
1)^{\frac{1}{2}}\right].$ (5.14)

Proof of Step 2. According to (5.2), we have that

$\begin{array}[]{l}\displaystyle{\int_{0}^{t}\|DR^{\epsilon}_{t-s}\gamma_{\epsilon,\delta}(v,s)\|_{\vartheta}\,ds\leq
c\epsilon^{-\frac{1}{2}}\int_{0}^{t}((t-s)\wedge
1)^{-\frac{1}{2}}e^{-\lambda(t-s)}\|\gamma_{\epsilon,\delta}(v,s)\|_{\vartheta}\,ds.}\end{array}$

Then, thanks to (5.8) and (5.10) we conclude

$\begin{array}[]{l}\displaystyle{\int_{0}^{t}\|DR^{\epsilon}_{t-s}\Gamma_{\epsilon,\delta}(v,s)\|_{\vartheta}\,ds\leq
c\,\epsilon^{-\frac{1}{2}}(t\wedge
1)^{-\varrho}\epsilon^{\frac{1}{2}-\varrho}\lambda_{\epsilon,\delta}(R,t),}\end{array}$

and (5.14) follows.

Step 3. We have

$(t\wedge
1)^{\varrho+\frac{1}{2}}\,\|D^{2}\Gamma_{\epsilon,\delta}(v)(t)\|_{\vartheta}\leq
c\,\epsilon^{-(\frac{1}{2}+\varrho)}\lambda_{\epsilon,\delta}(T,t).$ (5.15)

Proof of Step 3. By proceeding as in the proof of Step 2, we have

$\begin{array}[]{ll}\displaystyle{\|D^{2}\Gamma_{\epsilon,\delta}(v)(t)\|_{0}}&\displaystyle{\leq
c\epsilon^{-1+\frac{\vartheta}{2}}\int_{0}^{t}((t-s)\wedge
1)^{-1+\frac{\vartheta}{2}}e^{-\lambda(t-s)}\alpha_{\epsilon,\delta}(R,s)\,ds}\\\\[18.0pt]
&\displaystyle{\leq
c\,\epsilon^{-1+\frac{\vartheta}{2}}\epsilon^{\frac{1}{2}-\varrho}(t\wedge
1)^{-(\varrho+\frac{1}{2})+\frac{\vartheta}{2}}\lambda_{\epsilon,\delta}(R,t),}\end{array}$

and this implies

$(t\wedge
1)^{\varrho+\frac{1}{2}}\,\|D^{2}\Gamma_{\epsilon,\delta}(v)(t)\|_{0}\leq
c\,\epsilon^{-(\frac{1}{2}+\varrho)+\frac{\vartheta}{2}}\lambda_{\epsilon,\delta}(R,t)(t\wedge
1)^{\frac{\vartheta}{2}}.$ (5.16)

Now, for every $x,h\in\,H$ and $t\in\,[0,T]$, we have

$\begin{array}[]{l}\displaystyle{(t\wedge
1)^{\varrho+\frac{1}{2}}\|D^{2}\Gamma_{\epsilon,\delta}(v)(t,x+h)-D^{2}\Gamma_{\epsilon,\delta}(v)(t,x)\|_{\mathscr{L}(H)}\leq
c\,\epsilon^{-(\frac{1}{2}+\varrho)+\frac{\vartheta}{2}}\lambda_{\epsilon,\delta}(R,t)\,(t\wedge
1)^{\frac{\vartheta}{2}}.}\end{array}$

Hence, if we assume that $\|h\|_{H}^{2}>\epsilon\,t/2$ we get

$\begin{array}[]{l}\displaystyle{(t\wedge
1)^{\varrho+\frac{1}{2}}\,\|D^{2}\Gamma_{\epsilon,\delta}(v)(t,x+h)-D^{2}\Gamma_{\epsilon,\delta}(v)(t,x)\|_{\mathscr{L}(H)}\leq
c\,\epsilon^{-(\frac{1}{2}+\varrho)}\lambda_{\epsilon,\delta}(R,t)\,\|h\|_{H}^{\vartheta}.}\end{array}$
(5.17)

When $\|h\|_{H}^{2}\leq\epsilon\,t/2$ we have

$\begin{array}[]{ll}\displaystyle{D^{2}\Gamma_{\epsilon,\delta}(v)(t,x)}&\displaystyle{=\int_{0}^{\epsilon^{-1}\|h\|_{H}^{2}}D^{2}R^{\epsilon}_{s}\gamma_{\epsilon,\delta}(v,t-s)(x)\,ds+\int_{\epsilon^{-1}\|h\|_{H}^{2}}^{t}D^{2}R^{\epsilon}_{s}\gamma_{\epsilon,\delta}(v,t-s)(x)\,ds}\\\\[18.0pt]
&\displaystyle{=:a_{\epsilon,\delta}(h,t,x)+b_{\epsilon,\delta}(h,t,x).}\end{array}$

Due to (5.3), from (5.8) we have

$\begin{array}[]{l}\displaystyle{|a_{\epsilon,\delta}(h,t,x+h)-a_{\epsilon,\delta}(h,t,x)|\leq
c\,\epsilon^{-1+\frac{\vartheta}{2}}\int_{0}^{\epsilon^{-1}\|h\|_{H}^{2}}(s\wedge
1)^{-1+\frac{\vartheta}{2}}\|\gamma_{\epsilon,\delta}(v,t-s)\|_{\vartheta}\,ds}\\\\[18.0pt]
\displaystyle{\leq
c\,\epsilon^{-1+\frac{\vartheta}{2}}\int_{0}^{\epsilon^{-1}\|h\|_{H}^{2}}s^{-1+\frac{\vartheta}{2}}e^{-\lambda
s}\alpha_{\epsilon,\delta}(R,s)\,ds\leq
c\,\epsilon^{-(\frac{1}{2}+\varrho)}\lambda_{\epsilon,\delta}(R,t)\,(t\wedge
1)^{-(\varrho+\frac{1}{2})}\,\|h\|_{H}^{\vartheta}.}\end{array}$ (5.18)

As for $b_{\epsilon,\delta}(h,t,\cdot)$, we have

$\begin{array}[]{l}\displaystyle{b_{\epsilon,\delta}(h,t,x+h)-b_{\epsilon,\delta}(h,t,x)=\int_{\epsilon^{-1}\|h\|_{H}^{2}}^{t}\left[D^{2}R^{\epsilon}_{s}\gamma_{\epsilon,\delta}(v,t-s)(x+h)-D^{2}R^{\epsilon}_{s}\gamma_{\epsilon,\delta}(v,t-s)(x)\right]\,ds.}\end{array}$

Hence, due again to (5.3), we have

$\begin{array}[]{l}\displaystyle{\|b_{\epsilon,\delta}(h,t,x+h)-b_{\epsilon,\delta}(h,t,x)\|_{\mathscr{L}(H)}}\\\\[10.0pt]
\displaystyle{\leq\int_{\epsilon^{-1}\|h\|_{H}^{2}}^{t}\|D^{2}R^{\epsilon}_{s}\gamma_{\epsilon,\delta}(v,t-s)(x+h)-D^{2}R^{\epsilon}_{s}\gamma_{\epsilon,\delta}(v,t-s)(x)\|_{\mathscr{L}(H)}\,ds}\\\\[18.0pt]
\displaystyle{\leq
c\,\int_{\epsilon^{-1}\|h\|_{H}^{2}}^{t}\|R^{\epsilon}_{s}\gamma_{\epsilon,\delta}(v,t-s)\|_{3}\,ds\|h\|_{H}}\\\\[18.0pt]
\displaystyle{\leq
c\,\epsilon^{-\frac{3-\vartheta}{2}}\epsilon^{\frac{1}{2}-\varrho}\int_{\epsilon^{-1}\|h\|_{H}^{2}}^{t}(s\wedge
1)^{-\frac{3-\vartheta}{2}}e^{-2\lambda s}((t-s)\wedge
1)^{-(\varrho+\frac{1}{2})}\lambda_{\epsilon,\delta}(R,t-s)\,ds.}\end{array}$

Since we assuming $\|h\|_{H}^{2}\leq\epsilon\,t/2$, we have

$\begin{array}[]{l}\displaystyle{\int_{\epsilon^{-1}\|h\|_{H}^{2}}^{t}(s\wedge
1)^{-\frac{3-\vartheta}{2}}e^{-2\lambda s}((t-s)\wedge
1)^{-(\varrho+\frac{1}{2})})\,ds}\\\\[18.0pt]
\displaystyle{=\int_{\epsilon^{-1}\|h\|_{H}^{2}}^{t/2}(s\wedge
1)^{-\frac{3-\vartheta}{2}}e^{-2\lambda s}((t-s)\wedge
1)^{-(\varrho+\frac{1}{2})}\,ds+\int_{t/2}^{t}s^{-\frac{3-\vartheta}{2}}e^{-2\lambda
s}((t-s)\wedge 1)^{-(\varrho+\frac{1}{2})}\,ds}\\\\[18.0pt] \displaystyle{\leq
c\,t^{-(\varrho+\frac{1}{2})}\,\|h\|_{H}^{-1+\vartheta}+c\,t^{-\frac{3-\vartheta}{2}}\,t^{\frac{1}{2}-\varrho}=c\,t^{-\varrho}\left(\epsilon^{\frac{1-\vartheta}{2}}\|h\|_{H}^{-1+\vartheta}+t^{-\frac{1-\vartheta}{2}}\right)}\\\\[14.0pt]
\displaystyle{\leq c\,\epsilon^{\frac{1-\vartheta}{2}}(t\wedge
1)^{-(\varrho+\frac{1}{2})}\,\|h\|_{H}^{-1+\vartheta}.}\end{array}$

Moreover, in the same way we have

$\int_{\epsilon^{-1}\|h\|_{H}^{2}}^{t}s^{-\frac{3-\vartheta}{2}}e^{-2\lambda
s}((t-s)\wedge 1)^{-\varrho}\,ds\leq
c\,\epsilon^{\frac{1-\vartheta}{2}}(t\wedge
1)^{-\varrho}\,\|h\|_{H}^{-1+\vartheta},$

so that

$(t\wedge
1)^{\varrho+\frac{1}{2}}\,\|b_{\epsilon,\delta}(h,t,x+h)-b_{\epsilon,\delta}(h,t,x)\|_{\mathscr{L}(H)}\leq
c\,\epsilon^{-(\frac{1}{2}+\varrho)}\lambda_{\epsilon,\delta}(R,t)\|h\|_{H}^{\vartheta}.$

This, together with (5.18) and (5.17), implies that for every $h\in\,H$

$\begin{array}[]{l}\displaystyle{(t\wedge
1)^{\varrho+\frac{1}{2}}\|D^{2}\Gamma_{\epsilon,\delta}(v)(t,x+h)-D^{2}\Gamma_{\epsilon,\delta}(v)(t,x)\|_{\mathscr{L}(H)}\leq
c\,\epsilon^{-(\frac{1}{2}+\varrho)}\lambda_{\epsilon,\delta}(R,t)\,\|h\|_{H}^{\vartheta}.}\end{array}$

Thus, thanks to (5.16), we obtain (5.15).

Conclusion. Estimate (5.12) is a consequence of (5.13), (5.14) and (5.15). ∎

######  Remark 5.4.

From the proof of the previous lemma, we easily see that for every
$t\in\,(0,T]$ and $\epsilon\in\,(0,1)$

$\begin{array}[]{l}\displaystyle{\epsilon^{\frac{\eta-\vartheta}{2}}[\Gamma_{\epsilon,\delta}(v)(t)]_{\eta}+\epsilon^{\varrho}(t\wedge
1)^{\varrho}\|D\Gamma_{\epsilon,\delta}(v)(t)\|_{\vartheta}+\epsilon^{\frac{1}{2}+\varrho}(t\wedge
1)^{\varrho+\frac{1}{2}}\|D^{2}\Gamma_{\epsilon,\delta}(v)(t)\|_{\vartheta}}\\\\[18.0pt]
\displaystyle{\leq c\,\lambda_{\epsilon,\delta}(R,t)\left((t\wedge
1)^{\frac{1-(\eta-\vartheta)}{2}-\varrho}\epsilon^{\frac{1-(\eta-\vartheta)}{2}-\varrho}+1\right),}\end{array}$
(5.19)

for some constant $c>0$ independent of $T>0$. Actually, in view of (5.4) and
(5.8), we have

$\begin{array}[]{ll}\displaystyle{[\Gamma_{\epsilon,\delta}(v)(t)]_{\eta}}&\displaystyle{\leq
c\,\epsilon^{-\frac{\eta-\vartheta}{2}}\int_{0}^{t}e^{-\omega_{\vartheta,\eta}(t-s)}((t-s)\wedge
1)^{-\frac{\eta-\vartheta}{2}}[\gamma_{\epsilon,\delta}(v,s)]_{\vartheta}\,ds}\\\\[14.0pt]
&\displaystyle{\leq
c\,\epsilon^{-\frac{\eta-\vartheta}{2}}\int_{0}^{t}e^{-\omega_{\vartheta,\eta}(t-s)}((t-s)\wedge
1)^{-\frac{\eta-\vartheta}{2}}\alpha_{\epsilon,\delta}(R,s)\,ds}\\\\[18.0pt]
&\displaystyle{\leq
c\,\lambda_{\epsilon,\delta}(R,t)\left(\epsilon^{\frac{1-(\eta-\vartheta)}{2}-\varrho}(t\wedge
1)^{\frac{1-(\eta-\vartheta)}{2}-\varrho}+1\right).}\end{array}$

This, together with (5.14) and (5.15), implies (5.19).

Now we are ready to prove the existence of a local mild solution.

######  Theorem 5.5.

Fix $\eta>1/2$ and $\vartheta\in\,(0,(\eta-1/2)\wedge 1)$ and define

$\varrho:=\frac{1-(\eta-\vartheta)}{2}.$ (5.20)

Then, there exist $\delta_{1},T_{1}>0$ such that for every
$\epsilon\in\,(0,1)$ problem (3.17) has a mild solution $u_{\epsilon}$ in
$C_{\epsilon,\varrho,\eta}((0,T_{1}];C^{2+\vartheta}_{b}(H))$, for every
$\delta\leq\delta_{1}$.

###### Proof.

A function $u_{\epsilon}$ is a mild solution of equation (3.17) if and only if
it is a fixed point for the mapping $\Gamma_{\epsilon,\delta}^{g}$ defined by

$\Gamma_{\epsilon,\delta}^{g}(v)(t)=R^{\epsilon}_{t}g+\Gamma_{\epsilon,\delta}(v)(t),\
\ \ \ t\in\,[0,T].$

Thus, we will prove the existence of a local mild solution for equation (3.17)
by showing that there exist some $T_{1},R>0$ and $\delta_{1}>0$ such that
$\Gamma_{\epsilon,\delta}^{g}$ maps
$\mathscr{Y}^{\epsilon,R}_{\varrho,\vartheta,T_{1}}$ into itself as a
contraction, for every $\delta\leq\delta_{1}$.

Thanks to (5.5) we have

$\|R^{\epsilon}_{t}g\|_{\eta}\leq c\,\|g\|_{\eta}.$ (5.21)

Moreover, thanks to (5.3)

$\begin{array}[]{l}\displaystyle{\|DR^{\epsilon}_{t}g(v)(t)\|_{\vartheta}\leq
c\,\epsilon^{-\frac{1-(\eta-\vartheta)}{2}}(t\wedge
1)^{-\frac{1-(\eta-\vartheta)}{2}}\,e^{-\lambda t}\,\|g\|_{\eta},}\\\\[10.0pt]
\displaystyle{\|D^{2}R^{\epsilon}_{t}g(v)(t)\|_{\vartheta}\leq
c\,\epsilon^{-(1-\frac{\eta-\vartheta}{2})}(t\wedge
1)^{-(1-\frac{\eta-\vartheta}{2})}\,e^{-2\lambda
t}\,\|g\|_{\eta}.}\end{array}$ (5.22)

Therefore, if we define $\varrho$ as in (5.20), we have that
$\varrho\in\,(0,1/4)$ and from (5.21) and (5.22) it follows

$\|R^{\epsilon}_{t}g\|_{\eta}+\epsilon^{\varrho}(t\wedge
1)^{\varrho}\|DR^{\epsilon}_{t}g(t)\|_{\vartheta}+\epsilon^{\frac{1}{2}+\varrho}(t\wedge
1)^{\varrho+\frac{1}{2}}\|D^{2}R^{\epsilon}_{t}g(t)\|_{\vartheta}\leq
c\,\|g\|_{\eta}.$ (5.23)

With $\varrho$ defined as in (5.20), together with (5.12) this implies that
for every $v\in\,\mathscr{Y}^{\epsilon,R}_{\varrho,\eta,\vartheta,T}$

$\|\Gamma_{\epsilon,\delta}^{g}(v)\|_{\epsilon,\varrho,\eta,\vartheta,T}\leq
c\,\|g\|_{\eta}+c\,\left[\delta\,R(1+R)+\epsilon^{-\frac{1}{2}}R\,(T\wedge
1)^{\frac{1}{2}}\right](T\vee 1).$

In particular, if we first take $R:=3c\,\|g\|_{\eta}$ and $\delta^{\prime}>0$
small enough such that

$c\,\delta^{\prime}\,R(1+R)\leq\frac{R}{3},$

and then fix $T^{\prime}\leq 1$ small enough so that

$\epsilon^{-\frac{1}{2}}R(T^{\prime}\wedge 1)^{\frac{1}{2}}\leq\frac{R}{3},$

we conclude that for every $\delta\leq\delta^{\prime}$ and $T\leq T^{\prime}$

$\|\Gamma_{\epsilon,\delta}^{g}(v)\|_{\epsilon,\varrho,\eta,\vartheta,T}\leq
R,$

so that $\Gamma_{\epsilon,\delta}^{g}$ maps
$\mathscr{Y}^{\epsilon,R}_{\varrho,\eta,\vartheta,T}$ into itself.

Now, if we fix
$v_{1},v_{2}\in\,\mathscr{Y}^{\epsilon,R}_{\varrho,\eta,\vartheta,T}$, we have

$\begin{array}[]{l}\displaystyle{\|\Gamma_{\epsilon,\delta}^{g}(v_{1})-\Gamma_{\epsilon,\delta}^{g}(v_{2})\|_{\epsilon,\varrho,\eta,\vartheta,T}\leq
c\int_{0}^{T}(t-s)^{-\frac{\eta-\vartheta}{2}}\|\gamma_{\epsilon,\delta}(v_{1},s)-\gamma_{\epsilon,\delta}(v_{2},s)\|_{\vartheta}\,ds}\\\\[18.0pt]
\displaystyle{+\epsilon^{\varrho}\sup_{t\in\,(0,T]}(t\wedge
1)^{\varrho}\int_{0}^{t}\epsilon^{-\frac{1}{2}}((t-s)\wedge
1)^{-\frac{1}{2}}e^{-\lambda(t-s)}\|\gamma_{\epsilon,\delta}(v_{1},s)-\gamma_{\epsilon,\delta}(v_{2},s)\|_{\vartheta}\,ds}\\\\[18.0pt]
\displaystyle{+\epsilon^{\frac{1}{2}+\varrho}\sup_{t\in\,(0,T]}(t\wedge
1)^{\varrho+\frac{1}{2}}\int_{0}^{t}\epsilon^{-(1-\frac{\vartheta}{2})}(t-s)^{-(1-\frac{\vartheta}{2})}e^{-2\lambda(t-s)}\|\gamma_{\epsilon,\delta}(v_{1},s)-\gamma_{\epsilon,\delta}(v_{2},s)\|_{\vartheta}\,ds}\\\\[18.0pt]
\displaystyle{+\epsilon^{\frac{1}{2}+\varrho}\sup_{t\in\,(0,T]}(t\wedge
1)^{\varrho+\frac{1}{2}}\,[D^{2}\Gamma_{\epsilon,\delta}^{g}(v_{1})(t)-D^{2}\Gamma_{\epsilon,\delta}^{g}(v_{2})(t)]_{\vartheta}=:\sum_{i=1}^{4}I_{\delta,i}(\epsilon).}\end{array}$

Then, according to (5.9) and (5.10), we have

$\begin{array}[]{l}\displaystyle{I_{\delta,1}(\epsilon)\leq
c\,\mathfrak{l}_{\epsilon,\delta}(R,T)\,\|v_{1}-v_{2}\|_{\epsilon,\varrho,\eta,\vartheta,T}.}\end{array}$
(5.24)

In the same way,

$\begin{array}[]{ll}\displaystyle{I_{\delta,2}(\epsilon)}&\displaystyle{\leq
c\,\epsilon^{\varrho}\sup_{t\in\,(0,T]}(t\wedge
1)^{\varrho}\int_{0}^{t}\epsilon^{-\frac{1}{2}}(t-s)^{-\frac{1}{2}}e^{-\lambda(t-s)}\mathfrak{a}_{\epsilon,\delta}(R,s)\,ds\,\|v_{1}-v_{2}\|_{\epsilon,\varrho,\eta,\vartheta,T}}\\\\[18.0pt]
&\displaystyle{\leq
c\,\mathfrak{l}_{\epsilon,\delta}(R,T)\|v_{1}-v_{2}\|_{\epsilon,\varrho,\eta,\vartheta,T}.}\end{array}$
(5.25)

and

$\begin{array}[]{l}\displaystyle{I_{\delta,3}(\epsilon)}\\\\[14.0pt]
\displaystyle{\leq c\,\epsilon^{\frac{1}{2}+\varrho}\sup_{t\in\,(0,T]}(t\wedge
1)^{\varrho+\frac{1}{2}}\int_{0}^{t}\epsilon^{-(1-\frac{\vartheta}{2})}((t-s)\wedge
1)^{-(1-\frac{\vartheta}{2})}e^{-2\lambda(t-s)}\mathfrak{a}_{\epsilon,\delta}(R,s)\,ds\,\|v_{1}-v_{2}\|_{\epsilon,\varrho,\eta,\vartheta,T}}\\\\[18.0pt]
\displaystyle{\leq c\,\epsilon^{\frac{\vartheta}{2}}(T\wedge
1)^{\frac{\vartheta}{2}}\,\mathfrak{l}_{\epsilon,\delta}(R,T)\,\|v_{1}-v_{2}\|_{\epsilon,\varrho,\eta,\vartheta,T}.}\end{array}$
(5.26)

As for $I_{\delta,4}(\epsilon)$, due to (5.26), if we fix any $x,h\in\,H$ and
assume $\|h\|_{H}^{2}>\epsilon\,t/2$ we have

$\begin{array}[]{l}\displaystyle{\epsilon^{\varrho+\frac{1}{2}}(t\wedge
1)^{\varrho+\frac{1}{2}}\,\|D^{2}\Gamma_{\epsilon,\delta}^{g}(v_{1})(t,x+h)-D^{2}\Gamma_{\epsilon,\delta}^{g}(v_{2})(t,x)\|_{\mathscr{L}(H)}}\\\\[18.0pt]
\displaystyle{\leq c\,\epsilon^{\frac{\vartheta}{2}}(t\wedge
1)^{\frac{\vartheta}{2}}\mathfrak{l}_{\epsilon,\delta}(R,t)\|v_{1}-v_{2}\|_{\epsilon,\varrho,\eta,\vartheta,T}\leq
c\,\mathfrak{l}_{\epsilon,\delta}(R,t)\,\|v_{1}-v_{2}\|_{\epsilon,\varrho,\eta,\vartheta,T}\|h\|_{H}^{\vartheta}.}\end{array}$
(5.27)

On the other hand, if we assume that $\|h\|_{H}^{2}\leq\epsilon\,t/2$ we write

$D^{2}\Gamma_{\epsilon,\delta}^{g}(v_{1})(t,x)-D^{2}\Gamma_{\epsilon,\delta}^{g}(v_{2})(t,x)=a_{\epsilon,\delta}(h,t,x)+b_{\epsilon,\delta}(h,t,x),$

where

$a_{\epsilon,\delta}(h,t,x):=\int_{0}^{\epsilon^{-1}\|h\|_{H}^{2}}D^{2}R^{\epsilon}_{s}\left(\gamma_{\epsilon,\delta}(v_{1},t-s)-\gamma_{\epsilon,\delta}(v_{2},t-s)\right)\,ds,$

and

$b_{\epsilon,\delta}(h,t,x)=:\int_{\epsilon^{-1}\|h\|_{H}^{2}}^{t}D^{2}R^{\epsilon}_{s}\left(\gamma_{\epsilon,\delta}(v_{1},t-s)-\gamma_{\epsilon,\delta}(v_{2},t-s)\right)\,ds.$

Then, thanks to (5.9) and (5.10), we can proceed as in Step 3 of the proof of
Lemma 5.3 and we obtain that (5.27) holds also when $\|h\|_{H}^{2}\leq t/2$.
In particular, we obtain that

$\epsilon^{\varrho+\frac{1}{2}}(t\wedge
1)^{\varrho+\frac{1}{2}}\,[D^{2}\Gamma_{\epsilon,\delta}^{g}(v_{1})(t)-D^{2}\Gamma_{\epsilon,\delta}^{g}(v_{2})(t)]_{\vartheta}\leq
c\,\mathfrak{l}_{\epsilon,\delta}(R,t)\|v_{1}-v_{2}\|_{\epsilon,\varrho,\eta,\vartheta,T},$

so that

$I_{\delta,4}(\epsilon)\leq
c\,\mathfrak{l}_{\epsilon,\delta}(R,t)\|v_{1}-v_{2}\|_{\epsilon,\varrho,\eta,\vartheta,T}.$
(5.28)

Therefore, if we combine (5.24), (5.25), (5.26) and (5.28), we obtain that

$\|\Gamma_{\epsilon,\delta}^{g}(v_{1})-\Gamma_{\epsilon,\delta}^{g}(v_{2})\|_{\epsilon,\varrho,\eta,\vartheta,T}\leq
c\,\mathfrak{l}_{\epsilon,\delta}(R,t)\|v_{1}-v_{2}\|_{\epsilon,\varrho,\eta,\vartheta,T}.$

This means that if we first choose $\delta_{1}\leq\delta^{\prime}$ such that

$c\,\delta_{1}\,(1+R)^{2}<\frac{1}{2},$

and then $T_{1}\leq T^{\prime}$ such that

$c\,\epsilon^{-\frac{1}{2}}\,(T_{1}\wedge 1)^{\frac{1}{2}}<\frac{1}{2},$

we can conclude that $\Gamma_{\delta,g}$ maps
$\mathscr{Y}^{R}_{\varrho,\vartheta,T_{1}}$ into itself as a contraction, for
every $\delta\leq\delta_{1}.$ ∎

##  6\. Further properties of mild solutions of the quasi-linear problem

We will show that any mild solution $u_{\epsilon}$ of equation (3.17) that
belongs to $C_{\epsilon,\varrho,\eta}((0,T];C^{2+\vartheta}_{b}(H))$ is in
fact a classical solution in the sense of Definition 2. Moreover, by using its
probabilistic interpretation in terms of equation (3.19), we will prove that a
maximum principle holds for equation (3.17). This will imply that the local
mild solution we have found in Section 5 is the unique global classical
solution of Theorem 3.4.

We start by proving that $QD^{2}_{x}u_{\epsilon}(t,x)$ is a trace-class
operator.

######  Lemma 6.1.

For every $t\in\,(0,T]$ and $x\in\,H$, we have that
$QD^{2}_{x}u_{\epsilon}(t,x)\in\,\mathscr{L}_{1}(H)$.

###### Proof.

If $u_{\epsilon}$ is a mild solution, with the notations we have introduced in
Section 5 we have

$u_{\epsilon}(t,x)=R^{\epsilon}_{t}g(x)+\Gamma_{\epsilon,\delta}(u_{\epsilon})(t,x).$

According to (2.17) we have that
$QD^{2}_{x}R^{\epsilon}_{t}g(t,x)\in\,\mathscr{L}_{1}(H)$, for every
$\epsilon\in\,(0,1)$, $t>0$ and $x\in\,H$, and thanks to (2.17) and (3.14)

$\|QD^{2}_{x}R^{\epsilon}_{t}g(t)\|_{0}\leq\kappa_{\epsilon,\eta}(t/2)\,\|g\|_{\eta}.$

As far as $\Gamma_{\epsilon,\delta}(u)$ is concerned, if
$R=\|u_{\epsilon}\|_{\epsilon,\varrho,\eta,\vartheta,T}$, thanks to (2.17),
(3.14) and (5.8), we have

$\begin{array}[]{l}\displaystyle{\|QD^{2}\Gamma_{\epsilon,\delta}(u)(t,x)\|_{\mathscr{L}_{1}(H)}\leq\int_{0}^{t}\|QD^{2}_{x}R^{\epsilon}_{t-s}\gamma_{\epsilon,\delta}(u,s)(x)\|_{\mathscr{L}_{1}(H)}\,ds\leq\int_{0}^{t}\kappa_{\epsilon,\vartheta}(t-s)\|\gamma_{\epsilon,\delta}(s)\|_{\vartheta}\,ds}\\\\[18.0pt]
\displaystyle{\leq c\,e^{-\beta_{\vartheta}}\int_{0}^{t}((t-s)\wedge
1)^{-\beta_{\vartheta}}e^{-\alpha_{\vartheta}\frac{(t-s)}{2}}\epsilon^{-(\varrho+\frac{1}{2})}(s\wedge
1)^{-(\frac{1}{+}\varrho)}\lambda_{\epsilon,\delta}(R,s)\,ds\leq
c_{\epsilon}(R,T).}\end{array}$

This allows to conclude that
$QD^{2}_{x}u_{\epsilon}(t,x)\in\,\mathscr{L}_{1}(H)$ for every
$\epsilon\in\,(0,1)$, $t\in\,(0,T]$ and $x\in\,H$.

∎

Next, we show that $u_{\epsilon}$ is differentiable with respect to
$t\in\,(0,T]$ and $x\in\,D(A)$ and is a classical solution of equation (3.17).
In Subsection 2.3, we have seen that for every $\varphi\in\,B_{b}(H)$ and
$x\in\,D(A)$ the mapping

$t\in\,(0,+\infty)\mapsto R^{\epsilon}_{t}\varphi(x)\in\,\mathbb{R},$

is differentiable and

$D_{t}R^{\epsilon}_{t}\varphi(x)=\mathscr{L}_{\epsilon}R^{\epsilon}_{t}\varphi(x).$

Hence, since

$\begin{array}[]{ll}\displaystyle{u_{\epsilon}(t,x)}&\displaystyle{=R^{\epsilon}_{t}g(x)+\int_{0}^{t}R^{\epsilon}_{t-s}\left(\frac{\epsilon}{2}\text{Tr}\,[\delta\,F(u_{\epsilon}(s,\cdot))D^{2}_{x}u_{\epsilon}(s,\cdot)]+\langle
b(\cdot),Du_{\epsilon}(s,\cdot)\rangle_{H}\right)(x)\,ds}\\\\[18.0pt]
&\displaystyle{=R^{\epsilon}_{t}g(x)+\Gamma_{\epsilon,\delta}(u_{\epsilon})(t,x),}\end{array}$

thanks to Lemma 6.1, for every $x\in\,D(A)$ we can differentiate both sides
with respect to $t>0$, and we get

$\begin{array}[]{ll}\displaystyle{D_{t}u_{\epsilon}(t,x)}&\displaystyle{=\mathscr{L}_{\epsilon}R^{\epsilon}_{t}g(x)+\frac{\epsilon}{2}\text{Tr}\,[\delta\,F(u_{\epsilon}(t,x))D^{2}_{x}u_{\epsilon}(t,x)]+\langle
b(x),Du_{\epsilon}(t,x)\rangle_{H}+\mathscr{L}_{\epsilon}\Gamma_{\epsilon,\delta}(t)(t,x)}\\\\[18.0pt]
&\displaystyle{=\mathscr{L}_{\epsilon}u_{\epsilon}(t,x)+\frac{\epsilon}{2}\text{Tr}\,[\delta\,F(u_{\epsilon}(t,x))D^{2}_{x}u_{\epsilon}(t,x)]+\langle
b(x),Du_{\epsilon}(t,x)\rangle_{H}.}\end{array}$

Thus, we have proven the following result.

######  Theorem 6.2.

Under Hypotheses 1 to 4, if $u_{\epsilon}$ is a mild solution of equation
(3.17) that belongs to
$C_{\epsilon,\varrho,\eta}((0,T];C^{2+\vartheta}_{b}(H))$, then it is a
classical solution.

Next, we show how any solution of equation (3.17) is related to the stochastic
PDE (3.19).

######  Theorem 6.3.

Assume Hypotheses 1 to 4. Then if
$u_{\epsilon}\in\,C_{\epsilon,\varrho,\eta}((0,T];C^{2+\vartheta}_{b}(H))$ is
a solution of equation (3.17) and
$X_{\epsilon}^{t,x}\in\,L^{2}(\Omega;C([0,t];H))$ is a solution of equation
(3.19), we have

$u_{\epsilon}(t,x)=\mathbb{E}g(X_{\epsilon}^{t,x}(t)).$ (6.1)

###### Proof.

The natural way to prove (6.1) is by applying the Itô formula to the function
$(s,x)\in\,[0,t]\times H\mapsto u_{\epsilon}(t-s,x)$ and to the process
$X_{\epsilon}^{t,x}(s)$. However, we cannot do this directly first because
$u_{\epsilon}$ satisfies equation (3.17) in classical sense only for
$x\in\,D(A)$ and second because $X_{\epsilon}^{t,x}$ is only a mild solution
of equation (3.19), and not a strong solution, as required when Itô’s formula
is used. To overcome these difficulties, we introduce a suitable approximation
of $u_{\epsilon}$ and $X_{\epsilon}^{t,x}$, by adapting an argument introduced
in [7, Proof of Theorem 9.25].

For every $m\in\,\mathbb{N}$ we define $J_{m}=m(m-A)^{-1}$ and

$u_{\epsilon,m}(t,x)=u_{\epsilon}(t,J_{m}x),\ \ \ \ \ \ (t,x)\in\,[0,T]\times
H.$

Since $J_{m}x\to x$, as $m\to\infty$, and
$u_{\epsilon}\in\,C_{\epsilon,\varrho,\eta}((0,T];C^{2+\vartheta}_{b}(H))$, we
have that

$\lim_{m\to\infty}\sup_{t\in\,[0,T]}\|u_{\epsilon,m}(t,\cdot)-u(t,\cdot)\|_{0}=0.$
(6.2)

Moreover,

$D_{x}u_{\epsilon,m}(t,x)=J_{m}^{\star}D_{x}u(t,J_{m}x),\ \ \ \
D^{2}_{x}u_{\epsilon,m}(t,x)=J_{m}^{\star}D^{2}_{x}u(t,J_{m}x)J_{m}.$ (6.3)

Next, for every $m\in\,\mathbb{N}$ we introduce the stochastic PDE

$\left\\{\begin{array}[]{l}\displaystyle{dX_{\epsilon,m}^{t,x}(s)=\left[AX_{\epsilon,m}^{t,x}(s)+J_{m}b(X_{\epsilon,m}^{t,x}(s))\right]\,ds+J_{m}\Sigma_{\epsilon,t}(s,X_{\epsilon,m}^{t,x}(s))\,dW^{m}_{s},}\\\\[10.0pt]
\displaystyle{X^{t,x}(0)=J_{m}x,}\end{array}\right.$ (6.4)

where $\Sigma_{\epsilon,t}$ is the operator introduced in (4.2) and
$W^{m}_{t}$ is the projection of the cylindrical Wiener process $W_{t}$ onto
$H_{m}:=\text{span}\\{e_{1},\ldots,e_{m}\\}$. By proceeding as in Section 4,
we can prove that equation (6.4) admits a unique mild solution
$X^{t,x}_{\epsilon,m}\in\,L^{2}(\Omega;C([0,T];H))$. Since $J_{m}$ maps $H$
into $D(A)$ and $W^{m}_{t}$ is a finite dimensional noise, it is immediate to
check that $X^{t,x}_{\epsilon,m}$ is a strong solution. Namely

$X^{t,x}_{\epsilon,m}(s)=J_{m}x+\int_{0}^{s}AX^{t,x}_{\epsilon,m}(r)\,dr+\int_{0}^{s}J_{m}b(X_{\epsilon,m}^{t,x}(r))\,dr+\int_{0}^{s}J_{m}\Sigma_{\epsilon,t}(r,X_{\epsilon,m}^{t,x}(r))\,dW^{m}(r).$

At the end of this section we will prove that

$\lim_{m\to\infty}\sup_{s\in\,[0,t]}\mathbb{E}\|X^{t,x}_{\epsilon,m}(s)-X^{t,x}(s)\|_{H}=0.$
(6.5)

Now we apply Itô’s formula to $u_{\epsilon,m}$ and $X^{t,x}_{\epsilon,m}$ and
thanks to (6.3) we get

$\begin{array}[]{l}\displaystyle{d_{s}u_{\epsilon,m}(t-s,X^{t,x}_{\epsilon,m}(s))=-D_{t}u(t-s,J_{m}X^{t,x}_{\epsilon,m}(s))\,ds}\\\\[10.0pt]
\displaystyle{+\frac{1}{2}\text{Tr}\left[J_{m}^{\star}D^{2}_{x}u(t-s,J_{m}X^{t,x}_{\epsilon,m}(s))J_{m}(J_{m}\Sigma_{\epsilon,t}(r,X_{\epsilon,m}^{t,x}(r)))(J_{m}\Sigma_{\epsilon,t}(r,X_{\epsilon,m}^{t,x}(r)))^{\star}\right]\,ds}\\\\[18.0pt]
\displaystyle{+\langle
AJ_{m}X^{t,x}_{\epsilon,m}(s)+J_{m}^{2}b(X_{\epsilon,m}^{t,x}(s)),D_{x}u(t-s,J_{m}X^{t,x}_{\epsilon,m}(s))\rangle_{H}\,ds}\\\\[18.0pt]
\displaystyle{+\langle
J^{2}_{m}\Sigma_{\epsilon,t}(r,X_{\epsilon,m}^{t,x}(s))\,dW^{m}(s),D_{x}u(t-s,J_{m}X^{t,x}_{\epsilon,m}(s))\rangle_{H}.}\end{array}$

Therefore, recalling that $u(t,x)$ satisfies equation (5.1), for every
$x\in\,D(A)$, since $J_{m}X^{t,x}_{\epsilon,m}(s)\in\,D(A)$ we have

$\begin{array}[]{l}\displaystyle{d_{s}u_{\epsilon,m}(t-s,X^{t,x}_{\epsilon,m}(s))}\\\\[18.0pt]
\displaystyle{=\langle
J^{2}_{m}\Sigma_{\epsilon,t}(r,X_{\epsilon,m}^{t,x}(s))\,dW^{m}(s),D_{x}u(t-s,J_{m}X^{t,x}_{\epsilon,m}(s))\rangle_{H}+[I_{m,1}^{\epsilon}(s)+I_{m,2}(s)^{\epsilon}]\,ds,}\end{array}$
(6.6)

where

$\begin{array}[]{l}\displaystyle{I_{m,1}^{\epsilon}(s):=\frac{\epsilon}{2}\text{Tr}\left[J_{m}^{\star}D^{2}_{x}u(t-s,J_{m}X^{t,x}_{\epsilon,m}(s))J_{m}(J_{m}\Sigma_{\epsilon,t}(r,X_{\epsilon,m}^{t,x}(r)))(J_{m}\Sigma_{\epsilon,t}(r,X_{\epsilon,m}^{t,x}(r)))^{\star}\right]}\\\\[18.0pt]
\displaystyle{-\frac{\epsilon}{2}\text{Tr}\left[D^{2}_{x}u(t-s,J_{m}X^{t,x}_{\epsilon,m}(s))(\Sigma_{\epsilon,t}(r,J_{m}X_{\epsilon,m}^{t,x}(r)))(\Sigma_{\epsilon,t}(r,J_{m}X_{\epsilon,m}^{t,x}(r)))^{\star}\right],}\end{array}$

and

$I_{m,2}^{\epsilon}(s):=\begin{array}[]{l}\displaystyle{\langle
J_{m}^{2}b(X_{\epsilon,m}^{t,x}(s))-b(J_{m}X_{\epsilon,m}^{t,x}(s)),D_{x}u(t-s,J_{m}X^{t,x}_{\epsilon,m}(s))\rangle_{H}.}\end{array}$

If we take the expectation of both sides in (6.6) and integrate with respect
to $s\in\,[0,t]$ we get

$\mathbb{E}g(J_{m}X_{\epsilon,m}^{t,x}(t))=u_{\epsilon,m}(t,J_{m}x)+\int_{0}^{t}\mathbb{E}\left(I_{m,1}^{\epsilon}(s)+I_{m,2}^{\epsilon}(s)\right)\,ds.$
(6.7)

In view of (6.2) and (6.5), we have that

$\lim_{m\to\infty}\mathbb{E}g(J_{m}X_{\epsilon,m}^{t,x}(t))=\mathbb{E}g(X^{t,x}(t)),\
\ \ \ \ \ \ \ \ \lim_{m\to\infty}u_{\epsilon,m}(t,J_{m}x)=u(t,x).$

Moreover, since
$u_{\epsilon}\in\,C_{\epsilon,\varrho,\eta}((0,T];C^{2+\vartheta}_{b}(H))$, by
using again (6.2) and (6.5) it is not difficult to check that

$\lim_{m\to\infty}\int_{0}^{t}\mathbb{E}\left(|I_{m,1}^{\epsilon}(s)|+|I^{\epsilon}_{m,2}(s)|\right)\,ds=0.$

Therefore, if we take the limit of both sides in (6.7), as $m\to\infty$, we
obtain (6.1).

∎

######  Remark 6.4.

Thanks to the representation formula (6.1) of $u_{\epsilon}$, we have that

$\sup_{t\in\,[0,T]}\|u_{\epsilon}(t,\cdot)\|_{0}\leq\|g\|_{0}.$ (6.8)

Now, we conclude this section with the proof of (6.5).

######  Lemma 6.5.

If $X^{t,x}_{\epsilon,m}$ is the solution of problem (6.4), we have

$\lim_{m\to\infty}\sup_{s\in\,[0,t]}\mathbb{E}\|X^{t,x}_{\epsilon,m}(s)-X_{\epsilon}^{t,x}(s)\|^{2}_{H}=0.$
(6.9)

###### Proof.

If we denote
$\rho_{\epsilon,m}(s):=X^{t,x}_{\epsilon,m}(s)-X^{t,x}_{\epsilon}(s)$ and
$W_{m}(r)=W(r)-W^{m}(r)$, we have

$\begin{array}[]{ll}\displaystyle{\rho_{\epsilon,m}(s)=}&\displaystyle{e^{sA}(J_{m}x-x)+\int_{0}^{s}e^{(s-r)A}\left(J_{m}b(X^{t,x}_{\epsilon,m}(r))-b(X_{\epsilon}^{t,x}(r))\right)\,dr}\\\\[18.0pt]
&\displaystyle{+\int_{0}^{s}e^{(s-r)A}\left(J_{m}\Sigma_{\epsilon,t}(r,X^{t,x}_{\epsilon,m}(r))-\Sigma_{\epsilon,t}(r,X_{\epsilon}^{t,x}(r))\right)dW^{m}(r)}\\\\[18.0pt]
&\displaystyle{+\int_{0}^{s}e^{(s-r)A}\Sigma_{\epsilon,t}(r,X_{\epsilon}^{t,x}(r))dW_{m}(r).}\end{array}$

Therefore, since $\|J_{m}\|_{\mathscr{L}(H)}\leq 1$, we have

$\begin{array}[]{ll}\displaystyle{\mathbb{E}\|\rho_{\epsilon,m}(s)\|^{2}_{H}\leq}&\displaystyle{c\|J_{m}x-x\|^{2}_{H}+c_{t}\int_{0}^{s}\mathbb{E}\|\rho_{\epsilon,m}(r)\|^{2}_{H}\,dr}\\\\[18.0pt]
&\displaystyle{+c_{t}\int_{0}^{s}\mathbb{E}\|J_{m}b(X_{\epsilon}^{t,x}(r))-b(X_{\epsilon}^{t,x}(r))\|_{H}^{2}\,dr}\\\\[18.0pt]
&\displaystyle{+c\int_{0}^{s}\mathbb{E}\|e^{(s-r)A}J_{m}\left(\Sigma_{\epsilon,t}(r,X^{t,x}_{\epsilon,m}(r))-\Sigma_{\epsilon,t}(r,X_{\epsilon}^{t,x}(r))\right)\|_{\mathscr{L}_{2}(H)}^{2}\,dr}\\\\[18.0pt]
&\displaystyle{+c\int_{0}^{s}\mathbb{E}\|J_{m}e^{(s-r)A}\Sigma_{\epsilon,t}(r,X_{\epsilon}^{t,x}(r))-e^{(s-r)A}\Sigma_{\epsilon,t}(r,X_{\epsilon}^{t,x}(r))\|^{2}_{\mathscr{L}_{2}(H)}\,dr}\\\\[18.0pt]
&\displaystyle{+c\int_{0}^{s}\mathbb{E}\|e^{(s-r)A}\Sigma_{\epsilon,t}(r,X_{\epsilon}^{t,x}(r))S_{m}\|^{2}_{\mathscr{L}_{2}(H)}\,dr=:\sum_{i=1}^{6}I_{m,i}^{\epsilon}(s),}\end{array}$

where $S_{m}:=I-P_{m}$ is the projection of $H$ onto
$\text{span}\\{e_{m+1},e_{m+2},\ldots\\}$.

By proceeding as in Section 4, we have that

$I_{m,4}^{\epsilon}(s)\leq c(\epsilon)\int_{0}^{s}((s-r)\wedge
1)^{-\frac{1}{2}}\left(1+((t-r)\wedge
1)^{-2\varrho}\right)\mathbb{E}\|\rho_{\epsilon,m}(r)\|_{H}^{2}\,dr,$

so that

$\mathbb{E}\|\rho_{\epsilon,m}(s)\|^{2}_{H}\leq
c(\epsilon)\int_{0}^{s}((s-r)\wedge
1)^{-(2\varrho+\frac{1}{2})}\mathbb{E}\|\rho_{\epsilon,m}(r)\|_{H}^{2}\,dr+\Lambda_{\epsilon,m}(s),$

where

$\Lambda_{\epsilon,m}(s):=I_{m,1}^{\epsilon}(s)+I_{m,3}^{\epsilon}(s)+I_{m,5}^{\epsilon}(s)+I_{m,6}^{\epsilon}(s).$

Since $2\varrho+1/2<1$, thanks to a generalized Gronwall’s inequality (see
[16]), this implies that

$\mathbb{E}\|\rho_{\epsilon,m}(s)\|^{2}_{H}\leq
c_{\epsilon,t}\,\Lambda_{\epsilon,m}(s)\leq
c_{\epsilon,t}\,\Lambda_{\epsilon,m}(t),\\\ \ \ \ s\in\,[0,t],$

and (6.9) follows if we can prove that

$\lim_{m\to\infty}\Lambda_{\epsilon,m}(t)=0.$ (6.10)

It is immediate to check that

$\lim_{m\to\infty}I_{m,1}^{\epsilon}+I_{m,3}^{\epsilon}(t)=0.$ (6.11)

Moreover, according to Hypothesis 3 and to the fact that $u$ is bounded in
$[0,t]\times H$, for every $m\in\,\mathbb{N}$ we have

$\begin{array}[]{l}\displaystyle{\|J_{m}e^{(t-r)A}\Sigma_{\epsilon,t}(r,X_{\epsilon}^{t,x}(r))-e^{(t-r)A}\Sigma_{\epsilon,t}(r,X_{\epsilon}^{t,x}(r))\|_{\mathscr{L}_{2}(H)}}\\\\[18.0pt]
\displaystyle{\leq
2\,\|e^{(t-r)A}\sigma(X_{\epsilon}^{t,x}(r),u_{\epsilon}(t-r,X_{\epsilon}^{t,x}(r))\|_{\mathscr{L}_{2}(H)}\leq
c\,(t-r)^{-\frac{1}{4}}\left(1+\|X_{\epsilon}^{t,x}(r)\|_{H}\right).}\end{array}$

Then, since

$\lim_{m\to\infty}\|J_{m}e^{(t-r)A}\Sigma_{\epsilon,t}(r,X_{\epsilon}^{t,x}(r))-e^{(t-r)A}\Sigma_{\epsilon,t}(r,X_{\epsilon}^{t,x}(r))\|=0,$

and since the mapping

$s\in\,[0,t]\mapsto(t-s)^{-\frac{1}{4}}\left(1+\|X_{\epsilon}^{t,x}(s)\|_{H}\right)\in\,\mathbb{R},$

belongs to $L^{2}(\Omega;L^{2}([0,t]))$, by the dominated convergence theorem
we have that

$\lim_{m\to\infty}I_{m,5}^{\epsilon}(t)=0.$ (6.12)

In the same way, by the dominated convergence theorem we have also that

$\lim_{m\to\infty}I_{m,6}^{\epsilon}(t)=0.$ (6.13)

Therefore, combining together (6.11), (6.12) and (6.13), we obtain (6.10) and
(6.9) follows. ∎

##  7\. Existence and uniqueness of global classical solutions for the quasi-
linear problem

In Theorem 5.5 we have proved that for every $\eta>1/2$ and
$\vartheta\in\,(0,(\eta-1/2)\wedge 1)$, there exist $\delta_{1},T_{1}>0$ such
that problem (3.17) has a mild solution $u_{\epsilon}$ in
$C_{\epsilon,\varrho,\eta}((0,T_{1}];C^{2+\vartheta}_{b}(H))$. In Section 6 we
have shown that such mild solution is in fact a classical solution. Our
purpose here is first proving that $u_{\epsilon}$ is defined on the interval
$[0,T]$, for every $T>0$, and then proving that it is the unique solution.

We start with the following a-priori bound.

######  Lemma 7.1.

There exists $\delta_{2}\in\,(0,\delta_{1}]$, that depends only on
$\|g\|_{\eta}$, such that if $u_{\epsilon}$ is a mild solution of (3.17) for
some $\delta\leq\delta_{2}$, belonging to
$C_{\epsilon,\varrho,\eta}((0,T];C^{2+\vartheta}_{b}(H))$, then

$\|u\|_{\epsilon,\varrho,\eta,\vartheta,T}\leq c_{\epsilon}\,\|g\|_{\eta},\ \
\ \ \ \epsilon\in\,(0,1),$ (7.1)

for some constant $c_{\epsilon}$ independent of $T>0$.

###### Proof.

In what follows, for any function $v:[0,T]\times H\to\mathbb{R}$ we define

$N_{\epsilon}(v(t)):=[v(t,\cdot)]_{\eta}+\epsilon^{\varrho}(t\wedge
1)^{\varrho}\|Dv(t,\cdot)\|_{\vartheta}+\epsilon^{\varrho+\frac{1}{2}}(t\wedge
1)^{\varrho+\frac{1}{2}}\|D^{2}v(t,\cdot)\|_{\vartheta}.$

With the notations we have introduced in Section 5, thanks to (6.8) we have

$\|u_{\epsilon}(t,\cdot)\|_{0}+N_{\epsilon}(u_{\epsilon}(t))\leq\|g\|_{0}+N_{\epsilon}(R^{\epsilon}_{t}g)+N_{\epsilon}(\Gamma^{\epsilon}_{\delta,1}(u_{\epsilon})(t))+N_{\epsilon}(\Gamma^{\epsilon}_{2}(u_{\epsilon})(t)),$
(7.2)

where

$\begin{array}[]{ll}\displaystyle{\Gamma^{\epsilon}_{\delta,1}(u)(t,x)}&\displaystyle{:=\int_{0}^{t}R^{\epsilon}_{t-s}\,\gamma^{\epsilon}_{\delta,1}(u,s)(x)\,ds:=\frac{\epsilon}{2}\int_{0}^{t}R^{\epsilon}_{t-s}\text{Tr}\left[\delta
F(u)(s,\cdot)D^{2}u(s,\cdot)\right](x)\,ds}\end{array}$

and

$\begin{array}[]{l}\displaystyle{\Gamma^{\epsilon}_{2}(u)(t,x):=\int_{0}^{t}R^{\epsilon}_{t-s}\gamma_{2}(u,s)(x)\,ds:=\int_{0}^{t}R^{\epsilon}_{t-s}\langle
b,Du(s,\cdot)\rangle_{H}(x)\,ds.}\end{array}$

In (5.23) we have already shown that

$\sup_{t\in\,[0,T]}N_{\epsilon}(R^{\epsilon}_{t}g)\leq c\,\|g\|_{\eta}.$ (7.3)

Thus, in order to prove (7.1) we need to estimate
$N_{\epsilon}(\Gamma^{\epsilon}_{\delta,1}(u_{\epsilon})(t))$ and
$N_{\epsilon}(\Gamma^{\epsilon}_{2}(u_{\epsilon})(t))$.

Thanks to (3.8) and (6.8), we have

$\|F(u_{\epsilon}(s,\cdot))\|_{0}\leq
c\left(1+\|u_{\epsilon}(s,\cdot)\|_{0}\right)\leq c\left(1+\|g\|_{0}\right),$

and then

$\begin{array}[]{l}\displaystyle{\|\gamma^{\epsilon}_{\delta,1}(u_{\epsilon},s)\|_{0}\leq
c\,\epsilon\,\delta\|F(u_{\epsilon}(s,\cdot))\|_{0}\|D^{2}u_{\epsilon}(s,\cdot)\|_{0}\leq
c\,\epsilon\,\delta\left(1+\|g\|_{0}\right)\|D^{2}u_{\epsilon}(s,\cdot)\|_{0}.}\end{array}$
(7.4)

Moreover, due to (3.9) and (6.8) we have

$[F(u_{\epsilon}(s,\cdot))]_{\vartheta}\leq
c\left(1+\|u_{\epsilon}(s,\cdot)\|_{0}+[u_{\epsilon}(s,\cdot)]_{\vartheta}\right)\leq
c\left(1+\|g\|_{0}+[u_{\epsilon}(s,\cdot)]_{\vartheta}\right),$

so that

$\begin{array}[]{ll}\displaystyle{[\gamma^{\epsilon}_{\delta,1}(u_{\epsilon},s)]_{\vartheta}}&\displaystyle{\leq
c\,\epsilon\,\delta\,[F(u_{\epsilon}(s,\cdot))]_{\vartheta}\|D^{2}u_{\epsilon}(s,\cdot)\|_{0}+c\,\epsilon\,\delta\|F(u_{\epsilon}(s,\cdot))\|_{0}[D^{2}u_{\epsilon}(s,\cdot)]_{\vartheta}}\\\\[18.0pt]
&\displaystyle{\leq
c\,\epsilon\,\delta\left(1+\|g\|_{0}\right)\|D^{2}u_{\epsilon}(s,\cdot)\|_{\vartheta}+c\,\epsilon\,\delta\,[u_{\epsilon}(s,\cdot)]_{\vartheta}\|D^{2}u_{\epsilon}(s,\cdot)\|_{0}.}\end{array}$

According to (2.7) and (6.8), this implies

$\begin{array}[]{ll}\displaystyle{[\gamma^{\epsilon}_{\delta,1}(u_{\epsilon},s)]_{\vartheta}}&\displaystyle{\leq
c\,\epsilon\,\delta\left(1+\|g\|_{0}\right)\|D^{2}u_{\epsilon}(s,\cdot)\|_{\vartheta}+c\,\epsilon\,\delta\,\|u_{\epsilon}(s,\cdot)\|_{0}[D^{2}u_{\epsilon}(s,\cdot)]_{\vartheta}}\\\\[18.0pt]
&\displaystyle{\leq
c\,\epsilon\,\delta\left(1+\|g\|_{0}\right)\|D^{2}u_{\epsilon}(s,\cdot)\|_{\vartheta}.}\end{array}$
(7.5)

Therefore, if we combine together (7.4) and (7.5) we conclude that

$\begin{array}[]{ll}\displaystyle{\|\gamma^{\epsilon}_{\delta,1}(u_{\epsilon},s)\|_{\vartheta}}&\displaystyle{\leq
c\,\epsilon\,\delta\left(1+\|g\|_{0}\right)\|D^{2}u_{\epsilon}(s,\cdot)\|_{\vartheta}}\\\\[14.0pt]
&\displaystyle{\leq
c\,\epsilon\,\delta\left(1+\|g\|_{0}\right)\epsilon^{-(\varrho+\frac{1}{2})}(s\wedge
1)^{-(\varrho+\frac{1}{2})}\,N_{\epsilon}(u_{\epsilon}(s,\cdot)).}\end{array}$

By proceeding as in the proof of Lemma 5.3 (see also Remark 5.4), this allows
to conclude

$N_{\epsilon}(\Gamma^{\epsilon}_{\delta,1}(u_{\epsilon})(t))\leq
c\,\delta\left(1+\|g\|_{0}\right)\sup_{s\in\,[0,t]}N_{\epsilon}(u_{\epsilon}(s,\cdot)),\
\ \ \ t\in\,[0,T].$ (7.6)

Now, let us estimate $N_{\epsilon}(\Gamma^{\epsilon}_{2}(u_{\epsilon})(t))$.
We have

$\|\gamma_{2}(u_{\epsilon})(t)\|_{1}\leq
c\,\left(\|Du_{\epsilon}(t,\cdot)\|_{0}+\|D^{2}u_{\epsilon}(t,\cdot)\|_{0}\right).$

Thus, according to (2.5) and (2.8), thanks to (6.8) for every $\alpha>0$ there
exists $\kappa_{\alpha}>0$ such that

$\|\gamma_{2}(u_{\epsilon})(t)\|_{1}\leq\alpha\,[D^{2}u_{\epsilon}(s,\cdot)]_{\vartheta}+\kappa_{\alpha}\|u_{\epsilon}(s,\cdot)\|_{0}.$
(7.7)

In view of (2.15) and (7.7), there exists some $\lambda_{\vartheta}>0$ such
that for every $t\in\,[0,T]$

$\begin{array}[]{l}\displaystyle{N_{\epsilon}(\Gamma^{\epsilon}_{2}(u)(t))\leq\alpha\int_{0}^{t}e^{-\lambda_{\vartheta}(t-s)}[D^{2}u_{\epsilon}(s,\cdot)]_{\vartheta}\,ds}\\\\[14.0pt]
\displaystyle{+\alpha\,\epsilon^{\varrho}\,(t\wedge
1)^{\varrho}\int_{0}^{t}e^{-\lambda_{\vartheta}(t-s)}\epsilon^{-\frac{\vartheta}{2}}((t-s)\wedge
1)^{-\frac{\vartheta}{2}}[D^{2}u_{\epsilon}(s,\cdot)]_{\vartheta}\,ds}\\\\[14.0pt]
\displaystyle{+\alpha\,\epsilon^{\varrho+\frac{1}{2}}(t\wedge
1)^{\varrho+\frac{1}{2}}\int_{0}^{t}e^{-\lambda_{\vartheta}(t-s)}\epsilon^{-\frac{1+\vartheta}{2}}((t-s)\wedge
1)^{-\frac{1+\vartheta}{2}}[D^{2}u_{\epsilon}(s,\cdot)]_{\vartheta}\,ds}\\\\[14.0pt]
\displaystyle{+\kappa_{\alpha}\int_{0}^{t}e^{-\lambda_{\vartheta}(t-s)}\epsilon^{-\frac{1+\vartheta}{2}}((t-s)\wedge
1)^{-\frac{1+\vartheta}{2}}\,ds\|g\|_{0}}\\\\[14.0pt] \displaystyle{\leq
c\,\alpha\epsilon^{-\frac{1+\vartheta}{2}}\sup_{s\in\,[0,t]}N_{\epsilon}(u_{\epsilon}(s,\cdot))+c\,\kappa_{\alpha}\epsilon^{-\frac{1+\vartheta}{2}}\,\|g\|_{0}.}\end{array}$
(7.8)

Hence, if we plug (7.3), (7.6) and (7.8) into (7.2), we obtain

$\begin{array}[]{l}\displaystyle{\|u_{\epsilon}(t,\cdot)\|_{\eta}+N_{\epsilon}(u_{\epsilon}(t,\cdot))}\\\\[14.0pt]
\displaystyle{\leq
c\,\|g\|_{\eta}+c\left[\delta\left(1+\|g\|_{0}\right)+\alpha\epsilon^{-\frac{1+\vartheta}{2}}\right]\sup_{s\in\,[0,t]}N_{\epsilon}(u_{\epsilon}(s,\cdot))+c\,\kappa_{\alpha}\epsilon^{-\frac{1+\vartheta}{2}}\,\|g\|_{0}.}\end{array}$

In particular if take $\delta_{2}\leq\delta_{1}$ such that

$c\,\delta_{2}\left(2+\|g\|_{0}\right)<1/2,$

and $\alpha=\delta\epsilon^{\frac{1+\vartheta}{2}}$, we obtain (7.1) for every
$\delta\leq\delta_{2}$.

∎

###  7.1. Conclusion of the proof of Theorem 3.4

Thanks to (7.1), by standard arguments we have that for every
$\epsilon\in\,(0,1)$ the local solution we found in Theorem 5.5 is in fact a
global solution. Moreover, this global solution is unique. Actually, if
$u_{1},u_{2}\in\,C_{\epsilon,\varrho,\eta}((0,T];C^{2+\vartheta}_{b}(H))$ are
two solutions of equation (5.1), for some fixed $\delta\leq\delta_{2}$, we
assume that

$t_{0}:=\sup\left\\{t\in\,(0,T]\ :\ u_{1}(s)=u_{2}(s),\
s\in\,[0,t]\right\\}<T.$

With the same notations we have used in Section 5, we introduce the problem

$u(t)=\Gamma_{\epsilon,\delta}^{\varphi}(u)(t)=R^{\epsilon}_{t}\varphi+\Gamma_{\epsilon,\delta}(u)(t),\
\ \ \ \ t\geq t_{0},$ (7.9)

where $\varphi:=u_{1}(t_{0})=u_{2}(t_{0})$. Due to (7.1), we have that

$\|\varphi\|_{\eta}\leq c_{\epsilon,\delta}\,\|g\|_{\eta},$

for some constant $c_{\epsilon,\delta}>0$ independent of $T>0$.

As shown in Section 5, there exist $\bar{R}$, $\bar{\tau}>0$ and
$\bar{\delta}\leq\delta_{2}$ such that the mapping
$\Gamma_{\epsilon,\delta}^{\varphi}$ maps
$\mathscr{Y}^{\epsilon,\bar{R}}_{\varrho,\eta,\vartheta,t_{0},\bar{\tau}}$
into itself as a contraction, for every $\delta\leq\bar{\delta}$, where

$\mathscr{Y}^{\epsilon,R}_{\varrho,\eta,\vartheta,t_{0},\bar{\tau}}:=\left\\{\,u\in\,C_{\epsilon,\varrho,\eta}((t_{0},t_{0}+\bar{\tau}];C^{2+\vartheta}_{b}(H))\
:\
\|u\|_{\epsilon,\varrho,\eta,\vartheta,t_{0},\bar{\tau}}\leq\bar{R}\right\\},$

and
$C_{\epsilon,\varrho,\eta}((t_{0},t_{0}+\bar{\tau}];C^{2+\vartheta}_{b}(H))$
is the space of all functions $u$ belonging to
$C([t_{0},t_{0}+\bar{\tau}];C^{\eta}_{b}(H))\cap
C((t_{0},t_{0}+\bar{\tau}];C^{2+\vartheta}_{b}(H))$ such that the norm

$\begin{array}[]{l}\displaystyle{\|u\|_{\epsilon,\varrho,\eta,\vartheta,t_{0},\bar{\tau}}}\\\\[14.0pt]
\displaystyle{:=\sup_{t\in\,(t_{0},t_{0}+\bar{\tau}]}\left(\|u(t,\cdot)\|_{\eta}+\epsilon^{\varrho}((t-t_{0})\wedge
1)^{\varrho}\|D_{x}u(t,\cdot)\|_{\vartheta}+\epsilon^{\varrho+\frac{1}{2}}((t-t_{0})\wedge
1)^{\varrho+\frac{1}{2}}\|D^{2}_{x}u(t,\cdot)\|_{\vartheta}\right)}\end{array}$

is finite.

In particular $\Gamma_{\epsilon,\delta}^{\varphi}$ has a unique fixed point in
$\mathscr{Y}^{\epsilon,\bar{R}}_{\varrho,\eta,\vartheta,t_{0},\bar{\tau}}$ or,
equivalently, equation (7.9) has a unique solution on the interval
$[t_{0},t_{0}+\bar{\tau}]$. This implies that

$u_{1}(s)=u_{2}(s),\ \ \ \ \ s\in\,[0,t_{0}+\bar{\tau}],$

violating the definition of $[0,t_{0}]$ as the maximal interval where $u_{1}$
and $u_{2}$ coincide.

##  8\. The large deviation principle

In this last section we give a proof of Theorem 3.6. We follow the well-known
method based on weak convergence, as developed in [3]. To this purpose, we
need to introduce some notations.

For every $t>0$, we denote by $\mathscr{P}_{t}$ the set of predictable
processes in $L^{2}(\Omega\times[0,t];H)$, and for every $M>0$ we introduce
the sets

$\mathscr{S}_{t,M}:=\left\\{\varphi\in\,L^{2}_{w}(0,t;H)\ :\
\|\varphi\|_{L^{2}(0,t;H)}\leq M\right\\},$

and

$\Lambda_{t,M}:=\left\\{\varphi\in\,\mathscr{P}_{t}\ :\
\varphi\in\mathscr{S}_{t,M},\ \mathbb{P}-\text{a.s.}\right\\}.$

In Theorem 3.5 we have shown that for every $M,t>0$ and
$\varphi\in\,\Lambda_{t,M}$ and for every $x\in\,H$ and $\epsilon\in\,(0,1)$
there exists a unique mild solution
$X^{t,x}_{\varphi,\epsilon}\in\,L^{2}(\Omega;C([0,t];H))$ for equation (3.19).

Next, we consider the problem

$\frac{dX}{ds}(s)=AX(s)+b(X(s))+\sigma(X(s),g(Z^{X(s)}(t-s)))\varphi(s),\ \ \
\ X(0)=x,$ (8.1)

where, as we did in Section 3, for every $y\in\,H$ we denote by $Z^{y}$ the
solution of equation (3.22). In what follows, we show that the following
result holds.

######  Proposition 8.1.

Assume that $g:H\to\mathbb{R}$ is Lipschitz-continuous. Then, under the same
assumptions of Theorem 3.5, for every $t>0$ and $\varphi\in\,L^{2}(0,t;H)$ and
for every $x\in\,H$, there exists a unique mild solution
$X^{t,x}_{\varphi}\in\,C([0,t];H)$ for equation (8.1).

Once proved Theorem 3.5 and Proposition 8.1, we introduce the following two
conditions.

  1. C1.

Let $\\{\varphi_{\epsilon}\\}_{\epsilon>0}$ be an arbitrary family of
processes in $\Lambda_{t,M}$ such that

$\lim_{\epsilon\to 0}\varphi_{\epsilon}=\varphi,\ \ \ \ \text{in distribution
in}\ \ \ L_{w}^{2}(0,t;H),$

where $L^{2}_{w}(0,t;H)$ is the space $L^{2}(0,t;H)$ endowed with the weak
topology and $\varphi\in\,\Lambda_{t,M}$. Then we have

$\lim_{\epsilon\to 0}X^{t,x}_{\varphi_{\epsilon},\epsilon}=X^{t,x}_{\varphi},\
\ \ \ \text{in distribution}\ \ \ C([0,t],H).$

  2. C2.

For every $t,R>0$, the level sets $\Phi_{t,R}=\\{I_{t,x}\leq R\\}$ are compact
in the space $C([0,t];H)$.

As shown in [3], Conditions C1. and C2. imply that the family
$\\{X^{t,x}_{\epsilon}\\}_{\epsilon\in\,(0,1)}$ satisfies a Laplace principle
with action functional $I_{t,x}$ in the space $C([0,t];H)$ for the . Due to
the compactness of the level sets $\Phi_{t,R}$ stated in C2. this is
equivalent to the validity of Theorem 3.6.

###  8.1. Proof of Proposition 8.1

For every $y\in\,H$ and $s\in\,[0,t]$ we define

$\Sigma_{t}(y,s):=\sigma(y,g(Z^{y}(t-s))).$

With this notation, a function in $C([0,t];H)$ is a mild solution for equation
(8.1) if it is a fixed point of the mapping $\Lambda_{t}$ defined for every
$X\in\,C([0,t];H)$ by

$\Lambda_{t}(X)(s):=e^{sA}x+\int_{0}^{s}e^{(s-r)A}b(X(r))\,dr+\int_{0}^{s}e^{(s-r)A}\Sigma_{t}(X(r),r)\,\varphi(r)\,dr,\
\ \ \ s\in\,[0,t].$

It is immediate to check that there exists a continuous increasing function
$\kappa(s)$ such that for every $y_{1},y_{2}\in\,H$

$\|Z^{y_{1}}(s)-Z^{y_{2}}(s)\|_{H}\leq\kappa(s)\,\|y_{1}-y_{2}\|_{H},\ \ \ \ \
\ s\geq 0.$ (8.2)

Hence, since we are assuming that $g:H\to\mathbb{R}$ is Lipschitz-continuous,
according to Hypothesis 1 for every $y_{1},y_{2},h\in\,H$ we have

$\|[\Sigma_{t}(y_{1},r)-\Sigma_{t}(y_{2},r)]h\|_{H}\leq
c\,\left(1+\kappa(t-r)\right)\|y_{1}-y_{2}\|_{H}\|h\|_{H},\ \ \ \ \
r\in\,[0,t].$

In particular, for every $X_{1},X_{2}\in\,C([0,t];H)$ and $s\in\,[0,t]$ we
have

$\begin{array}[]{ll}\displaystyle{\|\Lambda_{t}(X_{1})(s)-\Lambda_{t}(X_{2})(s)\|_{H}}&\displaystyle{\leq
c\,\int_{0}^{s}\left(1+\left(1+\kappa(t-r)\right)\,\|\varphi(r)\|_{H}\right)\|X_{1}(r)-X_{2}(r)\|_{H}\,dr}\\\\[14.0pt]
&\displaystyle{\leq
c_{t}\left(\|\varphi\|_{L^{2}(0,t;H)}+1\right)\|X_{1}-X_{2}\|_{C([0,t];H)}.}\end{array}$

This implies that $\Lambda_{t}:C([0,t];H)\to C([0,t];H)$ is Lipschitz
continuous and by standard arguments we conclude that $\Lambda_{t}$ has a
unique fixed point.

###  8.2. Proof of the validity of Conditions C1 and C2

It is enough to prove Condition C1. Actually, from the proof of Condition C1
we will see that the mapping

$\varphi\in\,L^{2}_{w}(0,t;H)\mapsto Y^{t,x}_{\varphi}$

is continuous and

$\Phi_{t,R}=\\{I_{t,x}\leq R\\}=\\{Y^{t,x}_{\varphi}\ :\
\varphi\in\,\Lambda_{t,c(R)}\\},$

for some $c(R)>0$. Therefore, since the set $\Lambda_{t,M}$ is compact in
$L^{2}_{w}(0,t;H)$ for every $M>0$, we conclude that Condition C2 holds.

In order to prove Condition C1, we first need to prove the following
preliminary results

######  Lemma 8.2.

Under the same assumptions of Theorem 3.6, for every $p\geq 1$ we have

$\sup_{\epsilon\in\,(0,1)}\mathbb{E}\sup_{s\in\,[0,t]}\|X^{t,x}_{\varphi_{\epsilon},\epsilon}(s)\|_{H}^{p}\leq
c(t,M,p)\left(1+\|x\|^{p}_{H}\right).$ (8.3)

###### Proof.

We have

$\begin{array}[]{ll}\displaystyle{X^{t,x}_{\varphi_{\epsilon},\epsilon}(s):=}&\displaystyle{e^{sA}x+\int_{0}^{s}e^{(s-r)A}b(X^{t,x}_{\varphi_{\epsilon},\epsilon}(r))\,ds+\int_{0}^{s}e^{(s-r)A}\Sigma_{t,\epsilon}(r,X^{t,x}_{\varphi_{\epsilon},\epsilon}(r))\,\varphi_{\epsilon}(r)\,dr}\\\\[14.0pt]
&\displaystyle{+\sqrt{\epsilon}\int_{0}^{s}e^{(s-r)A}\Sigma_{t,\epsilon}(r,X^{t,x}_{\varphi_{\epsilon},\epsilon}(r))\,dW(r),}\end{array}$

where $\Sigma_{t,\epsilon}$ is the operator defined in (4.2). Hence, for every
$s\in\,[0,t]$ and $p\geq 1$ we have

$\begin{array}[]{ll}\displaystyle{\|X^{t,x}_{\varphi_{\epsilon},\epsilon}(s)\|_{H}^{p}\leq}&\displaystyle{c_{p}\,\|x\|_{H}^{p}+c_{p}\int_{0}^{s}\|X^{t,x}_{\varphi_{\epsilon},\epsilon}(r)\|_{H}^{p}\,dr+c_{p,M}\left(\int_{0}^{s}\|e^{(s-r)A}\Sigma_{t,\epsilon}(r,X^{t,x}_{\varphi_{\epsilon},\epsilon}(r))\|_{\mathscr{L}(H)}^{2}\,dr\right)^{\frac{p}{2}}}\\\\[14.0pt]
&\displaystyle{+c_{p}\left\|\int_{0}^{s}e^{(s-r)A}\Sigma_{t,\epsilon}(r,X^{t,x}_{\varphi_{\epsilon},\epsilon}(r))\,dW(r)\right\|_{H}^{p}+c_{p,t}.}\end{array}$
(8.4)

According to Hypothesis 3, for every $\tau>0$, $s\in\,[0,t]$ and $x\in\,H$ we
have

$\|e^{\tau A}\Sigma_{t,\epsilon}(s,x)\|^{2}_{\mathscr{L}_{2}(H)}\leq
c\,(\tau\wedge
1)^{-\frac{1}{2}}\left(\|x\|_{H}^{2}+|u_{\epsilon}(t-s,x)|^{2}+1\right).$

Moreover, according to (6.1), we have

$\sup_{(s,x)\in\,[0,t]\times H}|u_{\epsilon}(s,x)|\leq\|g\|_{0},\ \ \ \ \
\epsilon\in\,(0,1),$

so that

$\sup_{\epsilon\in\,(0,1)}\|e^{\tau
A}\Sigma_{t,\epsilon}(s,x)\|^{2}_{\mathscr{L}_{2}(H)}\leq c\,(\tau\wedge
1)^{-\frac{1}{2}}\left(\|x\|_{H}^{2}+1\right).$ (8.5)

In particular,

$\mathbb{E}\sup_{r\in\,[0,s]}\left(\int_{0}^{s}\|e^{(s-r)A}\Sigma_{t,\epsilon}(r,X^{t,x}_{\varphi_{\epsilon},\epsilon}(r))\|_{\mathscr{L}(H)}^{2}\,dr\right)^{\frac{p}{2}}\leq
c_{p,t}\,\left(\mathbb{E}\sup_{r\in\,[0,s]}\|X^{t,x}_{\varphi_{\epsilon},\epsilon}(r)\|_{H}^{p}+1\right).$
(8.6)

Now, if we fix $p>4$, we can find $\alpha<1/4$ such that
$(\alpha-1)p/(p-1)>-1$. By using a stochastic factorization argument, we have

$\begin{array}[]{l}\displaystyle{\int_{0}^{s}e^{(s-r)A}\Sigma_{t,\epsilon}(r,X^{t,x}_{\varphi_{\epsilon},\epsilon}(r))\,dW(r)=c_{\alpha}\int_{0}^{s}e^{(s-r)A}(s-r)^{\alpha-1}Y_{\alpha,\epsilon}(r)\,dr,}\end{array}$

where

$Y_{\alpha,\epsilon}(r):=\int_{0}^{r}e^{(r-\rho)A}(r-\rho)^{-\alpha}\Sigma_{t,\epsilon}(\rho,X^{t,x}_{\varphi_{\epsilon},\epsilon}(\rho))\,dW(\rho).$

Then, we obtain

$\begin{array}[]{l}\displaystyle{\left\|\int_{0}^{s}e^{(s-r)A}\Sigma_{t,\epsilon}(r,X^{t,x}_{\varphi_{\epsilon},\epsilon}(r))\,dW(r)\right\|^{p}_{H}\leq
c_{\alpha,p}\left(\int_{0}^{s}(s-r)^{\frac{(\alpha-1)p}{p-1}}\,dr\right)^{p-1}\int_{0}^{s}\|Y_{\alpha,\epsilon}(r)\|_{H}^{p}\,dr,}\end{array}$

so that, thanks to (8.5) and to the fact that $\alpha<1/4$

$\begin{array}[]{l}\displaystyle{\mathbb{E}\sup_{r\in\,[0,s]}\left\|\int_{0}^{s}e^{(s-r)A}\Sigma_{t,\epsilon}(r,X^{t,x}_{\varphi_{\epsilon},\epsilon}(r))\,dW(r)\right\|^{p}_{H}\leq
c_{\alpha,p,t}\int_{0}^{s}\mathbb{E}\|Y_{\alpha,\epsilon}(r)\|_{H}^{p}\,dr}\\\\[14.0pt]
\displaystyle{\leq
c_{\alpha,p,t}\int_{0}^{s}\left(\mathbb{E}\int_{0}^{r}(r-\rho)^{-(\frac{1}{2}+2\alpha)}\left(\|X^{t,x}_{\varphi_{\epsilon},\epsilon}(\rho)\|_{H}^{2}+1\right)\,d\rho\right)^{\frac{p}{2}}\,dr}\\\\[14.0pt]
\displaystyle{\leq
c_{\alpha,p,t}\left(\int_{0}^{s}\mathbb{E}\sup_{\rho\in\,[0,r]}\|X^{t,x}_{\varphi_{\epsilon},\epsilon}(\rho)\|_{H}^{p}\,dr+1\right).}\end{array}$
(8.7)

Therefore, thanks to (8.4), (8.6) and (8.7),

$\begin{array}[]{ll}\displaystyle{\mathbb{E}\sup_{r\in\,[0,s]}\|X^{t,x}_{\varphi_{\epsilon},\epsilon}(r)\|_{H}^{p}\leq}&\displaystyle{c_{t,M,p}\left(\|x\|_{H}^{p}+1\right)+c_{p,t}\int_{0}^{s}\mathbb{E}\sup_{\rho\in\,[0,r]}\|X^{t,x}_{\varphi_{\epsilon},\epsilon}(\rho)\|_{H}^{p}\,dr,}\end{array}$

and Gronwall’s Lemma allows to conclude.

∎

######  Lemma 8.3.

Under the same assumptions of Theorem 3.6, we have

$|u_{\epsilon}(s,x)-g(Z^{x}(s))|\leq
c_{t}\,\sqrt{\epsilon}\left(1+\|x\|_{H}\right),\ \ \ \ s\in\,[0,t].$ (8.8)

###### Proof.

Thanks to (6.1), we have

$u_{\epsilon}(s,x)-g(Z^{x}(s))=\mathbb{E}\left(g(X^{s,x}_{\epsilon}(s))-g(Z^{x}(s))\right),$

so that, since we are assuming that $g$ is Lipschitz-continuous

$|u_{\epsilon}(s,x)-g(Z^{x}(s))|\leq
c\,\mathbb{E}\,\|X^{s,x}_{\epsilon}(s)-Z^{x}(s)\|_{H}.$

Now, if we define $\rho_{\epsilon}^{x}(s):=X^{s,x}_{\epsilon}(s)-Z^{x}(s)$, we
have

$\begin{array}[]{ll}\displaystyle{\rho_{\epsilon}^{x}(s)=}&\displaystyle{\int_{0}^{s}e^{(s-r)A}\left(b(X^{s,x}_{\epsilon}(r))-b(Z^{x}(r))\right)\,dr+\sqrt{\epsilon}\int_{0}^{s}e^{(s-r)A}\Sigma_{s,\epsilon}(r,X^{s,x}_{\epsilon}(r))\,dW(r),}\end{array}$

where $\Sigma_{s,\epsilon}$ is the operator introduced in (4.2). Due to (8.3)
and (8.7), we have

$\mathbb{E}\|\rho_{\epsilon}(s)\|_{H}\leq
c\int_{0}^{s}\mathbb{E}\|\rho_{\epsilon}(r)\|_{H}\,dr+c_{t}\,\sqrt{\epsilon}\left(1+|x\|_{H}\right),$

and Gronwall’s lemma allows to conclude.

∎

Now, we are ready to prove condition C1. Let
$\\{\varphi_{\epsilon}\\}_{\epsilon>0}$ be an arbitrary family of processes in
$\Lambda_{t,M}$ converging in distribution, with respect to the weak topology
of $L^{2}(0,t;H)$, to some $\varphi\in\,\Lambda_{t,M}$. As a consequence of
Skorohod theorem, we can assume that the sequence
$\\{\varphi_{\epsilon}\\}_{\epsilon>0}$ converges $\mathbb{P}$-a.s. to
$\varphi$, with respect to the weak topology of $L^{2}(0,t;H)$. We will prove
that this implies that

$\lim_{\epsilon\to
0}\sup_{s\in\,[0,t]}\mathbb{E}\|X_{\varphi_{\epsilon},\epsilon}^{t,x}(s)-X^{t,x}_{\varphi}(s)\|_{H}^{2}=0.$
(8.9)

If we define

$\rho_{\epsilon}(s):=X_{\varphi_{\epsilon},\epsilon}^{t,x}(s)-X^{t,x}_{\varphi}(s),\
\ \ \ s\in\,[0,t],$

we have

$\begin{array}[]{l}\displaystyle{\rho_{\epsilon}(s)=\int_{0}^{s}e^{(s-r)A}\left[b(X_{\varphi_{\epsilon},\epsilon}^{t,x}(r))-b(X_{\varphi}^{t,x}(r))\right]\,dr}\\\\[14.0pt]
\displaystyle{+\int_{0}^{s}e^{(s-r)A}\left[\sigma(X_{\varphi_{\epsilon},\epsilon}^{t,x}(r),u_{\epsilon}(t-r,X_{\varphi_{\epsilon},\epsilon}^{t,x}(r)))\,\varphi_{\epsilon}(r)-\sigma(X_{\varphi}^{t,x}(r),g(Z^{X^{t,x}_{\varphi}(r)}(t-r)))\,\varphi(r)\right]\,dr}\\\\[14.0pt]
\displaystyle{+\sqrt{\epsilon}\,\int_{0}^{s}e^{(s-r)A}\Sigma_{t,\epsilon}(r,X_{\varphi,\epsilon}^{t,x}(r))\,dW_{r}=:\sum_{k=1}^{3}I_{k,\epsilon}(s).}\end{array}$
(8.10)

For $I_{1,\epsilon}(s)$, due to the Lipschitz continuity of $b$, we have

$\|I_{1,\epsilon}(s)\|^{2}_{H}\leq
c_{t}\int_{0}^{s}\|\rho_{\epsilon}(s)\|^{2}_{H}\,ds.$ (8.11)

Concerning $I_{2,\epsilon}(s)$, it can be written as

$\begin{array}[]{l}\displaystyle{\int_{0}^{s}e^{(s-r)A}\left[\sigma(X_{\varphi_{\epsilon},\epsilon}^{t,x}(r),u_{\epsilon}(t-r,X_{\varphi_{\epsilon},\epsilon}^{t,x}(r)))-\sigma(X_{\varphi}^{t,x}(r),g(Z^{X_{\varphi_{\epsilon},\epsilon}^{t,x}(r)}(t-r)))\right]\,\varphi_{\epsilon}(r)\,dr}\\\\[14.0pt]
\displaystyle{+\int_{0}^{s}e^{(s-r)A}\left[\sigma(X_{\varphi}^{t,x}(r),g(Z^{X_{\varphi_{\epsilon},\epsilon}^{t,x}(r)}(t-r)))-\sigma(X_{\varphi}^{t,x}(r),g(Z^{X^{t,x}_{\varphi}(r)}(t-r)))\right]\,\varphi_{\epsilon}(r)\,dr}\\\\[14.0pt]
\displaystyle{+\int_{0}^{s}e^{(s-r)A}\sigma(X_{\varphi}^{t,x}(r),g(Z^{X^{t,x}_{\varphi}(r)}(t-r)))\,(\varphi_{\epsilon}(r)-\varphi(r))\,dr=:\sum_{k=1}^{3}J_{k,\epsilon}(s).}\end{array}$

According to (8.8), we have

$\|J_{1,\epsilon}(s)\|_{H}\leq
c\int_{0}^{s}\left(\|\rho_{\epsilon}(r)\|_{H}+c_{t}\,\sqrt{\epsilon}\left(1+\|X_{\varphi_{\epsilon},\epsilon}^{t,x}(r)\|_{H}\right)\right)\|\varphi_{\epsilon}(r)\|_{H}\,dr,$

so that

$\|J_{1,\epsilon}(s)\|^{2}_{H}\leq
c_{t,M}\int_{0}^{s}\|\rho_{\epsilon}(r)\|^{2}_{H}\,dr+\epsilon\,c_{t,M}\left(1+\sup_{r\in\,[0,t]}\|X_{\varphi_{\epsilon},\epsilon}^{t,x}\|^{2}_{H}\right).$
(8.12)

Moreover, thanks to (8.2), we have

$\|J_{2,\epsilon}(s)\|^{2}_{H}\leq
c_{t,M}\int_{0}^{s}\|Z^{X_{\varphi_{\epsilon},\epsilon}^{t,x}(r)}(t-r)-Z^{X^{t,x}_{\varphi}(r)}(t-r)\|^{2}_{H}\,dr\leq
c_{t,M}\int_{0}^{s}\|\rho_{\epsilon}(r)\|^{2}_{H}\,dr.$ (8.13)

Finally, for $I_{3,\epsilon}(s)$, thanks to (8.3) and (8.7) we have

$\mathbb{E}\sup_{s\in\,[0,t]}\|I_{3,\epsilon}(s)\|^{2}_{H}\leq
c_{t}\epsilon\left(1+\|x\|^{2}_{H}\right).$ (8.14)

Therefore, if we plug (8.11), (8.12), (8.13) and (8.14) into (8.10), in view
of (8.3) we obtain

$\mathbb{E}\,\|\rho_{\epsilon}(s)\|_{H}^{2}\leq
c_{t,M}\int_{0}^{s}\mathbb{E}\|\rho_{\epsilon}(r)\|_{H}^{2}\,dr+c_{t,M}\,\epsilon\left(1+\|x\|_{H}^{2}\right)+\mathbb{E}\|J_{3,\epsilon}(s)\|^{2}_{H},$

and the Gronwall lemma gives

$\mathbb{E}\,\|\rho_{\epsilon}(s)\|_{H}^{2}\leq
c_{t,M}\,\epsilon\left(1+\|x\|_{H}^{2}\right)+c_{t,M}\int_{0}^{s}\mathbb{E}\|J_{3,\epsilon}(r)\|^{2}_{H}\,dr.$
(8.15)

Now, due to the $\mathbb{P}$-a.s. convergence of $\varphi_{\epsilon}$ to
$\varphi$ in $L^{2}_{w}(0,t;H)$, since
$\\{\varphi\\}_{\epsilon\in\,(0,1)}\subset\Lambda_{t,M}$ and
$\varphi\in\,\Lambda_{t,M}$ we can apply the dominated convergence theorem and
we get

$\lim_{\epsilon\to 0}\int_{0}^{s}\mathbb{E}\|J_{3,\epsilon}(r)\|^{2}_{H}\,dr.$

Therefore, by taking the limit as $\epsilon$ goes to zero in both sides of
(8.15) we obtain (8.9).

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