# Regularity of the law of solutions to the stochastic heat equation with non-
Lipschitz reaction term

Michael Salins and Samy Tindel  Michael Salins: Department of Mathematics and
Statistics, Boston University, 665 Commonwealth Ave, Boston, MA, 02215, USA
<EMAIL_ADDRESS>Samy Tindel: Department of Mathematics, Purdue University, 150
N. University Street, W. Lafayette, IN 47907, USA<EMAIL_ADDRESS>
###### Abstract.

We prove the existence of density for the solution to the multiplicative
semilinear stochastic heat equation on an unbounded spatial domain, with drift
term satisfying a half-Lipschitz type condition. The methodology is based on a
careful analysis of differentiability for a map defined on weighted functional
spaces.

##  1\. Introduction

In this paper we consider a stochastic heat equation on ${\mathbb{R}}^{d}$ of
the form

$\frac{\partial u}{\partial t}(t,x)=\frac{1}{2}\Delta
u(t,x)+f(u(t,x))+\sigma(u(t,x))\,\dot{W}(t,x)\,,\qquad
t\in[0,T],\,x\in{\mathbb{R}}^{d}.$ (1)

In (1), $\Delta$ denotes the Laplace operator, $\dot{W}$ is a centered
Gaussian noise which is white in time and whose covariance function satisfies
a standard assumption called strong Dalang condition (see Assumption 2.2 below
for a precise statement). The coefficient $\sigma$ in (1) is supposed to be
differentiable with bounded derivative. Our equation deviates from the
standard setting for stochastic pdes due to the drift coefficient $f$. This
coefficient is only assumed to verify a mild damping condition, that is we
suppose that $f$ is continuously differentiable and that $f^{\prime}$ is upper
bounded by a constant $\kappa\in{\mathbb{R}}$:

$f^{\prime}(u)\leq\kappa,\quad\text{ for all }\quad u\in\mathbb{R}.$ (2)

This condition will be referred to as half-Lipschitz in the sequel. As a
motivating example, any odd degree polynomial with a negative leading
coefficient such as $f(u)=-u^{3}+u$ will satisfy (2). Under this setting, we
investigate the law of the random field mild solution to (1), $u(t,x)$, at a
fixed time $t>0$ and a fixed point in space $x\in\mathbb{R}^{d}$. We prove
using Malliavin calculus that the law has a density.

Stochastic pdes have been primarily considered for globally Lipschitz
continuous coefficients $f$ and $\sigma$ (see e.g [7, 26]). However, since
multiple relevant physical systems involve polynomial type nonlinearities, a
substantial amount of effort has been devoted to that case over the past
decades. Among those contributions, one can single out the following:

  1. (a)

The case of $x\in D$ (with $D$ a bounded domain in ${\mathbb{R}}^{d}$), and
polynomial nonlinearities $f$ with negative leading terms, has been
investigated in [5, 16]. The techniques use localization arguments based on
stopping time methods and a priori bounds. The papers [5, 16] are all handling
the case of a colored noise $\dot{W}$ which can accommodate stochastic
integrals without a need for renormalization.

  2. (b)

The case of a stochastic heat equation defined on an unbounded spatial domain
$\mathbb{R}^{d}$ with $f$ satisfying (2) and with at most polynomial growth
was first investigated independently by Iwata [13] and Brzezniak and Peszat
[3]. Unlike in the bounded domain setting, the solutions to (1) are unbounded
in space if $\sigma$ is bounded away from zero. Specifically, for any $t>0$,
${\mathbb{P}}(\sup_{x\in\mathbb{R}^{d}}|u(t,x)|=+\infty)=1$.

  3. (c)

The variational methods of Röckner and collaborators can be applied when the
perturbing noise is trace-class, so that Itô formula methods are available
[18, 23, 15]. This theory allows for spdes that are not semilinear, such as
porous medium equations, but precludes rough perturbations like space-time
white noise.

  4. (d)

In case of a space-time white noise $\dot{W}$ (or even a spatial white noise
if $x\in{\mathbb{R}}^{d}$ with $d\geq 2$), renormalization tools are in order.
We cannot list all the relevant contributions in this direction. Let us just
mention [11] for the celebrated kpz equation and [21] for the $\Phi_{4}^{3}$
model. Notice that most of those systems only admit an additive noise, and
that the current techniques only yield local (in time) solutions.

Studies of densities for stochastic processes in non-Markovian settings have
been one of the great achievements of Malliavin calculus. However, due to a
methodology based on differentiation and integration by parts, Malliavin
calculus results usually require smooth and bounded coefficients in
differential systems like (1). This is certainly the case in classical
references concerning stochastic pdes [2, 19, 20] or systems driven by a
fractional Brownian motion [1, 6]. A more recent trend has been to adapt the
integration by parts technology to settings with little regularity or less
restrictive growth assumptions. One can quote the following studies, which are
close in spirit to our own contribution:

  1. (a)

The article [12] deals with a stochastic differential equation driven by an
additive Brownian motion, whose drift coefficient lies in a fractional Sobolev
space of the form $W^{\gamma,p}$ (with a regularity parameter
$\gamma\in(0,1]$). The computations therein combine Malliavin calculus and
Girsanov transform tools.

  2. (b)

For stochastic differential equations driven by a fractional Brownian motion
let us mention the paper [9], which handles the case of a Hölder drift. This
is achieved thanks to a smart limiting procedure taken on Euler schemes. More
recently, the preprint [14] explores densities for a drift coefficient $f$
which has linear growth and satisfies a mild damping condition. The main tools
in [14] is Girsanov’s transform, again due to the fact that an equation with
additive noise is considered. The density is then analyzed by importing
arguments from the regularization by noise literature and investigating a
functional for a fractional bridge.

  3. (c)

In [16, 17] the authors consider a spde of the form (1), satisfying an
assumption which is similar to (2). The main difference between this setting
and ours is twofold: first [16, 17] focuses on spatial variables in bounded
sets of ${\mathbb{R}}^{d}$, while our result is concerned with $x$ in the
whole space ${\mathbb{R}}^{d}$. Then [16, 17] is restricted to coefficients
$f$ in (1) having polynomial growth, while we can reach exponential growth in
the current paper. Notice that in [17] the strategy is based on a localization
procedure relying on Lipschitz approximations of the drift coefficient $f$.
This method is ruled out in our unbounded domain setting. Indeed, in our case
the field $\\{u(t,x);\,x\in{\mathbb{R}}^{d}\\}$ is unbounded for any fixed
$t>0$, even if $f$ is Lipschitz. The boundedness of $u(t,\cdot)$ whenever $f$
is Lipschitz was a crucial ingredient in [17].

As one can see, our result is thus the first one establishing existence of
density for a spde with drift whose first derivative is unbounded and that is
defined on a noncompact domain. On top of this novel aspect, we believe that
our method of proof is applicable to other settings. In some subsequent
publication we plan to apply the techniques developed here to the renormalized
frameworks mentioned above.

In future work, we also wish to remove the growth restriction on $f$. While
many previous works restricted the growth rate of $f$ to polynomial growth
like $|f(u)|\leq C(1+|u|^{p})$ [13, 3, 5, 15], we allow $f$ to grow as fast as
$|f(u)|\leq Ke^{K|u|^{\nu}}$ for any $K,\nu>0$ like in [24]. The exponential
growth restriction is helpful for proving that the integrals
$\int_{0}^{t}\int_{\mathbb{R}^{d}}G(t-s,x-y)f(u(s,y))dyds$ in the mild
solution are well-defined. These growth restrictions do not seem necessary,
however, and in future work we hope to prove that the half-Lipschitz condition
on $f$ (2) along with appropriate assumptions on $\sigma$ and the $\dot{W}$,
is sufficient to guarantee the existence and uniqueness of mild solutions and
the existence of a density. Such a generalization requires sensitive analysis
of the spatial growth rates of solutions and is outside of the scope of the
current manuscript.

As mentioned above, the solutions to (1) with $x\in{\mathbb{R}}^{d}$ are
unbounded and heat equations enjoy infinite propagation speed. Therefore the
localization arguments that are invoked in the bounded domain case [16, 17]
cannot be applied to the unbounded domain setting. To investigate properties
of unbounded solutions, many researchers have introduced a spatial weight. For
example, Iwata [13] and Brzezniak and Peszat [3] used exponential weights
$\sup_{x}e^{-\lambda|x|}|u(t,x)|$. The choice of exponential weights,
unfortunately, introduced a polynomial growth restriction in the literature.
With this observation in mind, the first author of this paper proposed in [24]
a new method to handle equations like (1). Roughly speaking, in this paper and
in [24] we use polynomial weights
$\sup_{x\in\mathbb{R}^{d}}\frac{|u(t,x)|}{1+|x-x_{0}|^{\theta}}$ for
arbitrarily small $\theta>0$. This choice of weights allows to prove the main
results for superlinear half-Lipschitz reaction terms that grow as fast as
$|f(u)|\leq K\exp(K|u|^{\gamma})$ for any $K,\gamma>0$.

In order to explain how we obtained the existence of a density for the
solution to (1), let us give a few details about the approach in [24]. The
scheme therein basically splits the dynamics in two pieces: first a stochastic
map $\mathcal{I}$ defined for a random field $\varphi$ by

$\mathcal{I}(t,x)=\mathcal{I}^{\varphi}(t,x)=\int_{0}^{t}\int_{\mathbb{R}^{d}}G(t-s,x-y)\varphi(s,y)W(ds\,dy)\,,$

where $G(t,x):=(2\pi)^{-\frac{d}{2}}\exp(-|x|^{2}/2)$ is the standard Gaussian
heat kernel. This map is properly introduced in (98) below. The second piece
of our dynamics is a deterministic map called $\mathcal{M}$ (see Definition
2.9) given for a continuous function $z$ defined on
$[0,T]\times{\mathbb{R}}^{d}$ as the solution of the following integral
equation

$\mathcal{M}(z)(t,x)=\int_{0}^{t}\int_{\mathbb{R}^{d}}G(t-s,x-y)f(\mathcal{M}(z)(s,y))\,dyds+z(t,x).$
(3)

The crucial point in [24] is that, despite the fact that $f$ in (3) in not
globally Lipschitz continuous, the map $\mathcal{M}$ is globally Lipschitz
continuous on weighted spaces of continuous functions on
$[0,T]\times{\mathbb{R}}^{d}$. Thanks to some thorough estimates for both
$\mathcal{I}$ and $\mathcal{M}$ and a Yosida type approximation procedure for
the function $f$, one can prove existence and uniqueness for mild solutions to
(1). More specifically, the mild solution of (1) is defined to be a random
field that solves the integral equation

$\displaystyle u(t,x)=$
$\displaystyle\int_{\mathbb{R}^{d}}G(t-s,x-y)u(0,y)dy+\int_{0}^{t}\int_{\mathbb{R}^{d}}G(t-s,x-y)f(u(s,y))dyds$
$\displaystyle+\int_{0}^{t}\int_{\mathbb{R}^{d}}G(t-s,x-y)\sigma(u(s,y))W(ds\,dy).$
(4)

Letting $U_{0}(t,x)=\int_{\mathbb{R}^{d}}G(t-s,x-y)u(0,y)dy$ and
$\mathcal{I}^{\varphi}(t,x)$ and $\mathcal{M}$ be the maps defined above,
equation (1) can be recast as

$u(t,x)=\mathcal{M}\left(U_{0}+\mathcal{I}^{\sigma(u)}\right)(t,x).$ (5)

The existence and uniqueness of the solution to this equation was then
established in [24] via a Picard iteration scheme. Namely, we can recursively
define

$u_{0}(t,x)=U_{0}(t,x),\ \ \ \ \
u_{n+1}=\mathcal{M}(U_{0}+\mathcal{I}^{\sigma(u_{n})}).$ (6)

By properly bounding both maps $\mathcal{I}^{\varphi}$ and $\mathcal{M}$, the
existence and uniqueness for equation (1) is proved thanks to a fixed point
argument.

We can now explain our global method for the existence of density result and
outline the structure of our paper. First in Section 2 we introduce the main
assumptions and recall the existence and uniqueness results for (1) from [24],
as well as Malliavin calculus results from [20]. Then we proceed to prove the
Malliavin differentiability of $u(t,x)$ via the approximation scheme (6), and
also by studying the Malliavin differentiability of the maps
$\mathcal{I}^{\varphi}$ and $\mathcal{M}$. To begin with, Section 3 proves
Proposition 3.1. This result states that if a random field
$z:[0,T]\times\mathbb{R}^{d}\times\Omega\to\mathbb{R}$ has the property that
$z(t,x)$ is Malliavin differentiable for any
$(t,x)\in[0,T]\times\mathbb{R}^{d}$ and additionally that $z(t,x)$ and
$Dz(t,x)$ satisfy certain polynomial growth assumptions in the spatial
variable, then the random field $\mathcal{M}(z)(t,x)$ is also Malliavin
differentiable for any $(t,x)\in[0,T]\times\mathbb{R}^{d}$. Furthermore,
Proposition 3.1 establishes a weighted supremum norm bound which holds with
probability one:

$\sup_{t\in[0,T]}\sup_{x\in\mathbb{R}^{d}}\frac{|D\mathcal{M}(z)(t,x)|_{{\mathcal{H}}_{T}}}{1+|x-x_{0}|^{\theta}}\leq
K\sup_{t\in[0,T]}\sup_{x\in\mathbb{R}^{d}}\frac{|Dz(t,x)|_{{\mathcal{H}}_{T}}}{1+|x-x_{0}|^{\theta}}\,,$
(7)

where $\mathcal{H}_{T}$ is the natural Cameron-Martin space related to our
colored noise (see (53) below for a proper definition of the inner product in
$\mathcal{H}_{T}$).

The Malliavin differentiability of the stochastic integrals
$\mathcal{I}^{\varphi}$ when $\varphi$ is Malliavin differentiable is a
standard result from the theory of Malliavin calculus (see Proposition 2.16
below). In Section 4 we prove that certain moment estimates of weighted
supremum norms, that applied to stochastic integrals
$\mathcal{I}^{\varphi}(t,x)$ when $\varphi$ is real-valued, will also hold in
the case where $\varphi$ is Hilbert-space valued. Specifically, we apply these
results to derive estimates on the Malliavin derivatives of the stochastic
integral terms. In Section 4.2 we apply these Malliavin differentiability
results about $\mathcal{I}^{\varphi}$ and $\mathcal{M}$ to the recursively
defined Picard iteration scheme introduced in (6). In particular, this allows
us to prove that $u_{n}(t,x)$ is Malliavin differentiable for all
$n\in\mathbb{N}$, $t\in[0,T]$, and $x\in\mathbb{R}^{d}$. Furthermore, we prove
that for any $p>0$ and $T>0$,

$\sup_{n}\sup_{x_{0}\in\mathbb{R}^{d}}{\mathbb{E}}\left|\sup_{t\in[0,T]}\sup_{x\in\mathbb{R}^{d}}\frac{|Du_{n}(t,x)|_{{\mathcal{H}}_{T}}}{1+|x-x_{0}|^{\theta}}\right|^{p}<+\infty\,.$
(8)

In particular, these weighted supremum moment bounds guarantee that
$\sup_{n}E|Du_{n}(t,x)|_{{\mathcal{H}}_{T}}^{2}$ is finite for any fixed $t>0$
and $x\in\mathbb{R}^{d}$. The classical result [20, Lemma 1.2.3] then
guarantees that $u(t,x)$ is Malliavin differentiable.

Finally, in Section 5 we prove that the Malliavin derivative of the mild
solution is positive almost surely, that is
${\mathbb{P}}(\|Du(t,x)\|_{{\mathcal{H}}_{T}}>0)=1$. By [20, Theorem 2.1.2],
this positivity property implies that the law of $u(t,x)$ is absolutely
continuous with respect to Lebesgue measure. We prove the positivity by
constructing a particular family of deterministic test functions
$h_{t,x,\delta}\in{\mathcal{H}}_{T}$ and proving that, with probability one,
the directional Malliavin derivative $D_{h_{t},x,\delta}u(t,x)$ is non-
negative for some small (and random) $\delta>0$. This analysis involves
writing the directional derivatives in a mild form

$D_{h}u(t,x)=\left<\phi_{t,x},h\right>_{{\mathcal{H}}_{T}}+A_{h}(t,x)+B_{h}(t,x)\,,$

where $A_{h}(t,x)$ is a Lebesgue integral and $B_{h}(t,x)$ is a stochastic
integral. We prove that when $\delta$ is sufficiently small, the integral
terms are much smaller than the leading term, implying that the directional
Malliavin derivative is non-negative.

##  2\. Approach to existence and uniqueness

In this section we will summarize the method employed in [24] in order to
solve an equation like (1) with a half-Lipschitz reaction term. The method is
based on a fixed point argument in an appropriate weighted Hölder space. We
also include a minimal set of Malliavin calculus tools necessary to carry out
our main computations.

###  2.1. Functional space, assumptions and existence result

We start by defining the weighted function spaces which will be used
throughout the paper.

######  Definition 2.1.

Let $\theta>0$ be a positive parameter and let $x_{0}\in\mathbb{R}^{d}$. The
space $\mathcal{C}_{\theta,x_{0}}([0,T]\times\mathbb{R}^{d})$ designates the
set of continuous functions

$\left\\{z\in\mathcal{C}([0,T]\times\mathbb{R}^{d}):\lim_{|x|\to\infty}\sup_{t\in[0,T]}\frac{|z(t,x)|}{1+|x-x_{0}|^{\theta}}=0\right\\}.$
(9)

The space is endowed with the weighted supremum norm

$|z|_{\mathcal{C}_{\theta,x_{0}}([0,T]\times\mathbb{R}^{d})}:=\sup_{t\in[0,T]}\sup_{x\in\mathbb{R}^{d}}\frac{|z(t,x)|}{1+|x-x_{0}|^{\theta}}.$
(10)

For fixed $\theta$, the spaces
$\mathcal{C}_{\theta,x_{0}}([0,T]\times\mathbb{R}^{d})$ all coincide, but it
is convenient to use different centers of the weight $x_{0}$.

Next we state the assumptions on the stochastic noise $\dot{W}$. All of the
random variables below are defined on a complete probability space
$(\Omega,\mathcal{F},{\mathbb{P}})$ equipped with a filtration
$\\{\mathcal{F}_{t}:t\geq 0\\}$.

######  Assumption 2.2.

The noise $\dot{W}$ in (1) is a centered Gaussian spatially homogeneous noise
which is white in time. There exists a positive and positive definite function
$\Lambda$ such that formally

${\mathbb{E}}[\dot{W}(t,x)\dot{W}(s,y)]=\delta(t-s)\Lambda(x-y).$ (11)

In the above expression, $\delta$ is the Dirac measure. The Fourier transform
of $\Lambda$ is a measure $\mu$ and we assume that there exists $\eta\in(0,1)$
such that

$\int_{\mathbb{R}^{d}}\frac{1}{1+|\xi|^{2(1-\eta)}}\mu(d\xi)<+\infty.$ (12)

The multiplicative noise coefficient $\sigma$ in (1) satisfies standard
differentiability and nondegeneracy assumptions.

######  Assumption 2.3.

The noise coefficient $\sigma:\mathbb{R}\to\mathbb{R}$ is differentiable and
its derivative is uniformly bounded. Moreover, we assume that there exists
$\alpha>0$ such that

$\sigma(u)\geq\alpha,\quad\text{ for all }\quad u\in\mathbb{R}.$ (13)

As mentioned in the introduction, our system (1) departs from the standard
stochastic pde setting due to the drift coefficient $f$. Namely we only
suppose that $f$ in (1) satisfies a half-Lipschitz condition, is
differentiable, and obeys a very mild growth condition. This is summarized in
the assumption below.

######  Assumption 2.4.

The reaction term $f:\mathbb{R}\to\mathbb{R}$ is continuously differentiable.
Moreover, there exists $\kappa\in\mathbb{R}$ such that the derivative is
uniformly bounded from above

$f^{\prime}(u)\leq\kappa,\quad\text{ for all }\quad u\in\mathbb{R}.$ (14)

We further assume that there exist $K>0$, $\nu>0$ such that

$|f^{\prime}(u)|\leq K\exp\left(K|u|^{\nu}\right)$ (15)

Notice that the upper bound on the first derivative (14) implies that
$f:\mathbb{R}\to\mathbb{R}$ is half-Lipschitz, meaning that for any
$u_{1}>u_{2}$,

$f(u_{1})-f(u_{2})\leq\kappa(u_{1}-u_{2}).$ (16)

We now label a standard assumption for the initial condition $u_{0}$ for our
equation of interest.

######  Assumption 2.5.

The initial condition for (1) is continuous and uniformly bounded, meaning
that there exists $M>0$ such that

$\sup_{x\in\mathbb{R}^{d}}|u_{0}(x)|\leq M.$ (17)

Equation (1) is solved in the so-called mild sense. Namely an adapted process
$u$ is said to solve (1) with initial condition $u_{0}$ if for all
$(t,x)\in[0,T]\times\mathbb{R}^{d}$ we have

$\displaystyle u(t,x)=$
$\displaystyle\int_{{\mathbb{R}}^{d}}G(t,x-y)u_{0}(y)dy+\int_{0}^{t}\int_{{\mathbb{R}}^{d}}G(t-s,x-y)f(u(s,y))dyds$
$\displaystyle+\int_{0}^{t}\int_{{\mathbb{R}}^{d}}G(t-s,x-y)\sigma(u(s,y))\,W(ds\,dy)\,,$
(18)

where $G(t,x)$ is the heat kernel related to the Laplace operator in (1). In
(2.1), the stochastic integral is interpreted in the Itô sense (see [7] for
further details). With this notion of solution and under the set of
assumptions spelled out above, we now recall the main existence and uniqueness
result from [24].

######  Theorem 2.6 (Theorem 2.6 of [24]).

Suppose Assumptions 2.3–2.5 are satisfied. Then there exists a unique mild
solution to (1). This solution lives in the space
$C_{\theta}([0,T]\times\mathbb{R}^{d})$ introduced in Definition 2.1.

###  2.2. Methodology

In this section we review the methods used to solve (1) in [24]. Those tools
will also play a prominent role in analyzing the Malliavin derivative of the
solution.

####  2.2.1. Yosida approximations

A crucial ingredient in the analysis of equation (2.1) is based on Yosida
approximations for the nonlinear forcing term $f$ satisfying Assumption 2.4.
That is for any function $f$ satisfying (14) or (16) it is easily seen (see
[24, Proposition 2.4]) that

$f(u)=\phi(u)+\kappa u\,,$ (19)

where $\phi$ is non-increasing. For $\phi:\mathbb{R}\to\mathbb{R}$ that is
non-increasing, we define the Yosida approximations for $\lambda>0$ by

$\phi_{\lambda}(u):=\frac{1}{\lambda}(J_{\lambda}(u)-u)\text{ where
}J_{\lambda}(u)=(I-\lambda\phi)^{-1}(u).$ (20)

The family $\\{\phi_{\lambda}:\lambda>0\\}$ is intended to be a smooth
approximation of $\phi$ under monotonicity conditions. We now summarize some
properties of the Yosida approximations, taken from [8, Appendix D].

######  Lemma 2.7.

Let $\phi:\mathbb{R}\to\mathbb{R}$ be a differentiable non-increasing function
and let $\\{\phi_{\lambda}:\lambda>0\\}$ be its Yosida approximations defined
by (20). Then the following are true.

  1. _(i)_

$|\phi_{\lambda}(u_{1})-\phi_{\lambda}(u_{2})|\leq\frac{2}{\lambda}|u_{1}-u_{2}|$,
for $u_{1},u_{2}\in{\mathbb{R}}$ and all $\lambda>0$.

  2. _(ii)_

$|\phi_{\lambda}(u)|\leq|\phi(u)|$, for $u\in{\mathbb{R}}$ and all
$\lambda>0$.

  3. _(iii)_

$u\mapsto\phi_{\lambda}(u)$ is nonincreasing, for all $\lambda>0$.

  4. _(iv)_

$\lim_{\lambda\to 0}\phi_{\lambda}(u)=\phi(u)$, for all $u\in{\mathbb{R}}$.

  5. _(v)_

$\lim_{\lambda\to 0}\phi_{\lambda}^{\prime}(u)=\phi^{\prime}(u)$, for all
$u\in{\mathbb{R}}$.

These properties of Yosida approximations are easily translated into
approximations for the half-Lipschitz function $f$.

######  Lemma 2.8.

Let $f$ satisfy Assumption 2.4 so that $f$ satisfies the decomposition (19).
Define a family $\\{f_{\lambda}:\lambda>0\\}$ by

$f_{\lambda}(u)=\phi_{\lambda}(u)+\kappa u$ (21)

where $\phi_{\lambda}$ are Yosida approximations of the non-increasing
function $\phi$. Then $f_{\lambda}$ satisfies the following properties.

  1. _(i)_

$|f_{\lambda}(u_{1})-f_{\lambda}(u_{2})|\leq\left(\frac{2}{\lambda}+\kappa\right)|u_{1}-u_{2}|$,
for $u_{1},u_{2}\in{\mathbb{R}}$ and all $\lambda>0$.

  2. _(ii)_

$|f_{\lambda}(u)|\leq(1+2\kappa)|f(u)|$, for $u\in{\mathbb{R}}$ and all
$\lambda>0$.

  3. _(iii)_

$(f_{\lambda}(u_{1})-f_{\lambda}(u_{2}))\,{\textnormal{sign}}(u_{1}-u_{2})\leq\kappa|u_{1}-u_{2}|$,
for $u_{1},u_{2}\in{\mathbb{R}}$ and all $\lambda>0$.

  4. _(iv)_

$\lim_{\lambda\to 0}f_{\lambda}(u)=f(u)$, for all $u\in{\mathbb{R}}$.

  5. _(v)_

$\lim_{\lambda\to 0}f_{\lambda}^{\prime}(u)=f^{\prime}(u)$, for all
$u\in{\mathbb{R}}$.

####  2.2.2. Mapping $\mathcal{M}$

The second ingredient we wish to highlight in the study of (1) is the
introduction of a functional mapping
$\mathcal{M}:\mathcal{C}_{\theta,x_{0}}([0,T]\times\mathbb{R}^{d})\to\mathcal{C}_{\theta,x_{0}}([0,T]\times\mathbb{R}^{d})$.

######  Definition 2.9.

For a continuous function
$z\in\mathcal{C}_{\theta,x_{0}}([0,T]\times\mathbb{R}^{d})$ let
$\mathcal{M}(z)$ be the solution to the following equation

$\mathcal{M}(z)(t,x)=\int_{0}^{t}\int_{\mathbb{R}^{d}}G(t-s,x-y)f(\mathcal{M}(z)(s,y))\,dyds+z(t,x).$
(22)

######  Remark 2.10.

In order to prove existence of the map $\mathcal{M}$ one uses an approximating
sequence $\\{\mathcal{M}_{\lambda};\,\lambda>0\\}$ defined as in (22), with
$f$ replaced by its Yosida approximation $f_{\lambda}$ given in (20). Then
some a priori estimates on $\mathcal{M}_{\lambda}(z)$ are provided in [24].
Those estimates allow to conclude the existence part, thanks to some
compactness arguments.

With our Malliavin calculus considerations in mind, we formulate a time and
space inhomogenous version of Theorem 5.6 of [24]. To this aim, we consider
$\varphi:[0,T]\times\mathbb{R}^{d}\times\mathbb{R}$ and assume that $\varphi$
is uniformly half-Lipschitz in the third argument. This means that there
exists $\kappa\in\mathbb{R}$ such that for any $t\in[0,T]$,
$x\in\mathbb{R}^{d}$, and $u_{1}>u_{2}\in\mathbb{R}$,

$\varphi(t,x,u_{1})-\varphi(t,x,u_{2})\leq\kappa(u_{1}-u_{2}).$ (23)

We also impose the growth restriction that there exist $K>0,\nu>0$,
$x_{0}\in{\mathbb{R}}^{d}$ and $\beta\in[0,2)$ such that for any $t\in[0,T]$

$|\varphi(t,x,u)|\leq Ke^{K(|x-x_{0}|^{\beta}+|u|^{\nu})}.$ (24)

We introduce a new functional mapping $\mathcal{L}$ in the following way.
Given $\varphi$ satisfying (23)–(24) and $z\in
C_{\theta,x_{0}}([0,T]\times\mathbb{R}^{d})$, let $\mathcal{L}(z)\in
C_{\theta,x_{0}}([0,T]\times{\mathbb{R}^{d}})$ denote the solution to

$\mathcal{L}(z)(t,x)=\int_{0}^{t}\int_{\mathbb{R}^{d}}G(t-s,x-y)\varphi(s,y,\mathcal{L}(z)(s,y))\,dyds+z(t,x).$
(25)

The growth restriction (24) guarantees that the above integral is finite if
$\mathcal{L}(z)\in\mathcal{C}_{\theta,x_{0}}([0,T]\times\mathbb{R}^{d})$ for
some $\theta\in(0,2/\nu)$.

######  Remark 2.11.

The existence of a solution $\mathcal{L}(z)\in
C_{\theta,x_{0}}([0,T]\times\mathbb{R}^{d})$ for any $z\in
C_{\theta,x_{0}}([0,T]\times\mathbb{R}^{d})$ can be proved via Yosida
approximations following the arguments of Theorem 5.2 of [24]. We will prove
the existence of Malliavin derivatives that solve (25) in Section 3 below, and
we have no need to prove the existence of $\mathcal{L}(z)$ in full generality.
We do need to prove that $\mathcal{L}$ features a global Lipschitz continuity
property on the domain where it exists and we will use this property
frequently in the sequel.

######  Theorem 2.12.

Consider a function $\varphi:[0,T]\times\mathbb{R}^{d}\times\mathbb{R}$
verifying (23)–(24) and a generic $x_{0}\in{\mathbb{R}}^{d}$. Let
$\theta\in(0,2/\nu)$. There exists $K=K(T,\theta,\kappa)>0$ such that if
$z_{1},z_{2}\in\mathcal{C}_{\theta,x_{0}}([0,T]\times\mathbb{R}^{d})$ and if
there exist
$\mathcal{L}(z_{1}),\mathcal{L}(z_{2})\in\mathcal{C}_{\theta,x_{0}}([0,T]\times\mathbb{R}^{d})$
that solve (25), then

$|\mathcal{L}(z_{1})-\mathcal{L}(z_{2})|_{\mathcal{C}_{\theta,x_{0}}([0,T]\times\mathbb{R}^{d})}\leq
K|z_{1}-z_{2}|_{\mathcal{C}_{\theta,x_{0}}([0,T]\times\mathbb{R}^{d})}.$ (26)

The constant $K$ does not depend on the center of the weight $x_{0}$ and only
depends on $\varphi$ through the parameter $\kappa$.

###### Proof.

Let $v_{i}(t,x):=\mathcal{L}(z_{i})(t,x)-z_{i}(t,x)$ for $i\in\\{1,2\\}$ and
let $\tilde{v}(t,x)=v_{1}(t,x)-v_{2}(t,x)$. The function $\tilde{v}$ is weakly
differentiable and

$\frac{\partial\tilde{v}}{\partial
t}(t,x)=\frac{1}{2}\Delta\tilde{v}(t,x)+\varphi(t,x,v_{1}(t,x)+z_{1}(t,x))-\varphi(t,x,v_{2}(t,x)+z_{2}(t,x)).$
(27)

Without loss of generality, we can assume that $\tilde{v}$ is strongly
differentiable by approximating $\tilde{v}$ using resolvent operators [4,
Proposition 6.2.2]. Let $\rho(x)=(1+|x-x_{0}|^{2})^{\frac{\theta}{2}}$ be a
twice-differentiable weight. Then the quotient
$\tilde{q}(t,x)=\frac{\tilde{v}(t,x)}{\rho(x)}$ satisfies

$\displaystyle\frac{\partial\tilde{q}}{\partial t}(t,x)=$
$\displaystyle\frac{1}{2}\Delta\tilde{q}(t,x)+\nabla\tilde{q}(t,x)\cdot\frac{\nabla\rho(x)}{\rho(x)}+\frac{1}{2}\tilde{q}(t,x)\frac{\Delta\rho(x)}{\rho(x)}$
$\displaystyle+\frac{\varphi(t,x,v_{1}(t,x)+z_{1}(t,x))-\varphi(t,x,v_{2}(t,x)+z_{2}(t,x))}{\rho(x)}.$
(28)

By the assumption that
$\tilde{v}\in\mathcal{C}_{\theta,x_{0}}([0,T]\times\mathbb{R}^{d})$, the
weighted difference $\tilde{q}$ sits in the space
$\mathcal{C}_{0}([0,T]\times\mathbb{R}^{d})$, meaning that
$\lim_{|x|\to\infty}\sup_{t\in[0,T]}|\tilde{q}(t,x)|=0.$ For any $t\in[0,T]$,
there exists at least one point $x_{t}\in\mathbb{R}^{d}$ where the supremum is
attained. Specifically,

$|\tilde{q}(t,x_{t})|=\sup_{x\in\mathbb{R}^{d}}|\tilde{q}(t,x)|.$ (29)

Furthermore, the upper-left derivative of the supremum is bounded by

$\frac{d^{-}}{dt}|\tilde{q}(t,\cdot)|_{\mathcal{C}_{0}}\leq\frac{\partial\tilde{q}}{\partial
t}(t,x_{t}){\textnormal{sign}}(\tilde{q}(t,x_{t})),$ (30)

where $x_{t}$ is any maximizer such that relation (29) holds true (see [24,
Proposition 3.5]). Therefore applying (30) and (2.2.2), the left derivative
above satisfies

$\frac{d^{-}}{dt}|\tilde{q}(t,\cdot)|_{\mathcal{C}_{0}}\leq\frac{1}{2}\Delta\tilde{q}(t,x_{t}){\textnormal{sign}}(\tilde{q}(t,x_{t}))+\nabla\tilde{q}(t,x_{t})\cdot\frac{\nabla\rho(x_{t})}{\rho(x_{t})}{\textnormal{sign}}(\tilde{q}(t,x_{t}))\\\
+\frac{1}{2}\tilde{q}(t,x_{t})\frac{\Delta\rho(x_{t})}{\rho(x_{t})}{\textnormal{sign}}(\tilde{q}(t,x_{t}))+\mathcal{Q}_{t}\,,$
(31)

where we have set

$\mathcal{Q}_{t}\equiv\frac{\varphi(t,x_{t},v_{1}(t,x_{t})+z_{1}(t,x_{t}))-\varphi(t,x_{t},v_{2}(t,x_{t})+z_{2}(t,x_{t}))}{\rho(x_{t})}{\textnormal{sign}}(\tilde{q}(t,x_{t}))$
(32)

We now examine the right hand side of (31).

  1. (i)

Because $x_{t}$ is a maximizer or minimizer for $\tilde{q}$, we have

$\nabla\tilde{q}(t,x_{t})=0.$ (33)

  2. (ii)

The convexity of a function at a local maximizer or minimizer guarantees that

$\Delta\tilde{q}(t,x_{t}){\textnormal{sign}}(\tilde{q}(t,x_{t}))\leq 0.$ (34)

  3. (iii)

Direct calculations verify that
$\sup_{x}\frac{\Delta\rho(x)}{\rho(x)}<+\infty$ so that

$\frac{1}{2}\tilde{q}(t,x_{t})\frac{\Delta\rho(x_{t})}{\rho(x_{t})}{\textnormal{sign}}(\tilde{q}(t,x_{t}))\leq
C|\tilde{q}(t,x_{t})|.$ (35)

  4. (iv)

We split the analysis of the $\mathcal{Q}_{t}$ term in (32) into two cases,
according to the relation
$|\tilde{q}(t,\cdot)|_{\mathcal{C}_{0}}>|\tilde{z}(t,\cdot)|_{\mathcal{C}_{0}}$
or
$|\tilde{q}(t,\cdot)|_{\mathcal{C}_{0}}\leq|\tilde{z}(t,\cdot)|_{\mathcal{C}_{0}}$.
Namely let $\tilde{z}(t,x)$ be the weighted difference

$\tilde{z}(t,x)=\frac{z_{1}(t,x)-z_{2}(t,x)}{\rho(x)}.$ (36)

If
$|\tilde{q}(t,\cdot)|_{\mathcal{C}_{0}}>|\tilde{z}(t,\cdot)|_{\mathcal{C}_{0}}$,
then

${\textnormal{sign}}(\tilde{q}(t,x_{t}))={\textnormal{sign}}(\tilde{q}(t,x_{t})+\tilde{z}(t,x_{t}))={\textnormal{sign}}(v_{1}(t,x_{t})+z_{1}(t,x_{t})-(v_{2}(t,x_{t})+z_{2}(t,x_{t}))).$
(37)

In this case, (23) guarantees that

$\mathcal{Q}_{t}\leq\kappa|\tilde{q}(t,\cdot)+\tilde{z}(t,\cdot)|_{\mathcal{C}_{0}}\leq
2\kappa|\tilde{q}(t,\cdot)|.$ (38)

Hence plugging (33)-(34)-(35) and (38) into (31), in the case where
$|\tilde{q}(t,\cdot)|_{\mathcal{C}_{0}}>|\tilde{z}(t,\cdot)|_{\mathcal{C}_{0}}$
we get

$\frac{d^{-}}{dt}|\tilde{q}(t,\cdot)|_{\mathcal{C}_{0}}\leq
C|\tilde{q}(t,\cdot)|_{\mathcal{C}_{0}},$ (39)

where the constant $C$ depends only on $\kappa$ and $\theta$.

On the other hand if
$|\tilde{z}(t,\cdot)|_{\mathcal{C}_{0}}>|\tilde{q}(t,\cdot)|_{\mathcal{C}_{0}}$,
then we cannot get a bound on the left derivative
$\frac{d^{-}}{dt}|\tilde{q}(t,\cdot)|_{\mathcal{C}_{0}}$, but this is not a
problem because in this case we have an explicit upper bound on
$|\tilde{q}(t,\cdot)|_{\mathcal{C}_{0}}$ itself. To deal with both of these
cases simultaneously, it is convenient to bound the left derivative of

$\max\left\\{|\tilde{q}(t,\cdot)|_{\mathcal{C}_{0}},M\right\\},\quad\text{where}\quad
M:=\sup_{s\in[0,T]}|\tilde{z}(t,\cdot)|_{\mathcal{C}_{0}}.$ (40)

Specifically, if $|\tilde{q}(t,\cdot)|_{\mathcal{C}_{0}}>M$, then the left
derivative of $\max\left\\{|\tilde{q}(t,\cdot)|_{\mathcal{C}_{0}},M\right\\}$
is (39), while if $|\tilde{q}(t,\cdot)|_{\mathcal{C}_{0}}\leq M$, then the
left derivative of
$\max\left\\{|\tilde{q}(t,\cdot)|_{\mathcal{C}_{0}},M\right\\}$ is $0$.

From (31)-(39)-(40) and the considerations above, we can see that for any
fixed $T>0$ and for any $t\in[0,T]$ we have

$\frac{d^{-}}{dt}\max\left\\{|\tilde{q}(t,\cdot)|,M\right\\}\leq
C|\tilde{q}(t,\cdot)|_{\mathcal{C}_{0}}\leq
C\max\left\\{|\tilde{q}(t,\cdot)|,M\right\\}.$ (41)

Using the fact that $\tilde{q}(0,x)\equiv 0$, one can integrate (41) in order
to get an exponential growth bound:

$\sup_{t\in[0,T]}\max\left\\{|\tilde{q}(t,\cdot)|,M\right\\}\leq Me^{CT}.$
(42)

Therefore because
$M=\sup_{t\in[0,T]}\sup_{x\in\mathbb{R}^{d}}|\tilde{z}(t,\cdot)|_{\mathcal{C}_{0}}$,

$\sup_{t\in[0,T]}\sup_{x\in\mathbb{R}^{d}}|\tilde{q}(t,x)|\leq
e^{CT}\sup_{t\in[0,T]}\sup_{x\in\mathbb{R}^{d}}|\tilde{z}(t,\cdot)|_{\mathcal{C}_{0}}.$
(43)

The definitions of $\tilde{q}$ and $\tilde{z}$ guarantee that

$\sup_{t\in[0,T]}\sup_{x\in\mathbb{R}^{d}}\frac{|v_{1}(t,x)-v_{2}(t,x)|}{1+|x-x_{0}|^{\theta}}\leq
e^{CT}\sup_{t\in[0,T]}\sup_{x\in\mathbb{R}^{d}}\frac{|z_{1}(t,x)-z_{2}(t,x)|}{1+|x-x_{0}|^{\theta}}.$
(44)

Our claim (26) follows because $\mathcal{L}(z_{i})=v_{i}+z_{i}$. ∎

As a particular case of Theorem 2.12 for a homogeneous function $\varphi$, we
get the fact that $\mathcal{M}$ in Definition 2.9 is a Lipschitz map on
$\mathcal{C}_{\theta,x_{0}}([0,T]\times\mathbb{R}^{d})$ (this was the content
of Theorem 5.6 in [24]). The existence of $\mathcal{M}$ was proved in Theorem
5.2 of [24].

######  Proposition 2.13.

Suppose Assumption 2.4 is satisfied. Let
$\theta\in\left(0,\frac{2}{\nu}\right)$ where $\nu$ is from (14) For any
$z\in\mathcal{C}_{\theta,x_{0}}([0,T]\times\mathbb{R}^{d})$ there exists a
unique solution
$\mathcal{M}(z)\in\mathcal{C}_{\theta,x_{0}}([0,T]\times\mathbb{R}^{d})$ to
(22). Furthermore, $\mathcal{M}$ is globally Lipschitz continuous.
Specifically, there exists a constant $K=K(T,\theta,\kappa)$, depending only
on the weight parameter $\theta$ and $\kappa$ from Assumption 2.4 such that
for any two functions $z_{1},z_{2}\in C_{\theta}([0,T]\times\mathbb{R}^{d})$,

$|\mathcal{M}(z_{2})-\mathcal{M}(z_{1})|_{\mathcal{C}_{\theta,x_{0}}([0,T]\times\mathbb{R}^{d})}\leq
K\,|z_{2}-z_{1}|_{\mathcal{C}_{\theta,x_{0}}([0,T]\times\mathbb{R}^{d})}.$
(45)

####  2.2.3. Approximation scheme for existence and uniqueness

With the preliminary results in Sections 2.2.2 and 2.2.1, the Picard
iterations approximating the solution of (2.1) are defined as follows in [24].

  1. _(i)_

Initiate the iterations by setting

$\displaystyle U_{0}(t,x)$ $\displaystyle:=$
$\displaystyle\int_{\mathbb{R}^{d}}G(t,x-y)u_{0}(y)dy$ (46) $\displaystyle
Z_{0}(t,x)$ $\displaystyle:=$ $\displaystyle 0$ (47)

  2. _(ii)_

Given $(u_{n},Z_{n})$, define

$Z_{n+1}(t,x):=\int_{0}^{t}\int_{{\mathbb{R}}^{d}}G(t-s,x-y)\sigma(u_{n}(s,y))W(ds\,dy).$
(48)

  3. _(iii)_

Once $Z_{n+1}$ is introduced, set

$u_{n+1}(t,x):=\mathcal{M}(U_{0}+Z_{n+1})(t,x).$ (49)

It is proved in [24, Theorem 5.2] that the sequences $\\{u_{n};\,n\geq 0\\}$
and $\\{Z_{n};\,n\geq 0\\}$ converge to the solution of (2.1). Specifically,
the following relation hold true for all $p\geq 1$:

$\displaystyle\lim_{n\to\infty}\sup_{x_{0}\in\mathbb{R}^{d}}{\mathbb{E}}\left|\sup_{t\in[0,T]}\sup_{x\in\mathbb{R}^{d}}\frac{|Z_{n}(t,x)-Z(t,x)|}{1+|x-x_{0}|^{\theta}}\right|^{p}=0.$
(50)
$\displaystyle\lim_{n\to\infty}\sup_{x_{0}\in\mathbb{R}^{d}}{\mathbb{E}}\left|\sup_{t\in[0,T]}\sup_{x\in\mathbb{R}^{d}}\frac{|u_{n}(t,x)-u(t,x)|}{1+|x-x_{0}|^{\theta}}\right|^{p}=0,$
(51)

where $u$ solves (2.1) and $Z$ is the stochastic convolution

$Z(t,x):=\int_{0}^{t}\int_{{\mathbb{R}}^{d}}G(t-s,x-y)\sigma(u(s,y))W(ds\,dy).$
(52)

Also recall that in item  _(iii)_ above, the mapping $\mathcal{M}$ is defined
through a limiting procedure involving the Yosida approximations $f_{\lambda}$
of $f$.

###  2.3. Malliavin Calculus

This section is devoted to review some elementary notions of Malliavin
calculus (mostly borrowed from [20]). We first recall that our noise $\dot{W}$
is a Gaussian centered field whose covariance is formally given by (11). One
can also look at $\dot{W}$ as a centered Gaussian family
$\\{W(\varphi);\,\varphi\in\mathcal{H}_{T}\\}$, where $\mathcal{H}_{T}$
denotes the Hilbert space with inner product

$\left<\phi,\psi\right>_{{\mathcal{H}}_{T}}=\int_{0}^{T}\int_{\mathbb{R}^{d}}\int_{\mathbb{R}^{d}}\phi(t,y_{1})\psi(t,y_{2})\Lambda(y_{1}-y_{2})\,dy_{1}dy_{2}dt.$
(53)

Let $\mathcal{S}$ be the set of smooth and cylindrical random variables of the
form

$F=g\left(W(h_{1}),\ldots,W(h_{N})\right),$

where $N\geq 1$, $g\in C_{b}^{\infty}(\mathbb{R}^{N})$ and
$h_{1},\ldots,h_{N}\in\mathcal{H}_{T}$. For every $\ell\in\mathcal{H}_{T}$,
the partial Malliavin derivative of $F$ in the direction of $\ell$ is defined
for $F\in\mathcal{S}$ as the random variable

$D_{\ell}F=\sum_{i}^{N}\frac{\partial g}{\partial
x_{i}}\left(W(h_{1}),\ldots,W(h_{N})\right)\left\langle
h_{i},\ell\right\rangle_{\mathcal{H}_{T}}.$ (54)

Relation (54) can also be seen as an equation for $\langle
DF,\,\ell\rangle_{\mathcal{H}_{T}}$, where $DF$ is now a
$\mathcal{H}_{T}$-valued random variable. We can iterate this procedure to
define higher order derivatives $D^{k}_{\ell_{1}\cdots\ell_{k}}F$, which
produces a $\mathcal{H}^{\otimes k}$-valued random variable. For any $p\geq 1$
and integer $k\geq 1$, we define the Sobolev space $\mathbb{D}^{k,p}$ as the
closure of $\mathcal{S}$ with respect to the norm

$\|F\|_{k,p}^{p}=\mathbb{E}[|F|^{p}]+\sum_{i=1}^{k}\mathbb{E}\left[\|D^{i}F\|^{p}_{\mathcal{H}^{\otimes
l}}\right].$ (55)

If $V$ is Hilbert space, $\mathbb{D}^{k,p}(V)$ denotes the corresponding
Sobolev space of $V$-valued random variables.

The existence of a density for $u(t,x)$, the mild solution of (1), is obtained
by appling the following criterion borrowed from [20, Theorem 2.1.2].

######  Proposition 2.14.

Let $F$ be a real-valued random variable in $\mathbb{D}^{1,p}$ for some $p>1$,
such that $\|DF\|_{{\mathcal{H}}_{T}}>0$ with probability one. Then the law of
$F$ is absolutely continuous with respect to the Lebesgue measure in
$\mathbb{R}$.

When proceeding by approximations, we will rely on a technical result
summarized below (see [20, Lemma 1.2.3]) in order to probe Malliavin
differentiability.

######  Proposition 2.15.

Let $\\{F_{\lambda}:\lambda>0\\}$ be a sequence of random variables such that

$F_{\lambda}\in\mathbb{D}^{1,2}\ \text{ for all
$\lambda>0$,}\quad\lim_{\lambda\to
0}{\mathbb{E}}\left[|F_{\lambda}-F|^{2}\right]=0,\quad\sup_{\lambda}{\mathbb{E}}\left[\|DF_{\lambda}\|_{{\mathcal{H}}_{T}}^{2}\right]<+\infty.$
(56)

Then $F\in\mathbb{D}^{1,2}$ and the sequence $\\{DF_{\lambda}:\lambda>0\\}$
converges weakly to $DF$ in $L^{2}(\Omega:{\mathcal{H}}_{T})$ as $\lambda\to
0$.

We now state a differentiation rule for stochastic integrals that will be
invoked to differentiate solutions to stochastic pdes.

######  Proposition 2.16 (Section 1.3.1 of [20]).

Let $X$ be an adapted random field in $\mathbb{D}^{1,2}(\mathcal{H}_{T})$, and
define the Itô stochastic integral

$\mathcal{J}_{T}(X)=\int_{0}^{T}\int_{{\mathbb{R}}^{d}}X(s,x)\,W(ds\,dy).$

Then $\mathcal{J}_{T}(X)$ is an element of $\mathbb{D}^{1,2}$, and for any
$h\in\mathcal{H}_{T}$ we have

$D_{h}\mathcal{J}_{T}(X)=\left\langle
X,\,h\right\rangle_{\mathcal{H}_{T}}+\int_{0}^{T}\int_{{\mathbb{R}}^{d}}D_{h}X(s,x)\,W(ds\,dy).$

The next result gives an easy to check condition that guarantees that a
stochastic convolution with the fundamental solution to the heat equation
satisfies the assumptions of Proposition 2.16.

######  Proposition 2.17.

Let $X$ be an adapted random field such that for any
$(t,x)\in[0,T]\times\mathbb{R}^{d}$, we have $X(t,x)\in\mathbb{D}^{1,2}$. In
addition, we assume that

$\sup_{t\in[0,T]}\sup_{x\in\mathbb{R}^{d}}{\mathbb{E}}\left[|X(t,x)|^{2}\right]<+\infty,$
(57)

and

$\sup_{t\in[0,T]}\sup_{x\in\mathbb{R}^{d}}{\mathbb{E}}\left[|DX(t,x)|_{{\mathcal{H}}_{T}}^{2}\right]<+\infty.$
(58)

Let $G$ be the fundamental solution of the heat equation. Define the
stochastic convolution

$\mathcal{I}(t,x)=\mathcal{I}^{X}(t,x)=\int_{0}^{t}\int_{\mathbb{R}^{d}}G(t-s,x-y)X(s,y)W(ds\,dy).$
(59)

Then for any $(t,x)\in[0,T]\times\mathbb{R}^{d}$, $\mathcal{I}(t,x)$ is an
element of $\mathbb{D}^{1,2}$, and for any $h\in\mathcal{H}_{T}$ we have

$D_{h}\mathcal{I}(t,x)=\left\langle
G(t-\cdot,x-\cdot)X(\cdot,\cdot),\,h\right\rangle_{\mathcal{H}_{T}}+\int_{0}^{t}\int_{{\mathbb{R}}^{d}}G(t-s,x-y)D_{h}X(s,y)\,W(ds\,dy).$
(60)

In the above expression $G(t-r,x-y)$ is defined to be $0$ is $r>t$.

###### Proof.

To apply Proposition 2.16, we need to first verify that for fixed
$(t,x)\in[0,T]\times\mathbb{R}^{d}$, the integrand $(s,y)\mapsto
G(t-s,x-y)X(s,y)$ is an element of $\mathbb{D}^{1,2}({\mathcal{H}}_{T})$.
Fortunately, a straightforward consequence of the fact that $G(\cdot,\cdot)$
and $\Lambda(\cdot)$ are positive is that

$\displaystyle{\mathbb{E}}|G(t-\cdot,x-\cdot)X(\cdot,\cdot)|_{{\mathcal{H}}_{T}}^{2}$
$\displaystyle={\mathbb{E}}\int_{0}^{t}\int_{\mathbb{R}^{d}}\int_{\mathbb{R}^{d}}G(t-s,x-y_{1})G(t-s,x-y_{2})X(s,y_{1})X(s,y_{2})\Lambda(y_{1}-y_{2})dy_{1}dy_{2}ds$
$\displaystyle\leq
C\left(\sup_{s\in[0,t]}\sup_{y\in\mathbb{R}^{d}}{\mathbb{E}}|X(s,y)|^{2}\right)Q_{\Lambda}(t)\,$
(61)

where we have set

$Q_{\Lambda}(t)=\int_{0}^{t}\int_{\mathbb{R}^{d}}\int_{\mathbb{R}^{d}}G(t-s,x-y_{1})G(t-s,x-y_{2})\Lambda(y_{1}-y_{2})dy_{1}dy_{2}ds\,.$
(62)

Along the same lines, we also have

${\mathbb{E}}|G(t-\cdot,x-\cdot)DX(\cdot,\cdot)|_{{\mathcal{H}}_{T}}^{2}={\mathbb{E}}\int_{0}^{t}\int_{\mathbb{R}^{d}}\int_{\mathbb{R}^{d}}G(t-s,x-y_{1})G(t-s,x-y_{2})\\\
\times\left<DX(s,y_{1}),DX(s,y_{2})\right>_{\mathcal{H}_{T}}\Lambda(y_{1}-y_{2})dy_{1}dy_{2}ds,$

and therefore

${\mathbb{E}}|G(t-\cdot,x-\cdot)DX(\cdot,\cdot)|_{{\mathcal{H}}_{T}}^{2}\leq
C\left(\sup_{s\in[0,t]}\sup_{y\in\mathbb{R}^{d}}{\mathbb{E}}|DX(s,y)|_{{\mathcal{H}}_{T}}^{2}\right)Q_{\Lambda}(t).$
(63)

Now taking Fourier transforms as in [7], Assumption 2.2 guarantees that

$Q_{\Lambda}(t)=\int_{0}^{t}\int_{\mathbb{R}^{d}}e^{-(t-s)|\xi|^{2}}\mu(d\xi)<+\infty.$
(64)

Plugging this relation into (2.3) and (63), then taking hypothesis (57)-(58)
into account, our claim is a direct consequence of Proposition 2.16. ∎

##  3\. Malliavin differentiability of $\mathcal{M}$

In Section 2.2.3 we gave the iteration scheme allowing to solve (2.1). We
shall now follow the very same scheme in order to show Malliavin
differentiability, and we start by analyzing the mapping $\mathcal{M}$ defined
by (22). Namely we showed in [24] that $\mathcal{M}$ is a Lipschitz continuous
map on the weighted spaces
$\mathcal{C}_{\theta,x_{0}}([0,T]\times\mathbb{R}^{d})$. In this section we
prove the following proposition about the Malliavin differentiability of
$\mathcal{M}(z)$.

######  Proposition 3.1.

Let $\theta\in\left(0,\frac{2}{\nu}\right)$ where $\nu$ is from (15), and pick
a generic $x_{0}\in\mathbb{R}^{d}$. Denote by
$L^{2}(\Omega:\mathcal{C}_{\theta,x_{0}}([0,T]\times\mathbb{R}^{d}))$ the set
of $\mathcal{C}_{\theta,x_{0}}([0,T]\times\mathbb{R}^{d}))$-valued square
integrable random variables. Consider $z\in
L^{2}(\Omega:\mathcal{C}_{\theta,x_{0}}([0,T]\times\mathbb{R}^{d}))$ which has
the properties that $z(t,x)$ is Malliavin differentiable for all
$(t,x)\in[0,T]\times\mathbb{R}^{d}$ and

${\mathbb{E}}\left|\sup_{t\in[0,T]}\sup_{x\in\mathbb{R}^{d}}\frac{|Dz(t,x)|_{{\mathcal{H}}_{T}}}{1+|x-x_{0}|^{\theta}}\right|^{2}<+\infty.$
(65)

Let $\mathcal{M}(z)$ be given as in Definition 2.9 and assume Assumptions
2.3–2.4 for $\sigma$ and $b$. Then $\mathcal{M}(z)(t,x)$ is also Malliavin
differentiable for all $(t,x)\in[0,T]\times\mathbb{R}^{d}$ and almost surely
we have

$\sup_{t\in[0,T]}\sup_{x\in\mathbb{R}^{d}}\frac{|D[\mathcal{M}(z)(t,x)]|_{{\mathcal{H}}_{T}}}{1+|x-x_{0}|^{\theta}}\leq
K\sup_{t\in[0,T]}\sup_{x\in\mathbb{R}^{d}}\frac{|Dz(t,x)|_{{\mathcal{H}}_{T}}}{1+|x-x_{0}|^{\theta}},$
(66)

where $K=K(T,\theta,\kappa)$ is also the Lipschitz constant of $\mathcal{M}$
in (45), which does not depend on $x_{0}\in\mathbb{R}^{d}$.

We prove Proposition 3.1 in several steps. First, we prove this in the simpler
case where $f$ is globally Lipschitz continuous.

######  Lemma 3.2.

Let $\theta\in\left(0,\frac{2}{\nu}\right)$ where $\nu$ is from (15). Let
$x_{0}\in\mathbb{R}^{d}$. Assume that $f:\mathbb{R}\to\mathbb{R}$ is
differentiable and globally Lipschitz continuous and that
$\sup_{u}f^{\prime}(u)\leq\kappa$. Then if $z\in
L^{2}(\Omega:\mathcal{C}_{\theta,x_{0}}([0,T]\times\mathbb{R}^{d}))$ has the
properties that $z(t,x)$ is Malliavin differentiable for all
$(t,x)\in[0,T]\times\mathbb{R}^{d}$ and

${\mathbb{E}}\Bigg{|}\sup_{t\in[0,T]}\sup_{x\in\mathbb{R}^{d}}\frac{|Dz(t,x)|_{{\mathcal{H}}_{T}}}{1+|x-x_{0}|^{\theta}}\Bigg{|}^{2}<+\infty,$
(67)

then $\mathcal{M}(z)$ is also Malliavin differentiable and with probability
one we have

$\sup_{t\in[0,T]}\sup_{x\in\mathbb{R}^{d}}\frac{|D[\mathcal{M}(z)(t,x)]|_{{\mathcal{H}}_{T}}}{1+|x-x_{0}|^{\theta}}\leq
K\sup_{t\in[0,T]}\sup_{x\in\mathbb{R}^{d}}\frac{|Dz(t,x)|_{{\mathcal{H}}_{T}}}{1+|x-x_{0}|^{\theta}}\,,$
(68)

where $K=K(T,\theta,\kappa)$ is also the Lipschitz constant of $\mathcal{M}$.
Notice that $K$ depends on $\kappa$, the upper bound of $f^{\prime}(u)$, but
does not depend on the lower bound of $f^{\prime}(u)$.

###### Proof.

We define a sequence of functions $\\{m_{n}:n\geq 1\\}$ by Picard iterations
by

$\displaystyle m_{1}(t,x)=z(t,x)$ $\displaystyle
m_{n+1}(t,x)=\int_{0}^{t}\int_{\mathbb{R}^{d}}G(t-s,x-y)f(m_{n}(s,y))dyds+z(t,x).$

Standard arguments based on the Lipschitz continuity of $f$ show that $m_{n}$
converge to $m:=\mathcal{M}(z)$ in the
$L^{2}(\mathcal{C}_{\theta,x_{0}}([0,T]\times\mathbb{R}^{d}))$ topology.
Furthermore, $m_{1}(t,x)$ is Malliavin differentiable by assumption. Then by
induction and using the fact that integration and Malliavin differentiability
commute, $m_{n}(t,x)$ is Malliavin differentiable for all $n\geq 1$ and

$Dm_{n+1}(t,x)=\int_{0}^{t}\int_{\mathbb{R}^{d}}G(t-s,x-y)f^{\prime}(m_{n}(s,y))Dm_{n}(s,y)dyds+Dz(t,x).$
(69)

Notice that in order to get (69), we imposed the additional assumption that
$f^{\prime}$ is bounded. Let $L:=\sup_{u\in\mathbb{R}}|f^{\prime}(u)|$. We
also set

$\Phi_{n}(t):=\sup_{x\in\mathbb{R}^{d}}\sup_{s\in[0,t]}\frac{|Dm_{n}(s,x)|_{{\mathcal{H}}_{T}}}{1+|x-x_{0}|^{\theta}}.$
(70)

We assumed in the statement of the lemma that $\Phi_{1}(t)$ is finite with
probability one, since $m_{1}=z$ and $z$ satisfies (67). From (69) we can see
that for any $n\geq 1$ and $(t,x)\in[0,T]\times{\mathbb{R}}^{d}$ we have

$|Dm_{n+1}(t,x)|_{{\mathcal{H}}_{T}}\leq\int_{0}^{t}\int_{\mathbb{R}^{d}}G(t-s,x-y)L|Dm_{n}(s,y)|_{{\mathcal{H}}_{T}}dyds+|Dz(t,x)|_{{\mathcal{H}}_{T}}.$
(71)

Next, we observe that $|Dm_{n}(s,y)|\leq\Phi_{n}(s)(1+|y|^{\theta})$ by the
definition (70) of $\Phi_{n}$. Furthermore, due to the fact that $G$ is a
Gaussian kernel, we have

$\int_{\mathbb{R}^{d}}G(t-s,x-y)(1+|y-x_{0}|^{\theta})dy\leq
C(1+(t-s)^{\frac{\theta}{2}}+|x-x_{0}|^{\theta}).$ (72)

Plugging (72) into (71), there exists $C_{T}>0$ such that for any $t\in[0,T]$

$\Phi_{n+1}(t)\leq
C_{T}\int_{0}^{t}\Phi_{n}(s)ds+\sup_{r\in[0,T]}\sup_{x\in\mathbb{R}^{d}}\frac{|Dz(r,x)|_{\mathcal{H}_{T}}}{1+|x-x_{0}|^{\theta}}.$
(73)

Hence it is easily seen by induction that $\Phi_{n}$ satisfies the following
inequality, uniformly in $n$,

$\sup_{t\in[0,T]}\Phi_{n}(t)\leq
e^{C_{T}T}\sup_{t\in[0,T]}\sup_{x\in\mathbb{R}^{d}}\frac{|Dz(t,x)|}{1+|x-x_{0}|^{\theta}},$
(74)

with probability one. In particular, if we fix
$(t,x)\in[0,T]\times\mathbb{R}^{d}$, we get

$\sup_{n}{\mathbb{E}}|Dm_{n}(t,x)|^{2}\leq
c_{T,x}\,{\mathbb{E}}\left|\sup_{r\in[0,T]}\sup_{x\in\mathbb{R}^{d}}\frac{|Dz(r,x)|}{1+|x-x_{0}|^{\theta}}\right|^{2}<+\infty.$
(75)

Therefore by Proposition 2.15 and the fact that
${\mathbb{E}}|m_{n}(t,x)-m(t,x)|^{2}\to 0,$ the random variable $m(t,x)$ is
Malliavin differentiable and $Dm_{n}(t,x)$ converges weakly to $Dm(t,x)$ in
$L^{2}(\Omega;{\mathcal{H}}_{T})$ as $n\to\infty$. Taking limits in (69)
thanks to a standard procedure, $Dm(t,x)$ must be the solution to

$Dm(t,x)=\int_{0}^{t}G(t-s,x-y)f^{\prime}(m(s,y))Dm(s,y)dyds+Dz(t,x),$ (76)

where we recall that (76) admits a unique solution if
$\sup_{u}|f^{\prime}(u)|=L<\infty$.

Now that we have shown that $Dm(t,x)$ exists and solves (76), we argue that we
can improve the bound on $|Dm(t,x)|_{{\mathcal{H}}_{T}}$ so that it depends
only on the upper bound $\kappa:=\sup_{u}f^{\prime}(u)$ and not on the
Lipschitz constant $L=\sup_{u}|f^{\prime}(u)|$. To this aim, by the linearity
of (76), for any $h\in{\mathcal{H}}_{T}$,

$D_{h}m(t,x)=\int_{0}^{t}G(t-s,x-y)f^{\prime}(m(s,y))D_{h}m(t,x)dyds+D_{h}z(t,x).$
(77)

Recall that $m=\mathcal{M}(z)$ is the unique solution to (22) and define the
function $\varphi:[0,T]\times\mathbb{R}^{d}\times\mathbb{R}\to\mathbb{R}$ by

$\varphi(t,x,V)=f^{\prime}(m(t,x))V.$ (78)

Because $\kappa:=\sup_{u}f^{\prime}(u)<+\infty$, the following inequality
holds for any $t\in[0,T]$, $x\in\mathbb{R}^{d}$, and $V_{1}>V_{2}$:

$\varphi(t,x,V_{1})-\varphi(t,x,V_{2})\leq\kappa(V_{1}-V_{2}).$ (79)

The growth condition (24) is fulfilled for $\varphi$ because
$m\in\mathcal{C}_{\theta,x_{0}}$ and $f$ is globally Lipschitz continuous by
assumption. Hence for $L=\sup_{u}|f^{\prime}(u)|$,

$|\varphi(t,x,V)|\leq L|V|.$ (80)

Therefore, $D_{h}m$ satisfies the assumptions of Theorem 2.12. Moreover, it is
readily checked that for $D_{h}z\equiv 0$, equation (77) admits
$D_{h}m(t,x)=0$ as a unique solution. Hence there exists
$K=K(T,\theta,\kappa)$ depending only on $T,\theta,$ and $\kappa$, but not
$L$, such that

$\sup_{t\in[0,T]}\sup_{x\in\mathbb{R}^{d}}\frac{|D_{h}m(t,x)-0|}{1+|x-x_{0}|^{\theta}}\leq
K\sup_{t\in[0,T]}\sup_{x\in\mathbb{R}^{d}}\frac{|D_{h}z(t,x)-0|}{1+|x-x_{0}|^{\theta}}\,,$
(81)

with probability one. In particular, for any $t\in[0,T],x\in\mathbb{R}^{d}$
and $|h|_{{\mathcal{H}}_{T}}=1$

$\frac{|D_{h}m(t,x)|}{1+|x-x_{0}|^{\theta}}\leq
K\sup_{t\in[0,T]}\sup_{x\in\mathbb{R}^{d}}\frac{|Dz(t,x)|_{{\mathcal{H}}_{T}}}{1+|x-x_{0}|^{\theta}}\,,$
(82)

with probability one. Taking the supremum over $|h|_{{\mathcal{H}}_{T}}=1$,
$t\in[0,T]$ and $x\in\mathbb{R}^{d}$,

$\sup_{t\in[0,T]}\sup_{x\in\mathbb{R}^{d}}\frac{|Dm(t,x)|_{{\mathcal{H}}_{T}}}{1+|x|^{\theta}}\leq
K\sup_{t\in[0,T]}\sup_{x\in\mathbb{R}^{d}}\frac{|Dz(t,x)|_{{\mathcal{H}}_{T}}}{1+|x-x_{0}|^{\theta}}.$
(83)

This proves our claim (68) under the assumption that $f$ is globally Lipschitz
continuous. ∎

Now we start a limiting procedure in order to prove Proposition 3.1. Namely
let $f$ be any force satisfying Assumption 2.4 and let $f_{\lambda}$ be the
Yosida approximation defined by (21). By Lemma 3.2, for each $\lambda>0$,
because each $f_{\lambda}$ is Lipschitz continuous, there exists a unique
$m_{\lambda}$ solving (see also item  _(iii)_ in Section 2.2.3):

$m_{\lambda}(t,x)=\int_{0}^{t}\int_{\mathbb{R}^{d}}G(t-s,x-y)f_{\lambda}(m_{\lambda}(s,y))dyds+z(t,x).$
(84)

Due to the fact that $f_{\lambda}$ satisfies _(iii)_ in Lemma 2.8 uniformly in
$\lambda$ (for a given $\kappa>0$), it follows from Theorem 2.12 (see also
[24, Corollary 5.5]) that $m_{\lambda}$ is such that

$\sup_{t\in[0,T]}\sup_{x\in\mathbb{R}^{d}}\frac{|m_{\lambda}(t,x)|}{1+|x-x_{0}|^{\theta}}\leq
K\left(1+\sup_{t\in[0,T]}\sup_{x\in\mathbb{R}^{d}}\frac{|z(t,x)|}{1+|x-x_{0}|^{\theta}}\right),$
(85)

where the constant $K$ depends only on $T,\theta$ and $\kappa$. Moreover,
Lemma 3.2 guarantees that $m_{\lambda}(t,x)$ is Malliavin differentiable.
According to (76), the Malliavin derivative satisfies

$D_{h}m_{\lambda}(t,x)=\int_{0}^{t}\int_{\mathbb{R}^{d}}G(t-s,x-y)f^{\prime}_{\lambda}(m_{\lambda}(s,y))D_{h}m_{\lambda}(s,y)dyds+D_{h}z(t,x).$
(86)

In this context, relation (68) can be read as

$\sup_{t\in[0,T]}\sup_{x\in\mathbb{R}^{d}}\frac{|Dm_{\lambda}(t,x)|_{{\mathcal{H}}_{T}}}{1+|x|^{\theta}}\leq
K\sup_{t\in[0,T]}\sup_{x\in\mathbb{R}^{d}}\frac{|Dz(t,x)|_{{\mathcal{H}}_{T}}}{1+|x|^{\theta}}\,,$
(87)

and the constant $K$ is like in (85). Notice again from Lemma 2.8 that all of
the $f_{\lambda}$ have the same half-Lipschitz constant $\kappa$.

In the following lemma we improve on the approximation results in [24], and
show that $m_{\lambda}(t,x)$ converges in $L^{2}(\Omega)$.

######  Lemma 3.3.

Let $m_{\lambda}$ be the Yosida approximations defined in (84) and assume that
$z$ satisfies the assumptions of Proposition 3.1. Then for any fixed
$(t,x)\in[0,T]\times\mathbb{R}^{d}$, we have

$\lim_{\lambda\to 0}{\mathbb{E}}|m_{\lambda}(t,x)-m(t,x)|^{2}=0\,,$ (88)

where $m(t,x)$ is the unique solution to

$m(t,x)=\int_{0}^{t}\int_{\mathbb{R}^{d}}G(t-s,x-y)f(m(s,y))dyds+z(t,x).$ (89)

###### Proof.

First, we prove that $m_{\lambda}(t,x)$ converges almost surely to $m(t,x)$.
Let $v_{\lambda}(t,x)=m_{\lambda}(t,x)-z(t,x)$. Then

$v_{\lambda}(t,x)=\int_{0}^{t}\int_{\mathbb{R}^{d}}G(t-s,x-y)f_{\lambda}(m_{\lambda}(s,y))dyds$
(90)

By the growth rate assumption (15) and the bound (85),

$|f_{\lambda}(m_{\lambda}(t,x))|\leq
C(1+|x-x_{0}|^{\theta})\exp(C(1+|x-x_{0}|^{\nu\theta}))\,,$ (91)

where $C=C(\omega)$ is some random constant that is independent of $\lambda$.
If $\nu\theta<2$, (91) along with standard properties of the heat kernel $G$
prove that for almost any fixed $\omega\in\Omega$ the family $\\{(t,x)\mapsto
v_{\lambda}(t,x):\lambda\in(0,1)\\}$ is equibounded and equicontinuous for
$(t,x)$ in any compact subset of $[0,T]\times\mathbb{R}^{d}$. The Arzela-
Ascoli theorem guarantees that there exists a subsequence $\lambda_{n}\to 0$
such that $v_{\lambda_{n}}(t,x)$ converges to a limit $v^{*}(t,x)$, uniformly
on compact subsets of $(t,x)\in[0,T]\times\mathbb{R}^{d}$. Because
$m_{\lambda}=v_{\lambda}+z$, invoking the bound (91), properties (ii) and (iv)
in Lemma 2.8 plus some standard dominated convergence arguments, we get that
this limit solves

$v^{*}(t,x)=\int_{0}^{t}\int_{\mathbb{R}^{d}}G(t-s,x-y)f(v^{*}(s,y)+z(s,y))dyds.$
(92)

Thus if we define $m^{*}(t,x):=v^{*}(t,x)+z(t,x)$, then $m^{*}$ solves

$m^{*}(t,x)=\int_{0}^{t}\int_{\mathbb{R}^{d}}G(t-s,x-y)f(m^{*}(s,y))dyds+z(t,x).$
(93)

The solution to (93) is unique by Proposition 2.13. Therefore, the same
Arzela-Ascoli argument proves that every subsequence of $m_{\lambda}$ has a
further subsequence that converges to $m^{*}$ and therefore $\lim_{\lambda\to
0}m_{\lambda}(t,x)=m^{*}(t,x)$ with probability one.

Finally, the $L^{2}(\Omega)$ convergence of $m_{\lambda}(t,x)$ to $m^{*}(t,x)$
is a consequence of the almost sure bound (85), the assumption that $z\in
L^{2}(\Omega:C_{\theta}([0,T]\times\mathbb{R}^{d}))$, and the dominated
convergence theorem. ∎

After this series of preliminary Lemmas, we are now ready to prove the main
result of this section.

###### Proof of Proposition 3.1.

Let $z$ satisfy the assumptions of Proposition 3.1 and let $m_{\lambda}$ be
defined in (84). By (87) and the fact that $z$ satisfies (65), for any fixed
$(t,x)\in[0,T]\times\mathbb{R}^{d}$,

$\sup_{\lambda\in(0,1)}{\mathbb{E}}|Dm_{\lambda}(t,x)|_{\mathcal{H}_{T}}^{2}<+\infty.$
(94)

Lemma 3.3 proves that

$\lim_{\lambda\to 0}{\mathbb{E}}\left|m_{\lambda}(t,x)-m(t,x)\right|^{2}=0.$
(95)

The assumptions of Proposition 2.15 are therefore satisfied. The limit
$m(t,x)$ is Malliavin differentiable and verifies (66). ∎

######  Remark 3.4.

Applying Proposition 2.15 as above, we get the weak convergence of
$Dm_{\lambda}(t,x)$ to $Dm(t,x)$. Combining this with the almost sure bounds
(85), (87) and the exponential growth assumption (15), similarly to what we
did for (93), guarantees that $Dm(t,x)$ solves

$Dm(t,x)=\int_{0}^{t}\int_{\mathbb{R}^{d}}G(t-s,x-y)f^{\prime}(m(s,y))Dm(s,y)dyds+Dz(t,x).$
(96)

where $m=\mathcal{M}(z)$.

##  4\. Malliavin differentiability of the mild solution

As mentioned at the beginning of Section 3, our analysis of the Malliavin
differentiability of $u$ (solution to (2.1)) follows the scheme outlined in
Section 2.2.3. In this section we focus our attention on the stochastic
convolution $Z$ given in the iteration step (48). We prove Malliavin
differentiability via an approximation procedure and Proposition 2.15. Some
preliminary results are presented in Section 4.1 and the Malliavin
differentiability is achieved in Section 4.2.

###  4.1. Some moment bounds

In [25, Theorem 2.1], it is proved that certain stochastic convolutions
$\mathcal{I}(t,x)=\int_{0}^{t}G(t-s,x-y)\varphi(s,y)W(ds\,dy)$ are Hölder
continuous in $t$ and $x$ as long as $\varphi$ is a real-valued adapted random
field satisfying
$\sup_{t\in[0,T]}\sup_{x\in\mathbb{R}^{d}}{\mathbb{E}}|\varphi(t,x)|^{p}<\infty$
for sufficiently large $p>1$. The following lemma claims that the same result
is true when $\varphi$ is Hilbert-space valued. Due to our Malliavin calculus
considerations, in the current paper we are mostly interested in the case
where the integrand is ${\mathcal{H}}_{T}$-valued.

######  Lemma 4.1.

Suppose that $\varphi$ is an adapted ${\mathcal{H}}_{T}$-valued adapted random
field defined on $[0,T]\times{\mathbb{R}}^{d}$. We assume

$\sup_{t\in[0,T]}\sup_{x\in\mathbb{R}^{d}}{\mathbb{E}}|\varphi(t,x)|_{{\mathcal{H}}_{T}}^{p}<\infty\,,$
(97)

for sufficiently large $p>1$. As in (59), let

$\mathcal{I}(t,x)=\mathcal{I}^{\varphi}(t,x)=\int_{0}^{t}\int_{\mathbb{R}^{d}}G(t-s,x-y)\varphi(s,y)W(ds\,dy)\,,$
(98)

and recall that $\eta$ is introduced in the strong Dalang condition (12). Then
for any $0<\gamma<\alpha<\eta/2$ and $p\geq 2$, there exist constants
$C_{\alpha,\gamma}>0$ such that for any $x,x_{1},x_{2}\in\mathbb{R}^{d}$ and
$t,t_{1},t_{2}\in[0,T]$,

$\displaystyle{\mathbb{E}}|\mathcal{I}(t,x)|^{p}_{{\mathcal{H}}_{T}}\leq
C_{\alpha,\gamma}T^{p\alpha}\sup_{s\in[0,T]}\sup_{y\in\mathbb{R}^{d}}{\mathbb{E}}|\varphi(s,y)|^{p}_{{\mathcal{H}}_{T}}$
(99)
$\displaystyle{\mathbb{E}}|\mathcal{I}(t,x_{1})-\mathcal{I}(t,x_{2})|^{p}_{{\mathcal{H}}_{T}}\leq
C_{\alpha,\gamma}|x_{1}-x_{2}|^{2\gamma
p}\,T^{p(\alpha-\gamma)}\sup_{s\in[0,T]}\sup_{y\in\mathbb{R}^{d}}{\mathbb{E}}|\varphi(s,y)|^{p}_{{\mathcal{H}}_{T}}$
(100)
$\displaystyle{\mathbb{E}}|\mathcal{I}(t_{1},x)-\mathcal{I}(t_{2},x)|^{p}_{{\mathcal{H}}_{T}}\leq
C_{\alpha,\gamma}|t_{1}-t_{2}|^{\gamma
p}\,T^{p(\alpha-\gamma)}\sup_{s\in[0,T]}\sup_{y\in\mathbb{R}^{d}}{\mathbb{E}}|\varphi(s,y)|^{p}_{{\mathcal{H}}_{T}}.$
(101)

###### Proof.

The only difference between the proof in [25] and this lemma is that $\varphi$
and $\mathcal{I}$ are Hilbert-space-valued. Theorem 4.36 of [8] guarantees
that the BDG inequality holds for Hilbert-space valued stochastic integrals
identically to how it holds for real-valued random variables. ∎

In the sequel we will need to bound stochastic integrals in the norms related
to Definition 2.1. Towards this aim, the following lemma generalizes Theorem
4.3 of [24] to the Hilbert-space setting of Lemma 4.1. Its proof is omitted
for sake of conciseness.

######  Lemma 4.2.

Let $T>0$. Consider $p>\frac{2(d+1)}{\eta}$ where $d$ is the spatial dimension
and $\eta\in(0,1)$ is still the parameter from (12). For any
$\theta>\frac{d+1}{p-(d+1)}$, there exists $C_{T,p,\theta}>0$ such that for
all adapted random fields
$\varphi:[0,T]\times\mathbb{R}^{d}\times\Omega\to{\mathcal{H}}_{T}$ verifying

$\sup_{t\in[0,T]}\sup_{x\in\mathbb{R}^{d}}{\mathbb{E}}|\varphi(t,x)|_{{\mathcal{H}}_{T}}^{p}<+\infty,$
(102)

the stochastic integral $\mathcal{I}=\mathcal{I}^{\varphi}$ defined by (98)
satisfies

$\sup_{x_{0}\in\mathbb{R}^{d}}{\mathbb{E}}\left|\sup_{t\in[0,T]}\sup_{x\in\mathbb{R}^{d}}\frac{|\mathcal{I}(t,x)|_{{\mathcal{H}}_{T}}}{1+|x-x_{0}|^{\theta}}\right|^{p}\leq
C_{T,p,\theta}\sup_{t\in[0,T]}\sup_{x_{0}\in\mathbb{R}^{d}}{\mathbb{E}}|\varphi(t,x_{0})|^{p}_{{\mathcal{H}}_{T}}\,.$
(103)

Moreover, the constant $C_{T,p,\theta}$ defined in (103) satisfies $\lim_{T\to
0}C_{T,p,\theta}=0$.

Next we need to bound weighted norms of $\mathcal{H}_{T}$-valued processes, in
a suitable way for our stochastic convolutions. A deterministic type result in
this direction is presented below.

######  Lemma 4.3.

Assume that $X$ is a function indexed by $[0,T]\times{\mathbb{R}}^{d}$ such
that there exists $\theta\geq 0$ and $x_{0}$ satisfying

$M:=\sup_{t\in[0,T]}\sup_{x\in\mathbb{R}^{d}}\frac{|X(t,x)|}{1+|x-x_{0}|^{\theta}}<+\infty.$
(104)

Then there exists $C=C(T,\theta)>0$ such that we have

$\sup_{t\in[0,T]}\sup_{x\in\mathbb{R}^{d}}\frac{\left|G(t-\cdot,x-\cdot)X(\cdot,\cdot)\mathbbm{1}_{[0,t]}\right|_{{\mathcal{H}}_{T}}^{2}}{(1+|x-x_{0}|^{\theta})^{2}}\leq
CM^{2}Q_{\Lambda}(T)\,$ (105)

where $Q_{\Lambda}$ is defined in (64). If $X$ is a random field, then (105)
holds with probability one.

###### Proof.

Assume that there exists $\theta\geq 0$ and $T>0$ such that (104) holds. the
definition of the ${\mathcal{H}}_{T}$ norm (53), we write

$\displaystyle\left|G(t-\cdot,x-\cdot)X(\cdot,\cdot)\mathbbm{1}_{[0,t]}\right|_{{\mathcal{H}}_{T}}^{2}$
$\displaystyle=\int_{0}^{t}\int_{\mathbb{R}^{d}}\int_{\mathbb{R}^{d}}G(t-s,x-y_{1})G(t-s,x-y_{2})X(s,y_{1})X(s,y_{2})\Lambda(y_{1}-y_{2})dy_{1}dy_{2}.$
(106)

Now owing to the definition of $M$ and the positivity of $G$ and $\Lambda$, an
upper bound for the right hand side of (4.1) is

$\left|G(t-\cdot,x-\cdot)X(\cdot,\cdot)\mathbbm{1}_{[0,t]}\right|_{{\mathcal{H}}_{T}}^{2}\leq
M^{2}\int_{0}^{t}\int_{\mathbb{R}^{d}}\int_{\mathbb{R}^{d}}G(t-s,x-y_{1})G(t-s,x-y_{2})\\\
\times(1+|y_{1}-x_{0}|^{\theta})(1+|y_{2}-x_{0}|^{\theta})\Lambda(y_{1}-y_{2})dy_{1}dy_{2}.$
(107)

Moreover, the quantities $1+|y_{1}-x_{0}|^{\theta}$ and
$1+|y_{2}-x_{0}|^{\theta}$ above satisfy

$1+|y_{i}-x_{0}|^{\theta}\leq
C(1+|y_{i}-x|^{\theta}+|x-x_{0}|^{\theta})=C(1+|x-x_{0}|^{\theta}+|y_{i}-x|^{\theta}).$
(108)

In addition, invoking the elementary relation
$\sup_{z\in{\mathbb{R}}}|z|^{\theta}\exp(-a\,x^{2})\leq c_{a}<\infty$, valid
for any arbitrary constant $a>0$, it is easy to verify that there exists
$C=C(\theta)$ such that

$|y|^{\theta}G(t,y)=t^{\frac{\theta}{2}}\left|\frac{y}{\sqrt{t}}\right|^{\theta}G(t,y)\leq
Ct^{\frac{\theta}{2}}G(t/2,y).$ (109)

Plugging (108) and (109) into (107), we thus get

$\displaystyle\left|G(t-\cdot,x-\cdot)X(\cdot,\cdot)\mathbbm{1}_{[0,t]}\right|_{{\mathcal{H}}_{T}}^{2}$
(110) $\displaystyle\leq
CM^{2}\left(1+|x-x_{0}|^{\theta}\right)^{2}\int_{0}^{t}\int_{\mathbb{R}^{d}}\int_{\mathbb{R}^{d}}G(t-s,x-y_{1})G(t-s,x-y_{2})\Lambda(y_{1}-y_{2})dy_{1}dy_{2}$
$\displaystyle\qquad+CM^{2}\int_{0}^{t}\int_{\mathbb{R}^{d}}\int_{\mathbb{R}^{d}}(t-s)^{\theta}G((t-s)/2,x-y_{1})G((t-s)/2,x-y_{2})\Lambda(y_{1}-y_{2})dy_{1}dy_{2}.$

The right hand side above can now be expressed in terms of Fourier transforms
and a change of variable $s:=t-s$ in the time integral. We end up with

$\left|G(t-\cdot,x-\cdot)X(\cdot,\cdot)\mathbbm{1}_{[0,t]}\right|_{{\mathcal{H}}_{T}}^{2}\\\
\leq
CM^{2}(1+|x-x_{0}|^{\theta})^{2}\int_{0}^{t}\int_{\mathbb{R}^{d}}e^{-s|\xi|^{2}}\mu(d\xi)ds+CM^{2}\int_{0}^{t}\int_{\mathbb{R}^{d}}s^{\theta}e^{-\frac{s|\xi|^{2}}{2}}\mu(d\xi)ds\,.$
(111)

With (64) in mind, it is thus readily checked that

$\left|G(t-\cdot,x-\cdot)X(\cdot,\cdot)\mathbbm{1}_{[0,t]}\right|_{{\mathcal{H}}_{T}}^{2}\leq
CM^{2}(1+|x-x_{0}|^{\theta})^{2}Q_{\Lambda}(t)+CM^{2}t^{\theta}{Q}_{\Lambda}(t)\,,$

from which our claim (105) is easily deduced. ∎

###  4.2. Malliavin differentiability

With the above preliminary results in hand, let us state our Malliavin
differentiability result for the mild solution to the spde (2.1).

######  Theorem 4.4.

Recall that our coefficients $b,\sigma$ satisfy Assumptions 2.3–2.4. Let $u$
be the unique solution to (2.1) and let $Z$ be the stochastic convolution
defined in (52). For any fixed $(t,x)\in[0,T]\times{\mathbb{R}}^{d}$, both
$u(t,x)$ and $Z(t,x)$ are Malliavin differentiable. For any time horizon $T>0$
and power $p>1$, the Malliavin derivatives satisfy

$\sup_{x_{0}\in\mathbb{R}^{d}}{\mathbb{E}}\left|\sup_{t\in[0,T]}\sup_{x\in\mathbb{R}^{d}}\frac{|DZ(t,x)|_{{\mathcal{H}}_{T}}}{1+|x-x_{0}|^{\theta}}\right|^{p}<+\infty$
(112)

and

$\sup_{x_{0}\in\mathbb{R}^{d}}{\mathbb{E}}\left|\sup_{t\in[0,T]}\sup_{x\in\mathbb{R}^{d}}\frac{|Du(t,x)|_{{\mathcal{H}}_{T}}|}{1+|x-x_{0}|^{\theta}}\right|^{p}<+\infty$
(113)

###### Proof.

We will prove this theorem via Proposition 2.15, using the approximating
sequences $\\{Z_{n};\,n\geq 0\\}$ and $\\{u_{n};\,n\geq 0\\}$ that are
respectively defined by (48) and (49). We proceed by induction to prove a
uniform bound on $Du_{n}$ and $DZ_{n}$ in an appropriate topology. For $n\geq
0$, we thus suppose that both $u_{n}(t,x)$ and $Z_{n}(t,x)$ are elements of
the space $\mathbb{D}^{1,p}$ given by (55) with $p>1$ and satisfy

$\displaystyle\sup_{x_{0}\in\mathbb{R}^{d}}{\mathbb{E}}\left|\sup_{t\in[0,T]}\sup_{x\in\mathbb{R}^{d}}\frac{|DZ_{n}(t,x)|_{{\mathcal{H}}_{T}}}{1+|x-x_{0}|^{\theta}}\right|^{p}<+\infty$
(114)
$\displaystyle\sup_{x_{0}\in\mathbb{R}^{d}}{\mathbb{E}}\left|\sup_{t\in[0,T]}\sup_{x\in\mathbb{R}^{d}}\frac{|Du_{n}(t,x)|_{{\mathcal{H}}_{T}}}{1+|x-x_{0}|^{\theta}}\right|^{p}<+\infty.$
(115)

The inductive assumption (114)–(115) is trivially true for the initial $U_{0}$
and $Z_{0}$, defined in (46)–(47) because they are non-random. We shall now
propagate this assumption. We first consider $Z_{n+1}$ defined by (48). Namely
we are assuming (115) holds for $u_{n}$ and we wish to apply Proposition 2.17
to prove that $Z_{n+1}$ defined by (48) is Malliavin differentiable. To this
aim, Proposition 2.17 will be applicable if $\sigma(u_{n})$ satisfies
(57)–(58). We proceed to check those assumptions below.

From (51), for each $n\in\mathbb{N}$,

$\sup_{x_{0}\in\mathbb{R}^{d}}{\mathbb{E}}\left|\sup_{t\in[0,T]}\sup_{x\in\mathbb{R}^{d}}\frac{|u_{n}(t,x)|}{1+|x-x_{0}|^{\theta}}\right|^{p}<+\infty.$
(116)

By setting $x=x_{0}$ in the above expression and moving the time supremum
outside of the expectation,

$\displaystyle\sup_{x_{0}\in\mathbb{R}^{d}}\sup_{t\in[0,T]}{\mathbb{E}}|u_{n}(t,x_{0})|^{p}$
$\displaystyle\leq$
$\displaystyle\sup_{x_{0}\in\mathbb{R}^{d}}{\mathbb{E}}\left|\sup_{t\in[0,T]}|u_{n}(t,x_{0})|\right|^{p}$
(117) $\displaystyle\leq$
$\displaystyle\sup_{x_{0}\in\mathbb{R}^{d}}{\mathbb{E}}\left|\sup_{t\in[0,T]}\sup_{x\in\mathbb{R}^{d}}\frac{|u_{n}(t,x)|}{1+|x-x_{0}|^{\theta}}\right|^{p}<+\infty.$

Therefore, $u_{n}$ satisfies (57). The same reasoning shows that (115) implies
that $Du_{n}$ satisfies relation (58). In summary,

$\sup_{t\in[0,T]}\sup_{x\in\mathbb{R}^{d}}{\mathbb{E}}|u_{n}(t,x)|^{2}<+\infty,\quad\text{and}\quad\sup_{t\in[0,T]}\sup_{x\in\mathbb{R}^{d}}{\mathbb{E}}|Du_{n}(t,x)|_{{\mathcal{H}}_{T}}^{2}<+\infty.$

Since $\sigma$ verifies Assumptions 2.3, we also have

$\sup_{t\in[0,T]}\sup_{x\in\mathbb{R}^{d}}{\mathbb{E}}|\sigma\left(u_{n}(t,x)\right)|^{2}<+\infty,\quad\text{and}\quad\sup_{t\in[0,T]}\sup_{x\in\mathbb{R}^{d}}{\mathbb{E}}|D\sigma\left(u_{n}(t,x)\right)|_{{\mathcal{H}}_{T}}^{2}<+\infty.$

Now observe that the stochastic integral $Z_{n+1}$ defined in (48) can be
written as $\mathcal{I}^{\sigma(u_{n})}$ using the notation of (59).
Therefore, Proposition 2.17 asserts that $Z_{n+1}(t,x)$ is an element of
$\mathbb{D}^{1,2}$ and

$\displaystyle D_{h}Z_{n+1}(t,x)=$
$\displaystyle\left<G(t-\cdot,x-\cdot)\sigma(u_{n}(\cdot,\cdot)),h\right>_{{\mathcal{H}}_{T}}$
$\displaystyle+\int_{0}^{t}\int_{\mathbb{R}^{d}}G(t-s,x-y)\sigma^{\prime}(u_{n}(s,y))D_{h}u_{n}(s,y)\,W(ds\,dy)\,.$
(118)

Having shown that $Z_{n+1}(t,x)\in\mathbb{D}^{1,2}$ for all
$(t,x)\in[0,T]\times{\mathbb{R}}^{d}$, we proceed to prove (112) and (113). To
this aim, observe that relation (4.2) can be written as

$D_{h}Z_{n+1}(t,x)=\left<G(t-\cdot,x-\cdot)\sigma(u_{n}),h\right>_{{\mathcal{H}}_{T}}+\mathcal{I}^{\sigma^{\prime}(u_{n})D_{h}u_{n}}(t,x)\,,$
(119)

where we invoked our notation (59) again. We can bound the first term on the
right-hand-side thanks to (105) and the second term on the right-hand side
(remember that $\sigma^{\prime}$ is uniformly bounded according to Assumption
2.3) with (103) to get

$\displaystyle\sup_{x_{0}\in\mathbb{R}^{d}}{\mathbb{E}}\left|\sup_{t\in[0,T]}\sup_{x\in\mathbb{R}^{d}}\frac{|DZ_{n+1}(t,x)|_{{\mathcal{H}}_{T}}}{1+|x-x_{0}|^{\theta}}\right|^{p}$
(120) $\displaystyle\leq
C(Q_{\Lambda}(T))^{\frac{p}{2}}\sup_{x_{0}\in\mathbb{R}^{d}}{\mathbb{E}}\left|\sup_{t\in[0,T]}\sup_{x\in\mathbb{R}^{d}}\frac{|\sigma(u_{n}(t,x))|}{1+|x-x_{0}|^{\theta}}\right|^{p}+C_{T,p,\theta}\sup_{x_{0}\in\mathbb{R}^{d}}{\mathbb{E}}\left|\sup_{t\in[0,T]}\sup_{x\in\mathbb{R}^{d}}\frac{|Du_{n}(t,x)|_{{\mathcal{H}}_{T}}}{1+|x-x_{0}|^{\theta}}\right|^{p}.$

In addition, invoking Lemma 4.2, we have

$\lim_{T\downarrow 0}C_{T,p,\theta}=0,$ (121)

and we also recall that $Q_{\Lambda}(T)$ is defined in (64).

We are now ready to propagate relation (115) for $Du_{n}$ on a small time
interval. Namely, because $u_{n+1}=\mathcal{M}(U_{0}+Z_{n+1})$, relations (66)
and (120) guarantee that

$\displaystyle\sup_{x_{0}\in\mathbb{R}^{d}}{\mathbb{E}}\left|\sup_{t\in[0,T]}\sup_{x\in\mathbb{R}^{d}}\frac{|Du_{n+1}(t,x)|_{{\mathcal{H}}_{T}}}{1+|x-x_{0}|^{\theta}}\right|^{p}\leq
C\sup_{x_{0}\in\mathbb{R}^{d}}{\mathbb{E}}\left|\sup_{t\in[0,T]}\sup_{x\in\mathbb{R}^{d}}\frac{|DU_{0}(t,x)|_{{\mathcal{H}}_{T}}}{1+|x-x_{0}|^{\theta}}\right|^{p}$
(122)
$\displaystyle+C(Q_{\Lambda}(T))^{\frac{p}{2}}\sup_{x_{0}\in\mathbb{R}^{d}}{\mathbb{E}}\left|\sup_{t\in[0,T]}\sup_{x\in\mathbb{R}^{d}}\frac{|\sigma(u_{n}(t,x))|}{1+|x-x_{0}|^{\theta}}\right|^{p}+C_{T,p,\theta}\sup_{x_{0}\in\mathbb{R}^{d}}{\mathbb{E}}\left|\sup_{t\in[0,T]}\sup_{x\in\mathbb{R}^{d}}\frac{|Du_{n}(t,x)|_{{\mathcal{H}}_{T}}}{1+|x-x_{0}|^{\theta}}\right|^{p}.$

In addition, if the initial data $u_{0}$ is non-random, then
$DU_{0}(t,x)\equiv 0$. We choose $T_{0}>0$ small enough so that the
coefficient $C_{T_{0},p,\theta}<1$. Then by induction, invoking (122) we get

$\sup_{n}\sup_{x_{0}\in\mathbb{R}^{d}}{\mathbb{E}}\left|\sup_{t\in[0,T_{0}]}\sup_{x\in\mathbb{R}^{d}}\frac{|Du_{n}(t,x)|_{{\mathcal{H}}_{T}}}{1+|x-x_{0}|^{\theta}}\right|^{p}\\\
\leq\frac{C(Q_{\Lambda}(T))^{\frac{p}{2}}}{1-C_{T_{0},p,\theta}}\sup_{n}\sup_{x_{0}\in\mathbb{R}^{d}}{\mathbb{E}}\left|\sup_{t\in[0,T]}\sup_{x\in\mathbb{R}^{d}}\frac{|\sigma(u_{n}(t,x))|}{1+|x-x_{0}|^{\theta}}\right|^{p}.$
(123)

The right hand side of the above expression is finite because of (51) and the
assumption that $\sigma$ is globally Lipschitz continuous. Therefore,

$\sup_{n}\sup_{x_{0}\in\mathbb{R}^{d}}{\mathbb{E}}\left|\sup_{t\in[0,T_{0}]}\sup_{x\in\mathbb{R}^{d}}\frac{|Du_{n}(t,x)|_{{\mathcal{H}}_{T}}}{1+|x-x_{0}|^{\theta}}\right|^{p}<+\infty.$
(124)

Then (120), (51), and (124) imply that

$\sup_{n}\sup_{x_{0}\in\mathbb{R}^{d}}{\mathbb{E}}\left|\sup_{t\in[0,T_{0}]}\sup_{x\in\mathbb{R}^{d}}\frac{|DZ_{n}(t,x)|_{{\mathcal{H}}_{T}}}{1+|x-x_{0}|^{\theta}}\right|^{p}<+\infty.$
(125)

In particular, for any fixed $(t,x)\in[0,T_{0}]\times\mathbb{R}^{d}$, we have
$\sup_{n}{\mathbb{E}}|Du_{n}(t,x)|_{{\mathcal{H}}_{T}}^{2}<+\infty$ and
$\sup_{n}{\mathbb{E}}|DZ_{n}(t,x)|_{{\mathcal{H}}_{T}}^{2}<+\infty$.
Approximations (50)–(51) and Proposition 2.15 guarantee that $u(t,x)$ and
$Z(t,x)$ are Malliavin differentiable for $t\in[0,T_{0}]$ and
$x\in\mathbb{R}^{d}$, where $T_{0}$ is the small parameter chosen above.
Furthermore, (112)–(113) hold for $T<T_{0}$ by Fatou’s lemma

We can extend this result to arbitrary time horizons $T>T_{0}$ by taking
advantage of the self-similarity of the process $W$. Namely for any
$n\in\mathbb{N}$ and $t>0$,

$\displaystyle u_{n+1}(T_{0}+t,x)=$
$\displaystyle\int_{\mathbb{R}^{d}}G(t,x-y)u_{n+1}(T_{0},y)dy+\int_{0}^{t}\int_{\mathbb{R}^{d}}G(t-s,x-y)f(u_{n}(T_{0}+s,y))dyds$
$\displaystyle+\int_{0}^{t}\int_{\mathbb{R}^{d}}G(t-s,x-y)\sigma(u_{n}(T_{0}+s))W(dy,(T_{0}+ds))$

This translated process has the property that

$u_{n+1}(T_{0}+t,x)=\mathcal{M}\left(U_{T_{0},n+1}+\tilde{Z}_{n+1}\right)(t,x)$

where we have set

$\displaystyle U_{T_{0},n+1}(t,x)$ $\displaystyle:=$
$\displaystyle\int_{\mathbb{R}^{d}}G(t,x-y)u_{n+1}(T_{0},y)dy$
$\displaystyle\tilde{Z}_{n+1}(t,x)$ $\displaystyle:=$
$\displaystyle\int_{0}^{t}\int_{\mathbb{R}^{d}}G(t-s,x-y)\sigma(u_{n}(T_{0}+s))W(dy,(T_{0}+ds))$

The $p$-th moment bounds (120)–(122) continue to hold with $U_{0}$ replaced by
$U_{T_{0},n+1}$ and $Z_{n}$ replaced by $\tilde{Z_{n}}$. The constants
$C_{T,p,\theta}$ and $Q_{\Lambda}(T)$ in these expressions do not change (they
only depend on the law of $W$, which is invariant by time shift) and we can
use the same value of $T_{0}$ as above. The only novelty is that $DU_{0}\equiv
0$, while $DU_{T_{0},n+1}$ is not zero. But we have a uniform bound on

$\sup_{n}\sup_{x_{0}\in\mathbb{R}^{d}}{\mathbb{E}}\left|\sup_{t\in[0,T_{0}]}\sup_{x\in\mathbb{R}^{d}}\frac{|DU_{T_{0},n+1}(t,x)|_{{\mathcal{H}}_{T}}}{1+|x-x_{0}|^{\theta}}\right|^{p}\,,$

as a consequence of (124) and properties of the heat kernel $G$. Therefore,
the same arguments that implied (124)–(125) imply

$\sup_{n}\sup_{x_{0}\in\mathbb{R}^{d}}{\mathbb{E}}\left|\sup_{t\in[T_{0},2T_{0}]}\sup_{x\in\mathbb{R}^{d}}\frac{|Du_{n}(t,x)|_{{\mathcal{H}}_{T}}}{1+|x-x_{0}|^{\theta}}\right|^{p}<+\infty.$
(126)

It follows immediately from (120) that

$\sup_{n}\sup_{x_{0}\in\mathbb{R}^{d}}{\mathbb{E}}\left|\sup_{t\in[T_{0},2T_{0}]}\sup_{x\in\mathbb{R}^{d}}\frac{|DZ_{n}(t,x)|_{{\mathcal{H}}_{T}}}{1+|x-x_{0}|^{\theta}}\right|^{p}<+\infty.$
(127)

Proposition 2.15 guarantees that $u(t,x)$ and $Z(t,x)$ are Malliavin
differentiable for $t\in[T_{0},2T_{0}]$. Repetition of this argument proves
the Malliavin differentiability of $u(t,x)$ and $Z(t,x)$ for
$t\in[2T_{0},3T_{0}],[3T_{0},4T_{0}]$, and so on. ∎

We close this section by identifying an integral equation satisfied by
directional Malliavin derivatives of $u$

######  Corollary 4.5.

Recall that ${\mathcal{H}}_{T}$ is the Hilbert space with inner product given
by (53) and assume the same hypotheses as in Theorem 4.4. For any $T>0$ and
$h\in{\mathcal{H}}_{T}$, and $(t,x)\in[0,T]\times\mathbb{R}^{d}$, the
Malliavin derivative of $u(t,x)$ solves the integral equation

$\displaystyle D_{h}u(t,x)=$
$\displaystyle\left<G(t-\cdot,x-\cdot)\sigma(u(\cdot,\cdot)),h\right>_{{\mathcal{H}}_{T}}$
$\displaystyle+\int_{0}^{t}\int_{\mathbb{R}^{d}}G(t-s,x-y)f^{\prime}(u(s,y))D_{h}u(s,y)dyds$
$\displaystyle+\int_{0}^{t}\int_{\mathbb{R}^{d}}G(t-s,x-y)\sigma^{\prime}(u(s,y))D_{h}u(s,y)W(ds\,dy).$
(128)

###### Proof.

The integral equation (4.5) is easily obtained by combining (96) with (125)
and Proposition 2.16. ∎

##  5\. Positivity of the Malliavin derivative

In this section we prove that the solution $u$ to (2.1) has the property that
$u(t,x)$ admits a density for $t>0$, $x\in\mathbb{R}^{d}$ by analyzing the
quantity $\|Du(t,x)\|_{\mathcal{H}_{T}}$. Our main theorem can be expressed as
below.

######  Theorem 5.1.

Let $u$ solution to (2.1), where we assume that the coefficients $b,\sigma$
satisfy Assumptions 2.3–2.4. We also suppose that the Gaussian noise $\dot{W}$
verifies Hypothesis 2.2. Then for $t\in(0,T]\times{\mathbb{R}}^{d}$, the
random variable $u(t,x)$ admits a density with respect to Lebesgue measure.

In order to prove Theorem 5.1, we start in Subsection 5.1 by explaining our
strategy, and we perform most of our computations in Subsections 5.2–5.5.

###  5.1. Strategy

According to Proposition 2.14, the existence of a density is established if we
can prove that $u(t,x)\in\mathbb{D}^{1,p}$ for some $p\geq 1$ and if the
Malliavin derivative has strictly positive ${\mathcal{H}}_{T}$ norm. Moreover,
Theorem 4.4 asserts that $u(t,x)$ belongs to the Sobolev space
$\mathbb{D}^{1,p}$ for all $p>1$. It remains to prove that for every $t>0$,
$x\in\mathbb{R}^{d}$,

${\mathbb{P}}\left(\|Du(t,x)\|_{{\mathcal{H}}_{T}}>0\right)=1.$ (129)

For convenience we will further reduce (129) to a statement about directional
derivatives. Namely condition (129) is implied if

${\mathbb{P}}\left(\textnormal{There exists }h\in{\mathcal{H}}_{T}\textnormal{
such that }D_{h}u(t,x)>0\right)=1.$ (130)

We now elaborate on (130). Recall the integral equation (4.5) for the
directional Malliavin derivative of $u(t,x)$. Namely for any
$h\in{\mathcal{H}}_{T}$, $t\in[0,T]$, and $x\in\mathbb{R}^{d}$, we have
obtained that

$D_{h}u(t,x)=\left<\phi_{t,x},h\right>_{{\mathcal{H}}_{T}}+A_{h}(t,x)+B_{h}(t,x),$
(131)

where the function $\phi_{t,x}$ is defined on $[0,T]\times\mathbb{R}^{d}$ by

$\phi_{t,x}(r,z)=G(t-r,x-z)\sigma(u(r,z))\mathbbm{1}_{[0,t]}(r)\,,$ (132)

and where the terms $A_{h}$ and $B_{h}$ are given by

$\displaystyle A_{h}(t,x)$ $\displaystyle=$
$\displaystyle\int_{0}^{t}\int_{\mathbb{R}^{d}}G(t-s,x-y)f^{\prime}(u(s,y))D_{h}u(s,y)dyds$
(133) $\displaystyle B_{h}(t,x)$ $\displaystyle=$
$\displaystyle\int_{0}^{t}\int_{\mathbb{R}^{d}}G(t-s,x-y)\sigma^{\prime}(u(s,y))D_{h}u(s,y)W(ds\,dy).$
(134)

Furthermore, one can also express the directional derivative of $u(t,x)$ as

$D_{h}u(t,x)=\mathcal{L}\left(\left<\phi_{\cdot,\cdot},h\right>_{{\mathcal{H}}_{T}}+B_{h}(\cdot,\cdot)\right)(t,x)\,,$
(135)

where $\mathcal{L}$ is the functional mapping defined in (25) with
$\varphi(t,x,V):=f^{\prime}(u(s,y))V$. This point of view allows us to apply
Theorem 2.12 to get useful upper bounds on $|D_{h}u(t,x)|$.

With (130) in mind, for fixed $t_{0}\in[0,T]$, $x_{0}\in\mathbb{R}$, and
$\delta\in(0,t_{0})$ we chose a specific test function
$h_{t_{0},x_{0},\delta}\in{\mathcal{H}}_{T}$. Specifically, let
$h_{t_{0},x_{0},\delta}$ be defined by

$h_{t_{0},x_{0},\delta}(r,z):=G(t_{0}-r,x_{0}-z)\mathbbm{1}_{[t_{0}-\delta,t_{0}]}(r).$
(136)

Because $h_{t_{0},x_{0},\delta}(r,z)$ is only nonzero when
$r\in[t_{0}-\delta,t_{0}]$ and $\phi_{t,x}(r,z)$ defined in (132) is only non-
zero when $r\in[0,t]$, the inner product
$\left<\phi_{t,x},h_{t_{0},x_{0},\delta}\right>_{{\mathcal{H}}_{T}}$ is zero
when $t\in[0,t_{0}-\delta)$. Then because the equations (131)–(134) are linear
with respect to $D_{h_{t_{0},x_{0},\delta}}u(t,x)$, it follows that
$D_{h_{t_{0},x_{0},\delta}}u(t,x)=0$ for all $t\in[0,t_{0}-\delta]$. This
means that the integral terms $A_{h_{t_{0},x_{0},\delta}}$ and
$B_{h_{t_{0},x_{0},\delta}}$ in (133) and (134) can be written as integrals
starting at $t_{0}-\delta$. Specifically, for any $t\in[t_{0}-\delta,t_{0}]$,
we have

$\displaystyle A_{h_{t_{0},x_{0},\delta}}(t,x)$ $\displaystyle=$
$\displaystyle\int_{t_{0}-\delta}^{t}\int_{\mathbb{R}^{d}}G(t-s,x-y)f^{\prime}(u(s,y))D_{h_{t_{0},x_{0},\delta}}u(s,y)dyds$
(137) $\displaystyle B_{h_{t_{0},x_{0},\delta}}(t,x)$ $\displaystyle=$
$\displaystyle\int_{t_{0}-\delta}^{t}\int_{\mathbb{R}^{d}}G(t-s,x-y)\sigma^{\prime}(u(s,y))D_{h_{t_{0},x_{0},\delta}}u(s,y)W(ds\,dy)\,.$
(138)

Moreover, for $t\in[0,t_{0}-\delta]$ both $A_{h_{t_{0},x_{0},\delta}}(t,x)$
and $B_{h_{t_{0},x_{0},\delta}}(t,x)$ are vanishing.

With those preliminary considerations in hand, using the decomposition (131),
we will achieve the desired property (130). That is we will be able to show
that with probability one there exists $\delta>0$ such that
$D_{h_{t_{0},x_{0},\delta}}u(t_{0},x_{0})>0$. This is accomplished by proving
that
$\left<\phi_{t_{0},x_{0}},h_{t_{0},x_{0},\delta}\right>_{\mathcal{H}_{T}}$ is
larger than
$A_{h_{t_{0},x_{0},\delta}}(t_{0},x_{0})+B_{h_{t_{0},x_{0},\delta}}(t_{0},x_{0})$
when $\delta$ is small, guaranteeing that (131) is positive. The main ideas
are the following.

  1. _(i)_

Invoking the non degeneracy of the kernel $G$ and of the coefficient $\sigma$,
we shall see that the inner product
$\left<\phi_{t_{0},x_{0}},h_{t_{0},x_{0},\delta}\right>_{\mathcal{H}_{T}}$ has
a lower bound that is proportional to $Q_{\Lambda}(\delta)$ defined in (64).

  2. _(ii)_

Using (135) along with the boundedness of $\sigma^{\prime}$, we will prove
that an upper bound on the
$L^{p}(\Omega:{\mathcal{C}_{\theta,x_{0}}([0,T]\times\mathbb{R}^{d})})$ norm
of $D_{h_{t_{0},x_{0},\delta}}u(t,x)$ is also proportional to
$Q_{\Lambda}(\delta)$.

  3. _(iii)_

Finally, the fact that $A_{h_{t_{0},x_{0},\delta}}(t_{0},x_{0})$, and
$B_{h_{t_{0},x_{0},\delta}}(t_{0},x_{0})$ in (137)–(138) are integrals whose
integrands have $L^{p}(\Omega)$ norms proportional to $Q_{\Lambda}(\delta)$,
but whose interval of integration are of size $\delta$, we can show that with
probability one there exists a subsequence $\delta_{k}\downarrow 0$ such that

$\lim_{k\to\infty}\frac{|A_{h_{t_{0},x_{0},\delta_{k}}}(t_{0},x_{0})|+|B_{h_{t_{0},x_{0},\delta_{k}}}(t_{0},x_{0})|}{Q_{\Lambda}(\delta_{k})}=0,\quad\text{
with probability one.}$ (139)

Putting together the 3 items above, one concludes that with probability one,
there exists a small enough (random) $\delta_{k}=\delta_{k}(\omega)$ such that

$D_{h_{t_{0},x_{0},\delta_{k}}}u(t_{0},x_{0})=\left<\phi_{t,x},h_{t_{0},x_{0},\delta_{k}}\right>_{{\mathcal{H}}_{T}}+A_{h_{t_{0},x_{0},\delta_{k}}}(t_{0},x_{0})+B_{h_{t_{0},x_{0},\delta_{k}}}(t_{0},x_{0})>0.$
(140)

This will prove (130) and therefore the existence of a density. The remainder
of the section is devoted to detail the 3 steps above.

###  5.2. Lower and upper bounds on
$\left<\phi_{t,x},h_{t_{0},x_{0},\delta}\right>_{\mathcal{H}_{T}}$

In this section we give details about item _(i)_ above. Our findings are
summarized in the following lemma.

######  Lemma 5.2.

Let $t_{0}>0,x_{0}\in\mathbb{R}^{d}$. For $\delta\in[0,t_{0}]$ we recall the
definition of $h_{t_{0},x_{0},\delta}$ in (136):

$h_{t_{0},x_{0},\delta}(r,z):=G(t_{0}-r,x_{0}-z)\mathbbm{1}_{[t_{0}-\delta,t_{0}]}(r).$
(141)

Also recall that $\phi_{t,x}$ is defined by (132) and $Q_{\Lambda}$ is
introduced in (62)–(64). Then we have the following lower and upper bounds for
$\left<\phi_{t,x},h_{t_{0},x_{0},\delta}\right>_{\mathcal{H}_{T}}$. For any
$T>0,t_{0}\in[0,T],x_{0}\in\mathbb{R}^{d}$ and $\delta\in[0,t_{0}]$

$\left<\phi_{t_{0},x_{0}},h_{t_{0},x_{0},\delta}\right>_{\mathcal{H}_{T}}\geq\alpha\,Q_{\Lambda}(\delta)\,,\quad\text{
with probability one}\,$ (142)

where $\alpha>0$ is from (13). In addition, there exists $C>0$ such that for
any $T>0,t_{0}\in[0,T],x_{0}\in\mathbb{R}^{d},y_{0}\in\mathbb{R}^{d}$, and
$\delta\in[0,t_{0}]$,

$\sup_{t\in[t_{0}-\delta,t_{0}]}\sup_{x\in\mathbb{R}^{d}}\frac{|\left<\phi_{t,x},h_{t_{0},x_{0},\delta}\right>_{\mathcal{H}_{T}}|}{1+|x-y_{0}|^{\theta}}\leq
CQ_{\Lambda}(\delta)\left(1+\sup_{s\in[t_{0}-\delta,t_{0}]}\sup_{y\in\mathbb{R}^{d}}\frac{|u(s,y)|}{1+|y-y_{0}|^{\theta}}\right).$
(143)

Furthermore, if $t\in[0,t_{0}-\delta],x_{0}\in\mathbb{R}^{d}$, and
$x\in\mathbb{R}^{d}$,

$\left<\phi_{t,x},h_{t_{0},x_{0},\delta}\right>_{\mathcal{H}_{T}}=0.$ (144)

###### Proof.

Recall the expressions (132) and (141) for $\phi_{t,x}$ and
$h_{t_{0},x_{0},\delta}$ as well as the definition (53) of the inner product
in ${\mathcal{H}}_{T}$. Then it is readily checked that

$\displaystyle\left<\phi_{t,x},h_{t_{0},x_{0},\delta}\right>_{{\mathcal{H}}_{T}}$
$\displaystyle=\int_{t_{0}-\delta}^{t}\int_{\mathbb{R}^{d}}\int_{\mathbb{R}^{d}}G(t-s,x-y_{1})\sigma(u(s,y_{1}))G(t-s,x_{0}-y_{2})\Lambda(y_{1}-y_{2})dy_{1}dy_{2}ds,$
(145)

for all $t\in[t_{0}-\delta,t_{0}]$. We can now prove (142) in the following
way: owing to the fact that $\sigma(u(s,y_{1}))>\alpha$ (see Assumption 2.3)
and thanks to the positivity of $G$ and $\Lambda$ we obtain

$\displaystyle\left<\phi_{t_{0},x_{0}},h_{t_{0},x_{0},\delta}\right>_{{\mathcal{H}}_{T}}$
$\displaystyle\geq\alpha\int_{t_{0}-\delta}^{t_{0}}\int_{\mathbb{R}^{d}}\int_{\mathbb{R}^{d}}G(t_{0}-s,x_{0}-y_{1})G(t_{0}-s,x_{0}-y_{2})\Lambda(y_{1}-y_{2})dy_{1}dy_{2}ds.$
(146)

The last term in the above expression is $Q_{\Lambda}(\delta)$, which was
defined in (62). We have thus obtained

$\left<\phi_{t_{0},x_{0}},h_{t_{0},x_{0},\delta}\right>_{H_{T}}\geq\alpha
Q_{\Lambda}(\delta),$ (147)

that is (142) holds true.

The proof of (143) is a consequence of Lemma 4.3. Namely start from the
expression (5.2) for the inner product
$\langle\phi_{t_{0},x_{0}},h_{t_{0},x_{0},\delta}\rangle_{{\mathcal{H}}_{T}}$.
Then notice that in the right hand side of (5.2), the time integrand goes from
$t_{0}-\delta$ to $t$. By the Cauchy Schwarz inequality, we get

$\left\langle\phi_{t_{0},x_{0}},h_{t_{0},x_{0},\delta}\right\rangle_{H_{T}}\leq(R_{1}(t,x))^{\frac{1}{2}}(R_{2}(t))^{\frac{1}{2}}\,,$
(148)

where we have set

$R_{1}(t,x)=\int_{t_{0}-\delta}^{t}\int_{\mathbb{R}^{d}}\int_{\mathbb{R}^{d}}G(t-s,x-y_{1})G(t-s,x-y_{2})\\\
\times\sigma(u(s,y_{1}))\sigma(u(s,y_{2}))\Lambda(y_{1}-y_{2})dy_{1}dy_{2}$
(149)

and

$R_{2}(t)=\int_{t_{0}-\delta}^{t}\int_{\mathbb{R}^{d}}\int_{\mathbb{R}^{d}}G(t-s,x_{0}-y_{1})G(t-s,x_{0}-y_{2})\Lambda(y_{1}-y_{2})dy_{1}dy_{2}\,.$
(150)

The term $R_{2}(t)$ in (150) is easily bounded above by

$R_{2}(t)\leq Q_{\Lambda}(\delta)\,.$ (151)

In fact $R_{2}(t)$ would be exactly $Q_{\Lambda}(\delta)$ if we had $t=t_{0}$.
The term $R_{1}(t,x)$ defined by (149) can be bounded using a time translated
modification of Lemma 4.3 with $T=\delta$ (the length of the time interval
$[t_{0}-\delta,t_{0}]$) and $X(t,x)=\sigma(u(t,x)$. Specifically, we get

$\sup_{t\in[t_{0}-\delta,t_{0}]}\sup_{x\in\mathbb{R}^{d}}\frac{R_{1}(t,x)}{(1+|x-x_{0}|^{\theta})^{2}}\leq
CQ_{\Lambda}(\delta)\left(\sup_{t\in[t_{0}-\delta,t_{0}]}\sup_{x\in\mathbb{R}^{d}}\frac{|\sigma(u(t,x))|^{2}}{(1+|x-x_{0}|^{\theta})^{2}}\right).$

Since $\sigma$ has linear growth, we thus end up with

$\sup_{t\in[t_{0}-\delta,t_{0}]}\sup_{x\in\mathbb{R}^{d}}\frac{R_{1}(t,x)}{(1+|x-x_{0}|^{\theta})^{2}}\leq
CQ_{\Lambda}(\delta)\left(1+\sup_{t\in[t_{0}-\delta,t_{0}]}\sup_{x\in\mathbb{R}^{d}}\frac{|u(t,x)|^{2}}{(1+|x-x_{0}|^{\theta})^{2}}\right).$
(152)

Then (143) follows by plugging (151) and (152) into (148).

Finally, if $t\in[0,t_{0}-\delta]$ then the supports of $\phi_{t,x}$ and
$h_{t_{0},x_{0},\delta}$ are disjoint and therefore relation (144) follows. ∎

###  5.3. Upper bounds on moments of the derivative of $\mathbf{u(t,x)}$

We have seen in (113) that the Malliavin derivative of $u(t,x)$ is bounded in
the $L^{p}$-sense for all $p>1$. Equation (123) can even be seen as a
quantitative bound on this derivative. In the current section we push forward
this analysis to derive an upper bound for the moments of the weighted
supremum norms

$\sup_{y_{0}\in\mathbb{R}^{d}}{\mathbb{E}}\sup_{t\in[t_{0}-\delta,t_{0}]}\sup_{x\in\mathbb{R}^{d}}\left|\frac{|D_{h_{t_{0},x_{0},\delta}}u(t,x)|}{1+|x-y_{0}|^{\theta}}\right|^{p}\,,$
(153)

and to show that these are proportional to $(Q_{\Lambda}(\delta))^{p}$. As
mentioned in our strategy Section 5.1, this is the contents of item _(ii)_
above. We will use (135) along with Theorem 2.12 to prove these upper bounds.
Our main estimate is contained in the following lemma.

######  Lemma 5.3.

For $\delta>0$ and $t_{0}\in(\delta,T]$, define a quantity
$\mathcal{Q}_{1,\delta,p}$ by

$\mathcal{Q}_{1,\delta,p}=\sup_{y_{0}\in\mathbb{R}^{d}}{\mathbb{E}}\left|\sup_{t\in[t_{0}-\delta,t_{0}]}\sup_{x\in\mathbb{R}^{d}}\frac{|D_{h_{t_{0},x_{0},\delta}}u(t,x)|}{1+|x-y_{0}|^{\theta}}\right|^{p}.$
(154)

Then under the conditions of Theorem 5.1, there exist a constant $C>0$,
depending on $p$ and $\theta$, but not $\delta$, and a parameter
$\delta_{0}>0$ such that for all $\delta\in(0,\delta_{0})$,

$\mathcal{Q}_{1,\delta,p}\leq C(Q_{\Lambda}(\delta))^{p}.$ (155)

###### Proof.

First we recall our upper bounds on the weighted supremum norms of the
stochastic integral $B_{h_{t_{0},x_{0},\delta}}(t,x)$, where we recall that
$B_{h_{t_{0},x_{0},\delta}}(t,x)$ is defined by (138). Notice that

$B_{h_{t_{0},x_{0},\delta}}(t,x)=\mathcal{I}^{\varphi}(t,x),\quad\text{with}\quad\varphi(s,y)=\sigma^{\prime}(u(s,y))D_{h_{t_{0},x_{0},\delta}}u(s,y)\,,$

where the notation $\mathcal{I}^{\varphi}$ comes from (59). If both $p>1$ and
$\theta>0$ are sufficiently large, then inequality (103) guarantees that there
exist $C_{\delta,p,\theta}$ such that $\lim_{\delta\to
0}C_{\delta,p,\theta}=0$ and such that

$\sup_{y_{0}\in\mathbb{R}^{d}}{\mathbb{E}}\left|\sup_{t\in[t_{0}-\delta,t_{0}]}\sup_{x\in\mathbb{R}^{d}}\frac{|B_{h_{t_{0},x_{0},\delta}}(t,x)|}{1+|x-y_{0}|^{\theta}}\right|^{p}\leq
C_{\delta,p,\theta}\,\mathcal{Q}_{1,\delta,p}\,,$ (156)

where we have just defined $\mathcal{Q}_{1,\delta,p}$ in (154). We know that
the right-hand-side of (156) is finite because of Theorem 4.4. In addition,
because
$D_{h_{t_{0},x_{0},\delta}}u=\mathcal{L}(\langle\phi,\,h_{t_{0},x_{0},\delta}\rangle_{{\mathcal{H}}_{T}}+B_{h_{t_{0},x_{0},\delta}})$,
Theorem 2.12 guarantees that there exists $C>0$ such that for all
$\delta\in[0,1]$,

$\sup_{y_{0}\in\mathbb{R}^{d}}{\mathbb{E}}\left|\sup_{t\in[t_{0}-\delta,t_{0}]}\sup_{x\in\mathbb{R}^{d}}\frac{|D_{h_{t_{0},x_{0},\delta}}u(t,x)|}{1+|x-y_{0}|^{\theta}}\right|^{p}\leq
C\sup_{y_{0}\in\mathbb{R}^{d}}{\mathbb{E}}\left|\sup_{t\in[t_{0}-\delta,t_{0}]}\sup_{x\in\mathbb{R}^{d}}\frac{|\left<\phi_{t,x},h_{t_{0},x_{0},\delta}\right>_{{\mathcal{H}}_{T}}|}{1+|x-y_{0}|^{\theta}}\right|^{p}\\\
+C\sup_{y_{0}\in\mathbb{R}^{d}}{\mathbb{E}}\left|\sup_{t\in[t_{0}-\delta,t_{0}]}\sup_{x\in\mathbb{R}^{d}}\frac{|B_{h_{t_{0},x_{0},\delta}}(t,x)|}{1+|x-y_{0}|^{\theta}}\right|^{p}.$
(157)

Then plugging (143) and (156) into the right hand side of (157), we get

$\sup_{y_{0}\in\mathbb{R}^{d}}{\mathbb{E}}\left|\sup_{t\in[t_{0}-\delta,t_{0}]}\sup_{x\in\mathbb{R}^{d}}\frac{|D_{h_{t_{0},x_{0},\delta}}u(t,x)|}{1+|x-y_{0}|^{\theta}}\right|^{p}\leq
C_{\delta,p,\theta}\mathcal{Q}_{1,\delta,p}+C(Q_{\Lambda}(\delta))^{p}\left(1+\mathcal{Q}_{2,\delta,p}\right)\,,$
(158)

where we have set

$\mathcal{Q}_{2,\delta,p}=\sup_{y_{0}\in\mathbb{R}^{d}}{\mathbb{E}}\left|\sup_{t\in[t_{0}-\delta,t_{0}]}\sup_{x\in\mathbb{R}^{d}}\frac{|u(t,x)|}{1+|x-y_{0}|^{\theta}}\right|^{p}\,.$
(159)

Observe that as an easy consequence of (51), we have
$\mathcal{Q}_{2,\delta,p}<+\infty$. Furthermore, recall that $\lim_{\delta\to
0}C_{\delta,p,\theta}=0$ in equation (156). Hence there exists $\delta_{0}>0$
small enough so that $C_{\delta,p,\theta}<\frac{1}{2}$. The left hand side of
(158) being also of the form $\mathcal{Q}_{1,\delta,p}$ according to our
definition (154), for all $\delta\in(0,\delta_{0})$ we obtain that there
exists a constant $C$ such that

$\sup_{y_{0}\in\mathbb{R}^{d}}{\mathbb{E}}\left|\sup_{t\in[t_{0}-\delta,t_{0}]}\sup_{x\in\mathbb{R}^{d}}\frac{|D_{h_{t_{0},x_{0},\delta}}u(t,x)|}{1+|x-y_{0}|^{\theta}}\right|^{p}\leq
C(Q_{\Lambda}(\delta))^{p}.$ (160)

This proves our claim (155). ∎

###  5.4. Almost sure upper bounds on the integrals $\mathbf{A}$ and
$\mathbf{B}$

In this section, we will prove that with probability one,
$A_{h_{t_{0},x_{0},\delta}}(t_{0},x_{0})$ and
$B_{h_{t_{0},x_{0},\delta}}(t_{0},x_{0})$ converge to zero much faster than
$Q_{\Lambda}(\delta)$ as $\delta\downarrow 0$. These results, along with
(142), will enable us to prove that with probability one there exists
$\delta>0$ such that $D_{h_{t_{0},x_{0},\delta}}u(t_{0},x_{0})>0$.

As a first step, we show that
${|B_{h_{t_{0},x_{0},\delta}}(t_{0},x_{0})|}/{Q_{\Lambda}(\delta)}$ converges
to zero in probability.

######  Lemma 5.4.

Let $(t_{0},x_{0})\in(0,T]\times{\mathbb{R}}^{d}$, and recall that the term
$B_{h_{t_{0},x_{0},\delta}}(t_{0},x_{0})$ is defined by (138). We work under
the conditions of Theorem 5.1. Then for any ${\varepsilon}>0$,

$\lim_{\delta\to
0}{\mathbb{P}}\left(\frac{|B_{h_{t_{0},x_{0},\delta}}(t_{0},x_{0})|}{Q_{\Lambda}(\delta)}>{\varepsilon}\right)=0\,.$
(161)

###### Proof.

We start by applying Chebyshev inequality in order to get

${\mathbb{P}}\left(\frac{|B_{h_{t_{0},x_{0},\delta}}(t_{0},x_{0})|}{Q_{\Lambda}(\delta)}>{\varepsilon}\right)\leq\frac{{\mathbb{E}}|B_{h_{t_{0},x_{0},\delta}}(t_{0},x_{0})|^{p}}{{\varepsilon}^{p}(Q_{\Lambda}(\delta))^{p}}\,.$

Next applying successively (156) and (155) we easily get that

${\mathbb{P}}\left(\frac{|B_{h_{t_{0},x_{0},\delta}}(t_{0},x_{0})|}{Q_{\Lambda}(\delta)}>{\varepsilon}\right)\leq\frac{C_{\delta,p,\theta}}{{\varepsilon}^{p}(Q_{\Lambda}(\delta))^{p}}\mathcal{Q}_{1,\delta,p}\leq\frac{C}{{\varepsilon}^{p}}C_{\delta,p,\theta}\,,$

where $C_{\delta,p,\theta}$ satisfies $\lim_{\delta\to
0}C_{\delta,p,\theta}=0$. We have thus achieved (161), which finishes the
proof. ∎

Let us state a corollary to Lemma 5.4 which will be important in our arguments
towards positivity of the Malliavin derivative. Its proof derives from
standard tools in probability theory and is omitted for sake of conciseness.

######  Corollary 5.5.

Let the assumptions of Lemma 5.4 prevail. Then for any sequence
$\delta_{k}\downarrow 0$, there exists a subsequence $\delta_{k_{i}}$ on which
${|B_{h_{t_{0},x_{0},\delta_{k_{i}}}}(t_{0},x_{0})|}/{Q_{\Lambda}(\delta_{k_{i}})}$
converges to zero almost surely.

The Lebesgue integral $A_{h_{t_{0},x_{0},\delta}}$ is more difficult to
analyze because of the presence of the $f^{\prime}(u(t,x))$ term in (137).
Assumption 2.4 limits the growth to $|f^{\prime}(u)|\leq K\exp(K|u|^{\nu})$,
but because we only have moment estimates on
$\sup_{t\in[0,T]}\sup_{x\in\mathbb{R}^{d}}\frac{|u(t,x)|}{1+|x-x_{0}|^{\theta}}$,
there is no reason to expect to have moment estimates on
${\mathbb{E}}|f^{\prime}(u(t,x))|^{p}$. On the other hand, the bounds on
$f^{\prime}(u)$ guarantee that the integral defined in (137) is convergent
with probability one as long as $\theta\nu<2$. Before controlling the size of
$A_{h_{t_{0},x_{0},\delta}}$, let us thus state an estimate on
$D_{h_{t_{0},x_{0},\delta}}u(t,x)$ that holds with probability one.

######  Lemma 5.6.

For $\delta>0$ and $(t_{0},x_{0})\in(0,T]\times{\mathbb{R}}^{d}$, let
$h_{t_{0},x_{0},\delta}$ be defined by (136). We suppose that the assumptions
of Theorem 5.1 hold. Consider the sequence $\\{\delta_{k};\,k\geq 1\\}$
defined by $\delta_{k}=2^{-k}$. Then we have

${\mathbb{P}}\left(\lim_{k\to\infty}\sup_{t\in[t_{0}-\delta_{k},t_{0}]}\sup_{x\in\mathbb{R}^{d}}\frac{\delta_{k}^{\frac{1}{2}}|D_{h_{t_{0},x_{0},\delta_{k}}}u(t,x)|}{\left(1+|x-x_{0}|^{\theta}\right)Q_{\Lambda}(\delta_{k})}=0\right)=1.$
(162)

######  Remark 5.7.

Notice that there is an extra $\delta_{k}^{\frac{1}{2}}$ in the numerator of
the expression in (162). This extra factor will help all of this converge to
zero.

###### Proof of Lemma 5.6.

Recall that ${\delta}_{k}=2^{-k}$. Like in the proof of Lemma 5.4, we first
apply Chebyshev’s inequality. Recalling our notation (154) for
$\mathcal{Q}_{1,\delta,p}$, we get

${\mathbb{P}}\left(\sup_{t\in[t_{0}-\delta_{k},t_{0}]}\sup_{x\in\mathbb{R}^{d}}\frac{\delta_{k}^{\frac{1}{2}}|D_{h_{t_{0},x_{0},\delta_{k}}}u(t,x)|}{(1+|x-x_{0}|^{\theta})Q_{\Lambda}(\delta_{k})}>\delta_{k}^{\frac{1}{4}}\right)\leq\frac{\delta_{k}^{p/2}\mathcal{Q}_{1,\delta,p}}{Q_{\Lambda}(\delta_{k})^{p}\delta_{k}^{p/4}}\,.$

Therefore due to (155) we have

${\mathbb{P}}\left(\sup_{t\in[t_{0}-\delta_{k},t_{0}]}\sup_{x\in\mathbb{R}^{d}}\frac{\delta_{k}^{\frac{1}{2}}|D_{h_{t_{0},x_{0},\delta_{k}}}u(t,x)|}{(1+|x-x_{0}|^{\theta})Q_{\Lambda}(\delta_{k})}>\delta_{k}^{\frac{1}{4}}\right)\leq
C\delta_{k}^{\frac{p}{4}}\leq 2^{-\frac{pk}{4}}.$ (163)

By the Borel-Cantelli Lemma, with probability one there exists $K(\omega)$
such that for all $k\geq K(\omega)$

$\sup_{t\in[t_{0}-\delta_{k},t_{0}]}\sup_{x\in\mathbb{R}^{d}}\frac{\delta_{k}^{\frac{1}{2}}|D_{h_{t_{0},x_{0},\delta_{k}}}u(t,x)|}{1+|x-x_{0}|^{\theta}Q_{\Lambda}(\delta_{k})}\leq\delta_{k}^{\frac{1}{4}},$
(164)

implying that

${\mathbb{P}}\left(\lim_{k\to\infty}\sup_{t\in[t_{0}-\delta_{k},t_{0}]}\sup_{x\in\mathbb{R}^{d}}\frac{\delta_{k}^{\frac{1}{2}}|D_{h_{t_{0},x_{0},\delta_{k}}}u(t,x)|}{1+|x-x_{0}|^{\theta}Q_{\Lambda}(\delta_{k})}=0\right)=1.$
(165)

This achieves the proof of (162). ∎

With this intermediate result on the behavior of $Du(t,x)$, we can now state a
lemma estimating the integral term $A$.

######  Lemma 5.8.

Recall that we have set $\delta_{k}=2^{-k}$ for $k\geq 1$. The integral
$A_{h_{t_{0},x_{0},\delta_{k}}}(t_{0},x_{0})$ is defined by (137), that is

$A_{h_{t_{0},x_{0},\delta_{k}}}(t_{0},x_{0})=\int_{t_{0}-\delta}^{t_{0}}\int_{\mathbb{R}^{d}}G(t_{0}-s,x_{0}-y)f^{\prime}(u(s,y))D_{h_{t_{0},x_{0},\delta_{k}}}u(s,y)dyds.$
(166)

Our assumptions are those of Theorem 5.1. Then we have

${\mathbb{P}}\left(\lim_{k\to\infty}\frac{|A_{h_{t_{0},x_{0},\delta_{k}}}(t_{0},x_{0})|}{Q_{\Lambda}(\delta_{k})}=0\right)=1.$
(167)

###### Proof.

We have seen that the random variable $\mathcal{Q}_{2,\delta,p}$ defined by
(159) is such that ${\mathbb{E}}[\mathcal{Q}_{2,\delta,p}]<\infty$, as an easy
consequence of (51). Hence with probability one there exists a (random)
constant $M(\omega)$ such that

$\sup_{t\in[0,t_{0}]}\sup_{x\in\mathbb{R}^{d}}\frac{|u(t,x)|}{1+|x-x_{0}|^{\theta}}\leq
M(\omega).$ (168)

Furthermore, by possibly increasing the value of $M(\omega)$, Lemma 5.6
guarantees that choosing $\delta_{k}=2^{-k}$, with probability one we have

$\sup_{t\in[t_{0}-\delta_{k},t_{0}]}\sup_{x\in\mathbb{R}^{d}}\frac{|D_{h_{t_{0},x_{0},\delta_{k}}}u(t,x)|}{1+|x-x_{0}|^{\theta}}\leq
M(\omega)\frac{Q_{\Lambda}(\delta_{k})}{\delta_{k}^{\frac{1}{2}}}\text{ for
all }k\in\mathbb{N}.$ (169)

In this proof, we allow the value of $M(\omega)$ to change from line to line
as long as its value remains independent of $\delta_{k}$.

Let us turn to the estimate on $A$ in (166). Owing to (15), (168) and (169),
we easily obtain

$|A_{h_{t_{0},x_{0},\delta_{k}}}(t_{0},x_{0})|\leq\int_{t_{0}-\delta_{k}}^{t_{0}}\int_{\mathbb{R}^{d}}G(t_{0}-s,x_{0}-y)\exp\left(M(\omega)|x_{0}-y|^{\theta\nu}\right)\\\
\times
M(\omega)(1+|y-x_{0}|^{\theta})\frac{Q_{\Lambda}(\delta_{k})}{\delta_{k}^{\frac{1}{2}}}dyds.$

By increasing the value of $M(\omega)$, the integrand in the above expression
is bounded by

$(2\pi(t_{0}-s))^{-\frac{d}{2}}M(\omega)\exp\left(-\frac{|y-x_{0}|^{2}}{2(t_{0}-s)}+M(\omega)|x_{0}-y|^{\theta\nu}\right)Q_{\Lambda}(\delta_{k})\delta_{k}^{-\frac{1}{2}}.$
(170)

We now bound the term

$\mathcal{R}_{\omega,\nu,\theta}\equiv-\frac{|y-x_{0}|^{2}}{2(t_{0}-s)}+M(\omega)|x_{0}-y|^{\theta\nu},$
(171)

which appears in the exponent in (170). Namely recast this exponent as

$\mathcal{R}_{\omega,\nu,\theta}=-\frac{|y-x_{0}|^{2}}{2(t_{0}-s)}+KM(\omega)(2(t_{0}-s))^{\frac{\theta\nu}{2}}\left(\frac{|x_{0}-y|}{\sqrt{2(t_{0}-s)}}\right)^{\theta\nu}\,.$

Next because $\theta\nu<2$, Young’s inequality with powers
$\frac{2}{\nu\theta}$ and $\frac{2}{2-\nu\theta}$ in the right hand side above
proves that $\mathcal{R}_{\omega,\nu,\theta}$ satisfies

$\mathcal{R}_{\omega,\nu,\theta}\leq-\frac{|y-x_{0}|^{2}}{2(t_{0}-s)}+\frac{\theta\nu|y-x_{0}|^{2}}{4(t_{0}-s)}+\frac{(2-\nu\theta)(M(\omega))^{\frac{2}{2-\nu\theta}}(2(t_{0}-s))^{\frac{\theta\nu}{2-\nu\theta}}}{2}.$

In addition, since $(t_{0}-s)<\delta_{k}\leq 1$, the third term in the above
expression is uniformly bounded with respect to $s$ and independent of $y$.
Therefore $|A_{h_{t_{0},x_{0},\delta_{k}}}(t_{0},x_{0})|$ is bounded by

$|A_{h_{t_{0},x_{0},\delta_{k}}}(t_{0},x_{0})|\\\ \leq
M(\omega)Q_{\Lambda}(\delta_{k})\delta_{k}^{-\frac{1}{2}}\int_{t_{0}-\delta_{k}}^{t_{0}}\int_{\mathbb{R}^{d}}(2\pi(t_{0}-s))^{-\frac{d}{2}}\exp\left(-\left(1-\frac{\theta\nu}{2}\right)\frac{|x_{0}-y|^{2}}{2(t_{0}-s)}\right)dyds.$

The above integral over $\mathbb{R}^{d}$ is the integral of a Gaussian density
and its value does not depend on $s$. Therefore, owing to the fact that the
time interval is length $\delta_{k}$, we end up with

$|A_{h_{t_{0},x_{0},\delta_{k}}}(t_{0},x_{0})|\leq
M(\omega)Q_{\Lambda}(\delta_{k})\delta_{k}^{\frac{1}{2}}\,,$ (172)

with probability one. This proves that

$\lim_{k\to\infty}\frac{|A_{h_{t_{0},x_{0},\delta_{k}}}(t_{0},x_{0})|}{Q_{\Lambda}(\delta_{k})}=0\,,\quad\text{
with probability one.}$ (173)

Our claim (167) is proved. ∎

Now we can establish the main result of this subsection, which is that there
exists a subsequence $\delta_{k}\downarrow 0$ such that with probability one,
for small values of $\delta_{k}$,
$|A_{h_{t_{0},x_{0},\delta_{k}}}(t_{0},x_{0})|$ and
$|B_{h{\delta_{k},t_{0},x_{0}}}(t_{0},x_{0})|$ are much smaller than
$Q_{\Lambda}(\delta_{k})$.

######  Proposition 5.9.

Consider $(t_{0},x_{0})\in(0,T]\times{\mathbb{R}}^{d}$ and the sequence
$\\{\delta_{k}=2^{-k};\,k\geq 1\\}$. The function $h_{t_{0},x_{0},\delta_{k}}$
is introduced in (141), and the terms $A,B$ are respectively given by
(137)-(138). We assume that the coefficients $b,\sigma$ satisfy Assumptions
2.3–2.4. Then there exists a subsequence of $\delta_{k}$, still denoted
$\delta_{k}$, such that

${\mathbb{P}}\left(\lim_{k\to\infty}\frac{|A_{h_{t_{0},x_{0},\delta_{k}}}(t_{0},x_{0})|+|B_{h_{t_{0},x_{0},\delta_{k}}}(t_{0},x_{0})|}{Q_{\Lambda}(\delta_{k})}=0\right)=1.$
(174)

###### Proof.

Let $\delta_{k}$ be a subsequence from Lemma 5.8 along which

${\mathbb{P}}\left(\lim_{k\to\infty}\frac{|A_{h_{t_{0},x_{0},\delta_{k}}}(t_{0},x_{0})|}{Q_{\Lambda}(\delta_{k})}=0\right)=1.$
(175)

Then thanks to Corollary 5.5, there is a subsequence of $\delta_{k}$
(relabeled as $\delta_{k}$) along which
$\lim_{k\to\infty}Q_{\Lambda}(\delta_{k})^{-1}|B_{h_{t_{0},x_{0},\delta_{k}}}(t_{0},x_{0})|=0$
with probability one. This proves our result. ∎

###  5.5. Positivity of the Malliavin derivative

In this section we prove that for any $t_{0}\in(0,T]$ and
$x_{0}\in\mathbb{R}^{d}$ we have $\|Du(t_{0},x_{0})\|_{{\mathcal{H}}_{T}}>0$
almost surely. This will allow to establish the existence of a density for the
random variable $u(t_{0},x_{0})$.

######  Theorem 5.10.

Let $t_{0}\in(0,T]$ and $x_{0}\in\mathbb{R}^{d}$. We assume that the
coefficients $b,\sigma$ satisfy Assumptions 2.3–2.4 and we consider the
solution $u$ to equation (2.1). Then the following holds true:

${\mathbb{P}}(\|Du(t_{0},x_{0})\|_{{\mathcal{H}}_{T}}>0)=1.$

###### Proof.

Let $h_{t_{0},x_{0},\delta}$ be defined by (141). In (131) we decomposed

$D_{h_{t_{0},x_{0},\delta}}u(t_{0},x_{0})=\left<\phi_{t_{0},x_{0}},h_{t_{0},x_{0},\delta}\right>_{{\mathcal{H}}_{T}}+A_{h_{t_{0},x_{0},\delta}}(t_{0},x_{0})+B_{h_{t_{0},x_{0},\delta}}(t_{0},x_{0}).$

Moreover, in (142), we proved that with probability one,

$\left<\phi_{t_{0},x_{0}},h_{t_{0},x_{0},\delta}\right>_{{\mathcal{H}}_{T}}\geq\alpha
Q_{\Lambda}(\delta).$

By Theorem 5.9 there exists a subsequence $\delta_{k}\downarrow 0$ such that

$\lim_{k\to\infty}\frac{|A_{h_{t_{0},x_{0},\delta_{k}}}(t_{0},x_{0})|+|B_{h_{t_{0},x_{0},\delta_{k}}}(t_{0},x_{0})|}{Q_{\Lambda}(\delta_{k})}=0.$

Therefore, with probability one, there exists a (random) $k(\omega)$ such that

$D_{h_{\delta_{k(\omega)},t_{0},x_{0}}}u(t_{0},x_{0})>0.$

This implies that $\|Du(t_{0},x_{0})\|_{{\mathcal{H}}_{T}}>0$ with probability
one, because at least one of the directional derivatives is nonzero. ∎

We conclude this paper by proving our main result for the density of
$u(t_{0},x_{0})$.

###### Proof of Theorem 5.1.

According to Proposition 2.14, we have to check that the random variable
$F=u(t_{0},x_{0})$ belongs to the Malliavin-Sobolev space $\mathbb{D}^{1,p}$,
and that ${\mathbb{P}}(\|Du(t_{0},x_{0})\|_{{\mathcal{H}}_{T}}>0)=1$. Now the
fact that $u(t_{0},x_{0})\in\mathbb{D}^{1,p}$ is established in Theorem 4.4,
while the condition
${\mathbb{P}}(\|Du(t_{0},x_{0})\|_{{\mathcal{H}}_{T}}>0)=1$ is the contents of
Theorem 5.10. This implies that the law of $u(t_{0},x_{0})$ is absolutely
continuous with Lebesgue measure. ∎

## Acknowledgements

The first author was partially supported by the Simon’s Foundation Award
962543. The second author was partially supported by the NSF grant
DMS-1952966.

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