# Pathwise regularization of the stochastic heat equation with multiplicative
noise through irregular perturbation

Rémi Catellier and Fabian A. Harang  Rémi Catellier: Université Côte d Azur,
CNRS, LJAD, France<EMAIL_ADDRESS>Fabian A. Harang:
Department of Mathematics, University of Oslo, P.O. box 1053, Blindern, 0316,
OSLO, Norway<EMAIL_ADDRESS>
###### Abstract.

Existence and uniqueness of solutions to the stochastic heat equation with
multiplicative spatial noise is studied. In the spirit of pathwise
regularization by noise, we show that a perturbation by a sufficiently
irregular continuous path establish wellposedness of such equations, even when
the drift and diffusion coefficients are given as generalized functions or
distributions. In addition we prove regularity of the averaged field
associated to a Lévy fractional stable motion, and use this as an example of a
perturbation regularizing the multiplicative stochastic heat equation.

###### Key words and phrases:

Pathwise regularization by noise, stochastic heat equation, generalized
parabolic Anderson model, fractional Lévy processes

###### 2010 Mathematics Subject Classification:

Primary 60H50, 60H15 ; Secondary 60L20

_Acknowledgments._ FH gratefully acknowledges financial support from the STORM
project 274410, funded by the Research Council of Norway. We would also thank
Nicolas Perkowski for several fruitful discussions on this topic.

###### Contents

  1. 1 Introduction
  2. 2 Non-linear Young-Volterra theory in Banach spaces
  3. 3 Averaged fields
  4. 4 Existence and uniqueness of the mSHE
  5. 5 Averaged fields with Lévy noise
  6. 6 Conclusion
  7. A Basic concepts of Besov spaces and properties of the heat kernel
  8. B Cauchy-Lipschitz theorem for mSHE in standard case.

##  1\. Introduction

The stochastic heat equation with multiplicative noise (mSHE), is given on the
form

$\partial_{t}u=\Delta u+b(u)+g(u)\xi,\qquad u_{0}\in\mathcal{C}^{\beta},$
(1.1)

where $\xi$ is a space time noise on $\mathbb{R}^{d}$ or $\mathbb{T}^{d}$ and
$\mathcal{C}^{\beta}$ is the Besov Hölder space of $\mathbb{R}^{d}$ or
$\mathbb{T}^{d}$, and $b$ and $g$ are sufficiently smooth functions. This
equation is a fundamental stochastic partial differential equation, and is
applied for modelling in a diverse selection of natural sciences, ranging from
chemistry and biology to physics. The existence and uniqueness of (1.1) is
typically proven under the condition that both $b$ and $g$ are Lipschitz
functions of linear growth (see e.g. [25]). When taking a pathwise approach to
a solution theory, even more regularity of these non linear functions may be
required (see for instance [3] where $g\in C^{3}$ is required and Appendix B
for a proof a pathwise wellposedness in a simple context). Of course if
$g\equiv 0$, then (1.1) is known as the (deterministic) non-linear heat
equation, for which uniqueness fails in general under weaker conditions on $b$
than Lipschitz and linear growth.

Motivated by this, a natural question to ask is if it is possible to prove
existence and uniqueness of (1.2) under weaker conditions on $b$ and $g$.
Inspired by the well known regularization by noise phenomena in stochastic
differential equations, one may think that the same principles of
regularization would extend to the case of stochastic partial differential
equations like (1.2). In this article, we aim at giving some insights into
this question by investigating (1.2) in a fully pathwise manner under
perturbation by a measurable time (only) dependent path. In fact, we will
prove that a perturbation by a sufficiently irregular path yields
wellposedness of the mSHE, even for distributional (generalized functions)
coefficients $g$ and $b$. In the next section we give a more detailed
description of the specific equation under consideration, and the techniques
that we apply in order to prove this regularizing effect of the perturbation.

###  1.1. Methodology

Inspired by the theory of regularization by noise for ordinary or stochastic
differential equations, we show that a suitably chosen measurable path
$w:[0,T]\rightarrow\mathbb{R}^{d}$ provides a regularizing effect on (1.1), by
considering the formal equation

$\partial_{t}u=\Delta u+b(u)+g(u)\xi+\dot{\omega}_{t},\qquad
u_{0}\in\mathcal{C}^{\beta},$ (1.2)

where $\xi$ is a spatial (distributional) noise taking values in
$\mathbb{R}^{d}$, and $\dot{\omega}_{t}$ is the distributional derivative of a
continuous path $\omega$. To this end, we formulate (1.2) in terms of the non-
linear Young framework, developed in [7, 15, 14]. We extend this framework to
the infinite dimensional setting adapted to Volterra type integrals appearing
when considering the mild formulation of (1.2). The integration framework
developed here is strongly based on the recently developed Volterra sewing
lemma of [24], and does not require any semi-group property of the Volterra
operator. The integral can therefore be applied to several different problems
relating to infinite dimensional Volterra integration, and thus we believe
that this construction is interesting in itself.

To motivate the methodology of the current paper, consider again (1.2) and set
$\theta=u-w$ with $\theta_{0}=u_{0}\in\mathcal{C}^{\beta}$ then formally
$\theta$ solves the following integral equation

$\theta_{t}=P_{t}\theta_{0}+\int_{0}^{t}P_{t-s}b(\theta_{s}+\omega_{s})\mathop{}\\!\mathrm{d}s+\int_{0}^{t}P_{t-s}\xi
g(\theta_{s}+\omega_{s})\mathop{}\\!\mathrm{d}s,$

where $P$ is the fundamental solution operator associated with the heat
equation, and the product $P_{t}\theta_{0}$ is interpreted as spatial
convolution. For simplicity, we will carry out most of our analysis with
$b\equiv 0$, as this term is indeed easier to handle than the term with
multiplicative noise. The equation we will then consider is given by

$\theta_{t}=P_{t}\theta_{0}+\int_{0}^{t}P_{t-s}\xi
g(\theta_{s}+\omega_{s})\mathop{}\\!\mathrm{d}s.$ (1.3)

In Section 4.2 we provide a detailed description of how our results can easily
be extended to include the drift term with distributional $b$, by simply
appealing to the the analysis carried out for the purely multiplicative
equation (1.3). Associated to the path $\omega$ and the distribution $g$,
define the averaged distribution
$T^{\omega}g:[0,T]\times\mathbb{R}^{d}\rightarrow\mathbb{R}^{d}$ by the
mapping

$(t,x)\mapsto\int_{0}^{t}g(x+\omega_{s})\mathop{}\\!\mathrm{d}s.$

After proving that for certain paths $\omega$ the distribution $T^{\omega}g$
is in fact a regular function, we then consider (1.3) as a non-linear Young
equation of the form

$\theta_{t}=P_{t}\theta_{0}+\int_{0}^{t}P_{t-s}\xi
T^{\omega}_{\mathop{}\\!\mathrm{d}s}g(\theta_{s}),\quad\theta_{0}\in\mathcal{C}^{\beta}.$
(1.4)

Here, the integral is interpreted in an infinite dimensional non-linear Young-
Volterra sense. That is, to suit our purpose, we extend the non-linear Young
integral to an infinite dimensional setting as well as allowing for the action
of a Volterra operator on the integrand. The integral is then constructed as
the Banach valued element

$\int_{0}^{t}P_{t-s}T^{\omega}_{\mathop{}\\!\mathrm{d}s}g(\theta_{s}):=\lim_{|\mathcal{P}|\rightarrow
0}\sum_{[u,v]\in\mathcal{P}}P_{t-u}\xi T^{\omega}_{u,v}g(\theta_{u}),$

where $T^{\omega}_{u,v}g:=T^{\omega}_{v}g-T^{\omega}_{u}g$. We stress that in
contrast to the non-linear Young integral used for example in [7, 15, 23, 14],
the above integral is truly an infinite dimensional object, and extra care
must be taken when building it from the averaged function $T^{\omega}g$.
Indeed, for each $t\geq 0$,
$\theta_{t}\in\mathcal{C}^{\beta}(\mathbb{R}^{d};\mathbb{R})$ and so the
function $T^{\omega}g$ is then lifted to be a functional on
$\mathcal{C}^{\beta}$. We show that this lift comes at the cost of an extra
degree of assumed regularity on the averaged function $T^{\omega}g$.
Furthermore, due to the assumption that $\xi\in\mathcal{C}^{-\vartheta}$ for
$\vartheta>0$ (i.e. $\xi$ is assumed to be truly distributional) we need to
make use of the product in Besov space in order to make the product of $\xi
T^{\omega}_{u,v}g(\theta_{u})$ well defined.

Similar to the theory of rough paths, our analysis can be divided into two
parts: (i) a probabilistic step, and (ii) a deterministic (analytic) step. We
give a short description of the two steps here:

  * (i)

Let $E$ be a separable Banach space. We develop an abstract framework of
existence and uniqueness of Banach valued equations

$\theta_{t}=p_{t}+\int_{0}^{t}S_{t-s}X_{\mathop{}\\!\mathrm{d}s}(\theta_{s}),$
(1.5)

where $S$ is a suitable (possibly singular) Volterra operator, and
$X:[0,T]\times E\rightarrow E$ is a function which is $\frac{1}{2}+$ Hölder
regular in time, and suitably regular in its spatial argument (to be specified
later), and $p:[0,T]\rightarrow E$ is a sufficiently regular function. To this
end, we use a simple extension of the Volterra sewing lemma developed in [24]
to construct the non-linear Young-Volterra integral appearing in (1.5) as the
following

$\int_{0}^{t}S_{t-s}X_{\mathop{}\\!\mathrm{d}s}(\theta_{s}):=\lim_{|\mathcal{P}|\rightarrow
0}\sum_{[u,v]\in\mathcal{P}[0,t]}S_{t-u}X_{u,v}(\theta_{u}),$

where $\mathcal{P}[0,t]$ is a partition of $[0,t]$ with mesh size
$|\mathcal{P}|$ converging to zero.

  * (ii)

The second step is then to consider $\\{\omega_{t}\\}_{t\in[0,T]}$ to be a
stochastic process on a probability space $(\Omega,\mathcal{F},\mathbb{P})$,
and we need to show that the averaged function $T^{\omega}g$ is indeed a
sufficiently regular function $\mathbb{P}$-a.s., even when $g$ is a true
distribution. This is done by probabilistic methods.

At last we relate the abstract function $X$ from (2.21) to the averaged
function $T^{\omega}$, and then a combination of the previous two steps gives
us existence and uniqueness of (1.4), and thus also (1.3), which then is used
to makes sense of (1.2) through the translation $u=v+\omega$.

###  1.2. Short overview of existing literature

In order to prove existence and uniqueness of (1.1), there is two main
directions to follow: the classical probabilistic setting based on Itô type
theory, or the pathwise approach based on rough paths or similar techniques.
Using the first approach, one typically require that $b$ and $g$ are Lipschitz
and of linear growth (see e.g. [25]). Note that in this context, at least in
dimension $d=1$, one can prove also prove existence and uniqueness with less
restrictive requirements on $g$ in certain cases (see [32, 33, 38]). In
particular in [32], the authors prove, using probabilistic arguments, that
there is a non-zero solution to the equation

$\partial_{t}u=\Delta u+|u|^{\gamma}\xi+\psi,\quad u(0,\cdot)=0,$

where $\xi$ is a space time white noise in dimension $1+1$, $\psi$ is a non-
zero, non-negative function smooth compactly supported function and
$0\leq\gamma<\frac{3}{4}$. They also prove that when $\gamma>\frac{3}{4}$
uniqueness holds. Note also that when $\xi$ is a deterministic function and
not a distribution, wellposedness is a well-know topic. In particular if $\xi$
is a non-negative continuous and bounded function, Fujita and Watanabe [13]
prove that Osgood condition on $g$ are "nearly" necessary and sufficient to
guaranty uniqueness. In particular when $g$ is only Hölder continuous, one can
not expect to have uniqueness. One can also consult [5] and the reference
therein for further attempt in that direction.

When using pathwise techniques to solve (stochastic) nonlinear heat equation,
as usual (see [9] for counterexamples in a rough path context), one typically
needs to require even higher regularity on $b$ and $g$ in order guarantee
existence and uniqueness (see e.g. [3] where three times differentiability is
assumed and the Appendix B for a simple proof).

To the best of our knowledge, little has been done in the direction of
investigating the regularizing effects obtained from measurable perturbations
of the heat equation. In the case when $g\equiv 0$ it has been proven in [34]
(see also [6]) that the additive stochastic heat equation on the form

$\partial_{t}u=\partial^{2}_{x}u+b(u)+\partial_{t}\partial_{x}\omega,\quad(t,x)\in[0,T]\times[0,1]$

exists uniquely, even when $b$ is only bounded and measurable and
$\partial_{t}\partial_{x}\omega$ is understood as a white noise on
$[0,T]\times[0,1]$. Thus the addition of noise seems to give similar
regularizing effects in the stochastic heat equation as is observed in SDEs.

Note that the recent publication [1] continue in the investigation when
$g\equiv 1$, and in particular the authors are able to recover results on skew
stochastic heat equation.

###  1.3. Main results

Before presenting our main results, let us first give a definition of what we
will call a solution to (1.2). Let us remind that the definition of admissible
weight and Besov spaces is written in Appendix A.2 and A.1, the precise
definition of non-linear Young-Volterra equation is given in Section 2.1 and
the definition of averaged field is given in Section 3. The equations
considered in the below results can either be considerd on $\mathbb{R}$ or
$\mathbb{T}$. Indeed, all the following results are true in both settings.
Furthermore, for the sake of using the space white noise, it might be
instructive to think to the space as the torus $\mathbb{T}$.

######  Definition 1.

Let $\omega:[0,T]\rightarrow\mathbb{R}$ be a measurable path, and consider a
$g\in\mathcal{S}^{\prime}$ such that the averaged field
$T^{\omega}b\in\mathcal{C}^{\gamma}_{T}\mathcal{C}^{\kappa}(w)$ for some
$\gamma>\frac{1}{2}$ and $\kappa\geq 3$ and an admissible weight
$w:\mathbb{R}^{d}\rightarrow\mathbb{R}$. Suppose that $0<\vartheta<\beta<1$
and suppose that $\xi$ is a spatial noise contained in
$\mathcal{C}^{-\vartheta}$. Take $\rho=\frac{\beta+\vartheta}{2}$ and suppose
that $\gamma-\rho>1-\gamma$. Let $u_{0}\in\mathcal{C}^{\beta}$. We say that
there exists a unique solution to the equation

$u_{t}=P_{t}u_{0}+\int_{0}^{t}P_{t-s}\xi
g(u_{s})\mathop{}\\!\mathrm{d}s+\omega_{t},\quad t\in[0,T]$ (1.6)

if $u\in\omega+\mathscr{C}^{\varsigma}_{T}\mathcal{C}^{\beta}$ for any
$1-\gamma<\varsigma<\gamma-\rho$, and there exists a unique
$\theta\in\mathscr{C}^{\varsigma}_{T}\mathcal{C}^{\beta}$ such that
$u=\omega+\theta$ and $\theta$ solves the non-linear Young equation

$\theta_{t}=P_{t}u_{0}+\int_{0}^{t}P_{t-s}\xi
T^{\omega}_{\mathop{}\\!\mathrm{d}s}g(\theta_{s}).$ (1.7)

Here the integral is understood as a non-linear Young–Volterra integral, as
constructed in Section 2.1 and the space $\mathscr{C}^{\varsigma}_{T}$ is
defined in Definition 6.

######  Theorem 2.

Let $\omega:[0,T]\rightarrow\mathbb{R}$ be a measurable path, and consider a
distribution $g\in\mathcal{S}^{\prime}(\mathbb{R})$ such that the associated
averaged field $T^{\omega}b\in\mathcal{C}^{\gamma}_{T}\mathcal{C}^{\kappa}(w)$
for some $\gamma>\frac{1}{2}$, $\kappa\geq 3$ and a admissible weight function
$w:\mathbb{R}^{d}\rightarrow\mathbb{R}_{+}$. Suppose that $0<\vartheta<1$ and
take $\vartheta<\beta<2-\vartheta$ and that $\xi$ is a spatial noise contained
in $\mathcal{C}^{-\vartheta}$. Let $\rho=\frac{\beta+\vartheta}{2}$ and assume
that $1-\gamma<\gamma-\rho$. Then there exists a time $\tau\in(0,T]$ such that
there exists a unique solution $u$ in the sense of Definition 1 to Equation
(1.6). If $w$ is globally bounded, then the solution is global, in the sense
that a unique solution $u$ to (1.6) is define on all $[0,\tau]$ for any
$\tau\in(0,T]$.

The unique solution can be interpreted to be a "physical" one, in the sense
that it is stable under approximations. We summarize this in the following
corollary.

######  Corollary 3.

Under the assumptions of Theorem 2, if $\\{g_{n}\\}_{n\in\mathbb{N}}$ is a
sequence of smooth functions converging to
$g\in\mathcal{S}^{\prime}(\mathbb{R})$ such that $T^{\omega}g_{n}\rightarrow
T^{\omega}g\in\mathcal{C}^{\gamma}_{T}\mathcal{C}^{\kappa}(w)$, then the
corresponding sequence of solutions
$\\{u_{n}\\}_{n\in\mathbb{N}}=\\{\omega+\theta^{n}\\}_{n\in\mathbb{N}}$ to
equation (1.7) converge to $u=w+\theta$ in the sense that
$\theta^{n}\to\theta$ in $\mathscr{C}^{\varsigma}_{\tau}\mathcal{C}^{\beta}$
for any $1-\gamma<\varsigma<\gamma-\rho$.

In applications, one is typically interested in the regularizing effects
provided by specific sample paths of stochastic processes $\omega$. Although
the class of regularizing paths is already well developed, we show here as an
example the regularizing effect of measurable sample paths of fractional Lévy
processes (see Section 5).

In this connection suppose that $\xi$ is a spatial white noise on the torus
$\mathbb{T}$, and we find a conditions on the distribution $g$ and the initial
data $u_{0}$ so that a unique solution to (1.6) exists.

######  Theorem 4.

Let $\alpha\in(0,2]$. Let $H\in(0,1)\cap\\{\alpha^{-1}\\}$. Let $L^{H}$ be a
Linear Fractional Lévy Process with Hurst parameter $H$ built from a symmetric
$\alpha$-stable Lévy process as defined in Section 5. Let $\vartheta\in(0,1)$
and let $\kappa>3-\frac{1-\vartheta}{2H}$. There exists $\varepsilon>0$ small
enough such that for all $\xi\in\mathcal{C}^{-\vartheta}$ and all
$g\in\mathcal{C}^{\kappa}(w)$ where $w$ is an admissible weight, almost surely
for all $u_{0}\in\mathcal{C}^{\vartheta+\varepsilon}$ there exists a unique
local solution to the mSHE in the sense of Definition 1. If $w$ is bounded,
then the solution exists globally.

One can specify the previous theorem by letting $\xi$ to be the space white
noise :

######  Corollary 5.

Let $d=1$ and let $\xi$ be a space white noise on the Torus $\mathbb{T}$. Let
$\alpha\in(0,2]$, let $H\in(0,1)\cap\\{\alpha^{-1}\\}$ and let
$\kappa>3-\frac{1}{4H}$. Let $w$ be an admissible weight. There exists
$\varepsilon>0$ small enough such that for all $g\in\mathcal{C}^{\kappa}(w)$,
almost surely for all $u_{0}\in\mathcal{C}^{\frac{1}{2}+\varepsilon}$ and let
there exists a unique solution (in the sense of Definition 1) to the mSHE

$\mathop{}\\!\mathrm{d}u=\Delta
u\mathop{}\\!\mathrm{d}t+g(u)\xi\mathop{}\\!\mathrm{d}t+\mathop{}\\!\mathrm{d}L^{H}_{t},\quad
u(0,\cdot)=u_{0},$

where $L^{H}$ is a linear fractional stable motion.

In particular taking $H<\frac{1}{4}$ allows us to take $\kappa<2$ and go
beyond the classical theory using Bony estimates for the product of
distribution in Besov spaces. When $H<\frac{1}{8}$ one can deal with non
Lipschitz-continuous $g$ and when $H<\frac{1}{12}$, one can deal with
distributional field $g$.

The proofs of the above theorems and corollary can be found in Sections 4 and
5.

###  1.4. Outline of the paper

The paper is structured as follows: In section 2.1 we extend the concept of
non–linear Young integration to the infinite dimensional setting, including a
Volterra operator, which in later sections will play the role of the inverse
Laplacian. We also give a result on existence and uniqueness of abstract
equations in Banach spaces.

In section 3 we give a short overview on the concept of averaged fields and
their properties. As this topic is by now well studied in the literature, we
only give here the necessary details and provide several references for
further information. We also show how the standard averaged field can be
viewed as an operator on certain Besov function spaces.

In section 4 we formulate the multiplicative stochastic heat equation in the
non-linear Young-Volterra integration framework, using the concept of averaged
fields. We prove existence and uniqueness of these equations, as well as
Theorem 2 and Corollary 3. In section 5 we investigate closer the regularity
of averaged fields associated with sample paths of fractional Levy processes,
and prove Theorem 4.

At last, in Section 6 we give a short reflection on the main results of the
article and provide some thoughts on future extensions of our results.

For thee sake of self-containedness, we have included an appendix with
preliminaries on weighted Besov spaces, and some results regarding the
regularity/singularity of the inverse Laplacian acting on Besov distributions.
In addition we give a short proof for existence and uniqueness of (1.2) in the
case of twice differentiable $g$.

###  1.5. Notation

For $\beta\in\mathbb{R}$ and $d\geq 1$, we define the Hölder Besov space

$\mathcal{C}^{\beta}=B^{\beta}_{\infty,\infty}(\mathbb{R}^{d};\mathbb{R})$

endowed with its usual norm built upon Paley-Littlewood blocks and denoted by
$\|\cdot\|_{\mathcal{C}^{\beta}}$ (see Appendix A.2 and especially Proposition
39 for more on weighted Besov spaces). Note that when working with a weight
$w$ and weighted spaces, we denote the spaces $\mathcal{C}^{\beta}(w)$ and the
norm $\|\cdot\|_{\mathcal{C}^{\beta}(w)}$. For $T>0$ and $\varsigma\in(0,1)$
and $E$ a Banach space with norm $\|\cdot\|_{E}$ and for $f:[0,T]\mapsto E$,

$[f]_{\varsigma;E}=\sup_{s\neq
t}\frac{\|f_{t}-f_{s}\|_{E}}{|t-s|^{\varsigma}}<\infty,$

and for $\varsigma>0$ we define

$\|f\|_{\varsigma;E}:=\sum_{k=0}^{[\varsigma]}\|f^{([k])}_{0}\|_{E}+\big{[}f^{([\varsigma])}\big{]}_{\varsigma-[\varsigma];E},$

and finally

$\mathcal{C}^{\varsigma}_{T}E=\mathcal{C}^{\varsigma}\big{(}[0,T];E\big{)}:=\left\\{f:[0,T]\to
E\,:\,\|f\|_{\varsigma;E}<\infty\right\\}.$

Whenever the underlying space $E$ is clear form the context we will use the
short hand notation $[f]_{\varsigma}$ etc. to denote the the Hölder semi norm.
We will frequently use the increment notation $f_{s,t}:=f_{t}-f_{s}$. We write
$f\lesssim g$ if there exists a constant $C>0$ such that $f\leq Cg$.
Furthermore to stress that the constant $C$ depends on a parameter $p$ we
write $l\lesssim_{p}g$. We write $f\simeq g$ is $f\lesssim g$ and $g\lesssim
f$. For $s\leq t$, we denote by $\mathcal{P}([s,t])$ a partition of the
interval $[s,t]$. We write
$|\mathcal{P}|=\max_{k\in\\{0,\cdots,n-1\\}}|t_{k+1}-t_{k}|$.

##  2\. Non-linear Young-Volterra theory in Banach spaces

For $\alpha\in(0,2]$, let $t\mapsto P_{t}^{\frac{\alpha}{2}}$ be the
fundamental solution associated to the $\alpha$-fractional heat equation

$\partial_{t}P^{\frac{\alpha}{2}}=-(-\Delta)^{\frac{\alpha}{2}}P^{\frac{\alpha}{2}},\quad
P^{\frac{\alpha}{2}}_{0}=\delta_{0}.$

For a function $y:\mathbb{R}^{d}\rightarrow\mathbb{R}$ let
$P^{\frac{\alpha}{2}}_{t}y$ denotes the convolution between the fundamental
solution at time $t\geq 0$ and $y$. Note that for $\alpha=2$, we obtain the
classical heat equation, and $P_{\cdot}:=P^{1}_{\cdot}$ then denotes the
convolution operator with the Gaussian kernel. Towards a pathwise analysis of
stochastic parabolic equations, one encounter the problem that for a function
$y\in\mathcal{C}^{\kappa}(\mathbb{R}^{d})$ with $\kappa\geq 0$, the mapping
$t\mapsto P_{t}^{\frac{\alpha}{2}}y$ is smooth in time everywhere except when
approaching $0$ where it is only continuous. In particular, from standard
(fractional) heat kernel estimates (see Corollary 45 in the Appendix) we know
that for $s\leq t\in[0,T]$, any $\theta\in[0,1]$ and $\rho\in(0,\alpha]$ the
following inequality holds

$\|(P_{t}^{\frac{\alpha}{2}}-P^{\frac{\alpha}{2}}_{s})y\|_{\mathcal{C}^{\varsigma+\alpha\rho}}\lesssim\|y\|_{\mathcal{C}^{\kappa}}|t-s|^{\theta}s^{-\theta-\rho}.$
(2.1)

Note the special case when $\rho=0$. Then $S_{t}$ is a linear operator on
$\mathcal{C}^{\kappa}$ which is $\theta$-Hölder continuous on an interval
$[\varepsilon,T]\subset[0,T]$ for any $\varepsilon>0$ and $\theta\in[0,1]$,
but might only continuous when approaching the point $t=0$. We therefore need
to extend the concept of Hölder spaces in order to take into account this type
of loss of regularity near the origin.

######  Definition 6.

Let $E$ be a Banach space. We define the space of continuous paths on $(0,T]$
which are Hölder of order $\varsigma\in(0,1)$ on $(0,T]$ and only continuous
at zero in the following way

$\mathscr{C}^{\varsigma}_{T}E:=\\{y:[0,T]\rightarrow
E\,|\,[y]_{\varsigma}<\infty\\}$

where we define the semi-norm

$[y]_{\varsigma}:=\sup_{s\leq
t\in[0,T];\,\zeta\in[0,\varsigma]}\frac{|y_{s,t}|_{E}}{|t-s|^{\zeta}s^{-\zeta}}.$

The space $\mathscr{C}^{\varsigma}_{T}E$ is a Banach space when equipped with
the norm $\|y\|_{\varsigma}:=|y_{0}|_{E}+[y]_{\varsigma}$.

Singular Hölder type spaces introduced above has recently been extensively
studied in [4]. The space introduced above can be seen as a special case of
the more general spaces considered there. We use the convention of taking
supremum over $\theta\in[0,\varsigma]$, as this will make computations in
subsequent sections simpler. It is well known that if a function
$y\in\mathcal{C}^{\gamma}_{T}E$ for some $\gamma\in[0,1)$, then
$y\in\mathcal{C}^{\theta}_{T}E$ for all $\theta\in[0,\gamma]$ (note that
$\theta=0$ implies that $y$ is bounded, which is always true for Hölder
continuous functions on bounded domains). Thus it follows that also

$\sup_{s\leq
t\in[0,T];\,\theta\in[0,\gamma]}\frac{|y_{s,t}|_{E}}{|t-s|^{\theta}}<\infty.$

######  Remark 7.

It is readily seen that $\mathscr{C}^{\varsigma}_{T}E$ consists of all
functions which are Hölder continuous on $(0,T]$, but is only continuous in
the point $\\{0\\}$. Note therefore that the following inclusion
$\mathcal{C}^{\varsigma}_{T}E\subset\mathscr{C}^{\varsigma}_{T}E$ holds.
Indeed, for any $s,t\in[0,T]$

$\frac{|y_{s,t}|_{E}}{|t-s|^{\varsigma}}=\frac{s^{-\varsigma}|y_{s,t}|_{E}}{s^{-\varsigma}|t-s|^{\varsigma}}$

and thus in particular,

$\frac{|y_{s,t}|_{E}}{s^{-\varsigma}|t-s|^{\varsigma}}\leq
T^{\varsigma}\frac{|y_{s,t}|_{E}}{|t-s|^{\varsigma}}.$

######  Remark 8.

It may be instructive for the reader to keep in mind that in subsequent
sections, we will take the Banach space $E$ to be the Besov-Hölder space
$\mathcal{C}^{\beta}(\mathbb{R}^{d})$ or $\mathcal{C}^{\beta}(\mathbb{T}^{d})$
for some $\beta\in\mathbb{R}$ and $d\geq 1$.

We will throughout this section work with general Volterra type operators
satisfying certain regularity assumptions. We therefore give the following
working hypothesis.

######  Hypothesis 9.

For each $t\in[0,T]$, let $S_{t}\in\mathcal{L}(E)$ be a linear operator on $E$
satisfying the following three regularity conditions for some $\rho>0$ and any
$\theta,\theta^{\prime}\in[0,1]$ and any $0\leq s\leq
t\leq\tau^{\prime}\leq\tau\leq T$

$\displaystyle{\rm(i)}$ $\displaystyle|S_{t}u|_{E}$ $\displaystyle\lesssim
t^{-\rho}|u|_{E}$ $\displaystyle{\rm(ii)}$ $\displaystyle|(S_{t}-S_{s})u|_{E}$
$\displaystyle\lesssim(t-s)^{\theta}s^{-\theta-\rho}|u|_{E}$
$\displaystyle{\rm(iii)}$
$\displaystyle\Big{|}\big{(}(S_{\tau-t}-S_{\tau-s})-(S_{\tau^{\prime}-t}-S_{\tau^{\prime}-s})\big{)}u\Big{|}_{E}$
$\displaystyle\lesssim(\tau-\tau^{\prime})^{\theta^{\prime}}(t-s)^{\theta}(\tau^{\prime}-t)^{-\theta-\theta^{\prime}-\rho}|u|_{E}$

We then say that $t\mapsto S_{t}$ is a $\rho$–singular operator.

###  2.1. Non-linear Young-Volterra integration

We are now ready to construct a non-linear Young-Volterra integral in Banach
spaces. If $X:[0,T]\times E\to E$ is a smooth function in time, and
$(P_{t})_{t\in[0,T]}$ is a (nice) linear operator of $E$ and if
$(y_{t})_{t\in[0,T]}$ is a continuous path from $[0,T]$ to $E$ itself, it is
quite standard to consider integrals of the following form :

$\int_{0}^{t}P_{t-r}\dot{X}_{r}(y_{r})\mathop{}\\!\mathrm{d}r.$

The aim this part is to extend the notion of integral for non-smooth drivers
$X$, and to solve integral equations using this extension of the integral; The
notion of $\rho$–singular operator will be useful in the following.

######  Lemma 10.

Consider parameters $\gamma>\frac{1}{2}$, $0\leq\rho\leq\gamma$ and
$0<\varsigma<\gamma-\rho$ and assume $\varsigma+\gamma>1$. Let $E$ be a Banach
space and let $y\in\mathscr{C}^{\varsigma}_{T}E$. Suppose that $X:[0,T]\times
E\rightarrow E$ satisfies for any $x,y\in E$ and $s\leq t\in[0,T]$

$\displaystyle{\rm(i)}$ $\displaystyle|X_{s,t}(x)|_{E}\lesssim
H(|x|_{E})|t-s|^{\gamma}$ $\displaystyle{\rm(ii)}$
$\displaystyle|X_{s,t}(x)-X_{s,t}(y)|_{E}\lesssim
H(|x|_{E}\vee|y|_{E})|x-y|_{E}|t-s|^{\gamma}$

where $H$ is a positive locally bounded function on $\mathbb{R}_{+}$. Let
$(S_{t})_{t\in[0,T]}\in\mathcal{L}(E)$ be a $\rho$–singular linear operator on
$E$, satisfying Hypothesis 9. We then define the non-linear Young-Volterra
integral by

$\Theta(y)_{t}:=\lim_{\begin{subarray}{c}\mathcal{P}\in\mathcal{P}([0,t])\\\
|\mathcal{P}|\rightarrow
0\end{subarray}}\sum_{[u,v]\in\mathcal{P}}S_{t-u}X_{u,v}(y_{u}).$ (2.2)

The integration map $\Theta$ is a continuous non-linear operator from
$\mathscr{C}^{\varsigma}_{T}E\rightarrow\mathscr{C}^{\varsigma}_{T}E$, and
there exists an $\varepsilon>0$ such that the following inequality holds

$|\Theta(y)_{t}-\Theta(y)_{s}|_{E}\lesssim\sup_{0\leq
z\leq\|y\|_{\infty}}H(z)(1+[y]_{\varsigma})(t-s)^{\varsigma}T^{\gamma-\rho+\varsigma},$
(2.3)

Furthermore, for a linear operator $A\in\mathcal{L}(E)$, the following
commutative property holds

$A\Theta(y)_{t}=\lim_{|\mathcal{P}|\rightarrow}\sum_{[u,v]\in\mathcal{P}}A\,S_{t-u}X_{u,v}(y_{u}).$

###### Proof.

Let us first assume that for any $0\leq s\leq t\leq\tau^{\prime}\leq\tau\leq
T$ the following operator is well-defined :

$\Theta_{s}^{t}(y)_{\tau}:=\lim_{\begin{subarray}{c}\mathcal{P}\in\mathcal{P}([s,t])\\\
|\mathcal{P}|\to
0\end{subarray}}\sum_{[u,v]\in\mathcal{P}}S_{\tau-u}X_{u,v}(y_{u}).$

Note that in this setting we have

$\Theta(y)_{t}=\Theta_{0}^{t}(y)_{t}$

and the increment satisfies

$\Theta(y)_{s,t}=\Theta_{s}^{t}(y)_{t}+\Theta_{0}^{s}(y)_{s,t}.$

Hence, in order to have the bound (2.3), it is enough to have a bound on
$\Theta_{s}^{t}(y)_{\tau}$ and on $\Theta_{s}^{t}(y)_{\tau^{\prime},\tau}$.

To this end, we will begin to show the existence of the integrals
$\Theta_{s}^{t}(y)_{\tau}$ and $\Theta_{s}^{t}(y)_{\tau^{\prime},\tau}$
together with the suitable bounds. Both these terms are constructed in the
same way, however since

$\Theta_{s}^{t}(y)_{\tau^{\prime},\tau}=\lim_{\begin{subarray}{c}\mathcal{P}\in\mathcal{P}([s,t])\\\
|\mathcal{P}|\to
0\end{subarray}}\sum_{[u,v]\in\mathcal{P}}\left(S_{\tau-u}-S_{\tau^{\prime}-u}\right)X_{u,v}(y_{u}),$
(2.4)

this term is a bit more involved as it has an increment of the kernel $P$ in
the summand. We will therefore show existence as well as a suitable bound for
this term and leave the specifics of the first term as a simple exercise for
the reader. Everything will be proven in a similar manner as the sewing lemma
from the theory of rough paths. More specifically, the recently developed
Volterra sewing lemma from [24] provides the correct techniques to this
specific setting. The uniqueness, and additivity (i.e. that
$\Theta(y)_{s,t}=\Theta_{s}^{t}(y)_{t}+\Theta_{0}^{s}(y)_{s,t}$) of the
mapping follows directly from the standard arguments given for example in [24]
or [12, Lem. 4.2].

Consider now a dyadic partition $\mathcal{P}^{n}$ of $[s,t]$ defined
iteratively such that $\mathcal{P}^{0}=\\{[s,t]\\}$, and for $n\geq 0$

$\mathcal{P}^{n+1}:=\bigcup_{[u,v]\in\mathcal{P}^{n}}\\{[u,m],[m,v]\\},$

where $m:=\frac{u+v}{2}$. It follows that $\mathcal{P}^{n}$ consists of
$2^{n}$ sub-intervals $[u,v]$, each of length $2^{-n}|t-s|$. Define the
approximating sum

$\mathcal{I}_{n}:=\sum_{[u,v]\in\mathcal{P}^{n}}(S_{\tau-u}-S_{\tau^{\prime}-u})X_{u,v}(y_{u}),$
(2.5)

and observe that for $n\in\mathbb{N}$, we have

$\mathcal{I}_{n+1}-\mathcal{I}_{n}=-\sum_{[u,v]\in\mathcal{P}^{n}}\delta_{m}\left[(S_{\tau-u}-S_{\tau^{\prime}-u})X_{u,v}(y_{u})\right],$
(2.6)

where $m=\frac{u+v}{2}$ and for a two variable function $f$, we use that
$\delta_{m}f_{u,v}:=f_{u,v}-f_{u,m}-f_{m,v}$. By elementary algebraic
manipulations we see that

$\delta_{m}\left[(S_{\tau-u}-S_{t-u})X_{u,v}(y_{u})\right]\\\
=(S_{\tau-u}-S_{\tau^{\prime}-u})(X_{m,v}(y_{u})-X_{m,v}(y_{m}))+(S_{\tau-u}-S_{\tau^{\prime}-u}-S_{\tau-m}+S_{\tau^{\prime}-m})X_{m,v}(y_{u}).$
(2.7)

We first investigate the second term on the right hand side above. Invoking
(iii) of Hypothesis 9 and invoking assumption (i) on $X$, we observe that for
any $\theta,\theta^{\prime}\in[0,1]$

$|(S_{\tau-u}-S_{t-u}-S_{\tau-m}+S_{t-m})X_{m,v}(y_{u})|_{E}\\\ \lesssim
H(|y_{u}|_{E})|\tau-\tau^{\prime}|^{\theta^{\prime}}|\tau^{\prime}-m|^{-\theta^{\prime}-\rho-\theta}|m-u|^{\theta}|v-m|^{\gamma}.$

Let us now fixe $\theta^{\prime}=\varsigma\in[0,\gamma-\rho)$, and choose
$\theta\in[0,1]$ such that $\gamma+\theta>1$ and $\theta+\varsigma+\rho<1$.
Note that this is always possible due to the fact that
$\gamma-\rho-\varsigma>0$. Furthermore, we note that for any partition
$\mathcal{P}$ of $[s,t]$ we have

$\sum_{[u,v]\in\mathcal{P}}|\tau^{\prime}-m|^{-\theta-\rho-\varsigma}|v-m|\lesssim\int_{s}^{t}|\tau^{\prime}-r|^{-\theta-\rho-\varsigma}\mathop{}\\!\mathrm{d}r\lesssim|t-s|^{1-\varsigma-\rho-\theta},$
(2.8)

where we have used that $m=(u+v)/2$. From this, it follows that for any
$\theta\in[0,\varsigma]$ the following inequality holds

$\sum_{[u,v]\in\mathcal{P}^{n}}|(S_{\tau-u}-S_{t-u}-S_{\tau-m}+S_{t-m})X_{m,v}(y_{u})|_{E}\\\
\lesssim\sup_{0\leq
z\leq\|y\|_{\infty}}H(z)|\mathcal{P}^{n}|^{\gamma+\theta-1}|\tau-\tau^{\prime}|^{\varsigma}|t-s|^{1-\theta-\rho-\varsigma}.$
(2.9)

Let us now move on to the first term in (2.7). By invoking the bounds on $P$
from (ii) of Hypothesis 9 and assumption (ii) on $X$, we observe that for any
$\theta\geq 0$ and any $0\leq\zeta\leq\varsigma$,

$|(S_{\tau-u}-S_{\tau^{\prime}-u})(X_{m,v}(y_{u})-X_{m,v}(y_{m}))|_{E}\\\
\lesssim|\tau-\tau^{\prime}|^{\theta}|\tau^{\prime}-u|^{-\rho-\theta}|m-v|^{\gamma}|m-u|^{\zeta}u^{-\zeta}H(|y_{u}|_{E}\vee|y_{m}|_{E})[y]_{\varsigma}.$

Similarly as shown in (2.9), we now take $\theta=\zeta=\varsigma$ and we
consider a sum over a partition $\mathcal{P}$ over $[s,t]\subset[0,T]$, and
see that since $\varsigma+\gamma>1$, and for $\rho+\varsigma<1$,

$\sum_{[u,v]\in\mathcal{P}}|\tau-\tau^{\prime}|^{\varsigma}|\tau^{\prime}-m|^{-\rho-\varsigma}|m-v|^{\gamma}|m-u|^{\varsigma}u^{-\varsigma}\leq|\mathcal{P}|^{\varsigma+\gamma-1}|\tau-\tau^{\prime}|^{\varsigma}\int_{s}^{t}|\tau^{\prime}-r|^{-\rho-\varsigma}r^{-\varsigma}\mathop{}\\!\mathrm{d}r$

where again $m=(u+v)/2$. Furthermore, when $s<t<\tau^{\prime}$, we have

$\displaystyle\int_{s}^{t}|\tau^{\prime}-r|^{-\rho-\varsigma}r^{-\varsigma}\mathop{}\\!\mathrm{d}r\leq$
$\displaystyle\int_{s}^{t}|\tau^{\prime}-r|^{-\rho-\varsigma}r^{-\varsigma}\mathop{}\\!\mathrm{d}r$
$\displaystyle\leq$
$\displaystyle(t-s)^{1-\rho-2\varsigma}\int_{0}^{1}(1-r)^{-(\rho+\varsigma)}r^{-\varsigma}\mathop{}\\!\mathrm{d}r$
$\displaystyle\lesssim$ $\displaystyle(t-s)^{1-\rho-2\varsigma}.$

We therefore obtain when specifying $\theta=\varsigma$,

$\sum_{[u,v]\in\mathcal{P}^{n}}|(S_{\tau-u}-S_{\tau^{\prime}-u})(X_{m,v}(y_{u})-X_{m,v}(y_{m}))|_{E}\\\
\lesssim|\mathcal{P}^{n}|^{\varsigma+\gamma-1}|\tau-\tau^{\prime}|^{\varsigma}|t-s|^{1-\rho-2\varsigma}\sup_{0\leq
z\|y\|_{\infty}}H(z)[y]_{\varsigma}.$ (2.10)

Combining (2.9) and (2.10), and using that for
$|\mathcal{P}^{n}|=2^{-n}|t-s|$, it follows from (2.6) that

$|\mathcal{I}_{n+1}(s,t)-\mathcal{I}_{n}(s,t)|_{E}\lesssim\sup_{0\leq
z\leq\|y\|_{\infty,[s,t]}}H(z)(1+[y]_{\varsigma})2^{-n(\gamma-(\rho+\varsigma))}(\tau-\tau^{\prime})^{\varsigma}(t-s)^{\gamma-\rho-\varsigma},$

For $m>n\in\mathbb{N}$ thanks to the triangle inequality, and the estimate
above, we get

$\|\mathcal{I}_{m}(s,t)-\mathcal{I}_{n}(s,t)\|_{E}\lesssim\sup_{0\leq
z\|y\|_{\infty,[s,t]}}H(z)(1+[y]_{\varsigma})(\tau-\tau^{\prime})^{\varsigma}(t-s)^{\gamma-\rho-\varsigma}\psi_{n,m},$
(2.11)

where $\psi_{n,m}=\sum_{i=n}^{m}2^{-i(\gamma-(\rho+\varsigma))}$, and it
follows that $\\{\mathcal{I}_{n}\\}_{n\in\mathbb{N}}$ is Cauchy in $E$. It
follows that there exists a limit
$\mathcal{I}=\lim_{n\rightarrow\infty}\mathcal{I}_{n}$ $E$. Moreover, from
(2.11) we find that the following inequality holds

$|\mathcal{I}-(S_{\tau-s}-S_{\tau^{\prime}-s})X_{s,t}(y_{s})|_{E}\lesssim\sup_{0\leq
z\leq\|y\|_{\infty}}H(z)(1+[y]_{\varsigma})(\tau-\tau^{\prime})^{\varsigma}(t-s)^{\gamma-\rho-\varsigma}\psi_{0,\infty}.$
(2.12)

the proof that $\mathcal{I}$ is equal to
$\Theta_{s}^{t}(y)_{\tau,\tau^{\prime}}$ defined in (2.4) (where in particular
$\Theta(y)$ is defined independent of the partition $\mathcal{P}$ of $[s,t]$)
follows by standard arguments for the sewing lemma, see [24] in the Volterra
case. Finally, we get for any $0\leq s\leq t\leq\tau^{\prime}\leq\tau\leq T$,

$|\Theta_{s}^{t}(y)_{\tau^{\prime},\tau}-(S_{\tau-s}-S_{\tau^{\prime}-s})X_{s,t}(y_{s})|_{E}\lesssim\sup_{0\leq
z\leq\|y\|_{\infty}}H(z)T^{\gamma-\rho+\varsigma}|t-s|^{\varsigma}.$ (2.13)

Using the same techniques, in order to prove that $\Theta_{s}^{t}(y)_{t}$
exists for all $0\leq s\leq t\leq\tau\leq T$ one has to control

$S_{\tau-m}(X_{m,v}(y_{m})-X_{m,v}(y_{u}))+(S_{\tau-u}-S_{\tau-m})X_{m,v}(y_{u}).$

Performing exactly the same computation, we have

$|\Theta_{s}^{t}(y)_{\tau}-S_{\tau-s}X_{s,t}(y_{s})|_{E}\lesssim\sup_{0\leq
z\leq\|y\|_{\infty}}H(z)T^{\gamma-\varsigma+\rho}|t-s|^{\varsigma}(1+[h]_{\varsigma}).$
(2.14)

Combining (2.13) and (2.14), and using the fact that

$\Theta(y)_{t}-\Theta(y)_{s}=\Theta_{s,t}(y)_{t}+\Theta_{0}^{s}(y)_{s,t}$
$|\Theta(y)_{t,s}-S_{t-s}X_{s,t}(y_{s})-(S_{t}-S_{s})X_{0,s}(y_{0})|_{E}\\\
\leq|\Theta_{s}^{t}(y)_{t}-(S_{t}-S_{s})X_{s,t}(y_{s})|_{E}+|\Theta_{0}^{s}(y)_{s,t}-S_{t-s}\xi
X_{0,s}(y_{0})|_{E}$

it is readily checked that the following inequality holds

$|\Theta(y)_{t}-\Theta(y)_{s}-S_{t-s}X_{s,t}(y_{s})-(S_{t}-S_{s})X_{0,s}(y_{0})|_{E}\lesssim\sup_{0\leq\leq\|y\|_{\infty}}H(z)(1+[y]_{\varsigma})T^{\gamma-\rho+\varsigma}(t-s)^{\varsigma}.$

Finally note that since $\varsigma<\gamma-\rho$,

$|S_{t-s}X_{s,t}(y_{s})|_{E}\lesssim H(|y_{s}|)|t-s|^{\gamma-\rho}\lesssim
H(|y_{s}|)T^{\gamma-\rho+\varsigma}|t-s|^{\varsigma},$

and

$|(S_{t}-S_{s})X_{0,s}(y_{0})|_{E}\lesssim
H(|y_{0}|)s^{\gamma-\rho-\varsigma}|t-s|^{\varsigma}.$

For the last claim, if $A\in\mathcal{L}(E)$ is a linear operator, then one re-
define $\mathcal{I}_{n}$ in (2.5) to be given as

$\mathcal{I}_{n}(A):=\sum_{[u,v]\in\mathcal{P}^{n}}A(S_{\tau-u}-S_{t-u})X_{u,v}(y_{u}),$

and by linearity of $A$ we see that $\mathcal{I}_{n}(A)=A\mathcal{I}_{n}$.
Taking the limits, using the above established inequalities, we find that
$\lim_{n\rightarrow\infty}\|\mathcal{I}_{n}(A)-A\mathcal{I}_{n}\|_{E}=0$. ∎

With the construction of the non-linear Young Volterra integral, we will in
later applications need certain stability estimates.

######  Proposition 11 (Stability of $\Theta$).

Let $\gamma,\varsigma,\rho$ be given as in of Lemma 10. Assume that for
$i=1,2$, $X^{i}:[0,T]\times E\rightarrow E$ satisfies for any $x,y\in E$ and
$s\leq t\in[0,T]$

$\displaystyle{\rm(i)}\qquad$ $\displaystyle|X^{i}_{s,t}(x)|_{E}+\|\nabla
X^{i}_{s,t}(x)|_{\mathcal{L}(E)}$ $\displaystyle\lesssim
H(|x|_{E})|t-s|^{\gamma}$ (2.15) $\displaystyle{\rm(ii)}\qquad$
$\displaystyle|X^{i}_{s,t}(x)-X^{i}_{s,t}(y)|_{E}$ $\displaystyle\lesssim
H(|x|_{E}\vee|y|_{E})|x-y|_{E}|t-s|^{\gamma}$ $\displaystyle{\rm(iii)}\qquad$
$\displaystyle|\nabla X^{i}_{s,t}(x)-\nabla X^{i}_{s,t}(y)|_{\mathcal{L}(E)}$
$\displaystyle\lesssim H(|x|_{E}\vee|y|_{E})|x-y|_{E}|t-s|^{\gamma},$

where $H$ is a positive locally bounded function, and $\nabla$ is understood
as a linear operator on $E$ in the Fréchet sense. Furthermore, suppose there
exists a positive and locally bounded function $H_{X^{1}-X^{2}}$ such that

$\displaystyle{\rm(i)}\qquad$
$\displaystyle|X^{1}_{s,t}(x)-X^{2}_{s,t}(x)|_{E}$ $\displaystyle\lesssim
H_{X^{1}-X^{2}}(|x|_{E})|t-s|^{\gamma}$ (2.16) $\displaystyle{\rm(ii)}\qquad$
$\displaystyle|(X^{1}_{s,t}-X^{2}_{s,t})(x)-(X^{1}_{s,t}-X^{2}_{s,t})(y)|_{E}$
$\displaystyle\lesssim
H_{X^{1}-X^{2}}(|x|_{E}\vee|y|_{E})|x-y|_{E}|t-s|^{\gamma}$

Let $\Theta^{1}$ denote the non-linear integral operator constructed in Lemma
10 with respect to $X^{1}$, and similarly let $\Theta^{2}$ denote the integral
operator with respect to $X^{2}$. Then for two paths
$y,\tilde{y}\in\mathscr{C}^{\varsigma}_{T}\mathcal{C}^{\beta+2\rho}$,

$[\Theta^{1}(y^{1})-\Theta^{2}(y^{2})]_{\varsigma}\lesssim_{P}\bigg{[}\sup_{0\leq
z\leq\|y^{1}\|\vee\|y^{2}\|}H(z)\left([y^{1}]_{\varsigma}+[y^{2}]_{\varsigma}\right)(|y^{1}_{0}-y^{2}_{0}|_{E}+[y^{1}-y^{2}]_{\varsigma})\\\
+\sup_{0\leq
z\leq\|y^{1}\|_{\infty}\vee\|y^{2}\|_{\infty}}H_{X^{1}-X^{2}}(z)[y^{1}]_{\varsigma}\vee[y^{2}]_{\varsigma}\bigg{]}T^{\gamma-\rho+\varsigma}.$
(2.17)

###### Proof.

Define the following two functions

$\Theta_{s,t}^{1}(y^{1})-\Theta_{s,t}^{2}(y^{2})=(\Theta_{s,t}^{1}(y^{1})-\Theta_{s,t}^{2}(y^{1}))+(\Theta_{s,t}^{2}(y^{1})-\Theta_{s,t}^{2}(y^{2}))=:D_{X^{1},X^{2}}(s,t)+D_{y^{1},y^{2}}(s,t).$

We treat $D_{X^{1},X^{2}}$ and $D_{y^{1},y^{2}}$ separately, and begin to
consider $D_{y^{1},y^{2}}$. Since $X^{i}$ is differentiable and satisfies
(i)-(iii) in (2.15), for $i=1,2$ we have

$X^{i}_{s,t}(y^{1}_{s})-X^{i}_{s,t}(y^{2}_{s})=\mathcal{X}^{i}_{s,t}(y^{1}_{s},y^{2}_{s})(y^{1}_{s}-y^{2}_{s}),$

where $\mathcal{X}^{i}(y^{1}_{s},y^{2}_{s}):=\int_{0}^{1}\nabla
X^{i}_{s,t}(qy^{1}_{s}+(1-q)y^{2}_{s})\mathop{}\\!\mathrm{d}q$. In order to
prove (2.17), we proceed with the exact same strategy as outlined in the proof
of Lemma 10. That is, we use the same proof as the proof of Lemma 10 to first
prove appropriate bounds for $D_{y^{1},y^{2}}(s,t)$ and similarly for
$D_{X^{1},X^{2}}(s,t)$ afterwards. To this end, changing the integrand in
(2.5) so that

$\mathcal{I}_{n}(s,t):=\sum_{[u,v]\in\mathcal{P}^{n}[s,t]}(S_{\tau-u}-S_{t-u})\mathcal{X}_{u,v}^{i}(y^{1}_{u},y^{2}_{u})(y^{1}_{u}-y^{2}_{u}),$

we continue along the lines of the proof in Lemma 10 to show that
$\mathcal{I}^{n}$ is Cauchy. As the strategy of this proof is identical to
that of Lemma 10 we will here only point out the important differences. By
appealing to the condition (iii) in (2.15),we observe in particular that

$|\mathcal{X}^{i}(y^{1}_{s},y^{2}_{s})-\mathcal{X}^{i}(y^{1}_{u},y^{2}_{u})|_{\mathcal{L}(E)}\lesssim|t-s|^{\gamma}\sup_{0\leq
z\leq\|y^{1}\|_{\infty}\vee\|y^{2}\|_{\infty}}H(z)\left([y^{1}]_{\varsigma}+[y^{2}]_{\varsigma}\right).$

Furthermore, it is readily checked that

$|y^{1}_{s}-y^{2}_{s}|_{E}\lesssim|y^{1}_{0}-y^{2}_{0}|_{E}+[y-\tilde{y}]_{\varsigma}T^{\varsigma}.$

Following along the lines of the proof of Lemma 10, one can then check that
for $m>n\in\mathbb{N}$

$|\mathcal{I}_{m}(s,t)-\mathcal{I}_{n}(s,t)|_{E}\\\ \lesssim\sup_{0\leq
z\leq\|y^{1}\|_{\infty}\vee\|y^{2}\|_{\infty}}H(z)\left([y^{1}]_{\varsigma}+[y^{2}]_{\varsigma}\right)(|y^{1}_{0}-y^{2}_{0}|_{E}+[y^{1}-y^{2}]_{\varsigma}T^{\varsigma})(\tau-\tau^{\prime})^{\varsigma}(t-s)^{\gamma-\rho-\varsigma}\psi_{n,m},$
(2.18)

where $\psi_{n,m}$ is defined as below (2.11). With this inequality at hand,
the remainder of the proof can be verified in a similar way as in the proof of
Lemma 10, and we obtain from this lemma that

$\|D_{y^{1},y^{2}}\|_{\mathscr{C}^{\varsigma}_{T}E}\lesssim C\sup_{0\leq
z\leq\|y^{1}\|_{\infty}\vee\|y^{2}\|_{\infty}}(z)\left([y^{1}]_{\varsigma}+[y^{2}]_{\varsigma}\right)(\|y^{1}_{0}-y^{2}_{0}\|_{E}+[y^{1}-y^{2}]_{\varsigma}T^{\varsigma}).$

Next we move on to prove a similar bound of $D_{X^{1},X^{2}}$ as defined in
(2.1). Set $Z=X^{1}-X^{2}$. By (2.16) it follows that $Z$ satisfies the
conditions of Lemma 10, and then from (2.3) it follows that

$[D_{X_{1},X^{2}}]_{\varsigma}\lesssim\sup_{0\leq
z\leq\|y^{1}\|_{\infty}\vee\|y^{2}\|_{\infty}}H_{X^{1}-X^{2}}(z)[y^{1}]_{\varsigma}\vee[y^{2}]_{\varsigma}T^{\gamma-\varsigma-\rho}.$

∎

###  2.2. Existence and uniqueness

We begin to prove local existence and uniqueness for an abstract type of
equation with values in a Banach space. The equation in itself does not
require the use of the non-linear Young-Volterra integral, and is formulated
for general operators
$\Theta:[0,T]\times\mathscr{C}^{\varsigma}_{T}E\rightarrow\mathscr{C}^{\varsigma}_{T}E$
satisfying certain regularity conditions. We will apply these results in later
sections in combination with the non-linear Young-Volterra integral operator
$\Theta$ created in the previous section, and thus the reader is welcome to
already think of $\Theta$ as being a non-linear Young integral operator as
constructed in Lemma 10.

######  Theorem 12 (Local existence and uniqueness).

Let
$\Theta:[0,T]\times\mathscr{C}^{\varsigma}_{T}E\rightarrow\mathscr{C}^{\varsigma}_{T}E$
be a function which satisfies for $y,\tilde{y}\in\mathscr{C}^{\varsigma}_{T}E$
and some $\epsilon>0$

$\displaystyle\left[\Theta(y)\right]_{\varsigma}$ $\displaystyle\leq
C(\|y\|_{\infty})(1+[y]_{\varsigma})T^{\varepsilon}$ (2.19)
$\displaystyle[\Theta(y)-\Theta(\tilde{y})]_{\varsigma}$ $\displaystyle\leq
C(\|y\|_{\infty}\vee\|\tilde{y}\|_{\infty})([y]_{\varsigma}+[\tilde{y}]_{\varsigma})(|y_{0}-\tilde{y}_{0}|_{E}+[y-\tilde{y}]_{\varsigma})T^{\varepsilon},$

where $C$ is a positive and increasing locally bounded function. Consider
$p\in\mathscr{C}^{\varsigma}_{T}E$ and let $\tau>0$ be such that

$\tau\leq\left[4(1+[p]_{\varsigma})C(1+|p_{0}|+[p]_{\varsigma})\right]^{-\frac{1}{\varepsilon}}.$
(2.20)

Then there exists a unique solution to the equation

$y_{t}=p_{t}+\Theta(y)_{t},\qquad p\in\mathscr{C}^{\varsigma}_{T}E$ (2.21)

in $\mathcal{B}_{\tau}(p)$, where $\mathcal{B}_{\tau}(p)$ is a unit ball in
$\mathscr{C}^{\varsigma}_{\tau}E$, centered at $p$.

###### Proof.

To prove existence and uniqueness, we will apply a standard fixed point
argument. Define the solution map
$\Gamma_{\tau}:\mathscr{C}^{\varsigma}_{\tau}E\rightarrow\mathscr{C}^{\varsigma}_{\tau}E$
given by

$\Gamma_{\tau}(y):=\\{p_{t}+\Theta(y)_{t}|\,t\in[0,\tau]\\}.$

Since
$\Theta:\mathscr{C}^{\varsigma}_{\tau}E\rightarrow\mathscr{C}^{\varsigma}_{\tau}E$
and $p\in\mathscr{C}^{\varsigma}_{\tau}E$ it follows that
$\Gamma_{\tau}(y)\in\mathscr{C}^{\varsigma}_{\tau}E$. We will now prove that
the solution map $\Gamma_{\tau}$ is an invariant map and a contraction on a
unit ball $\mathcal{B}_{\tau}(p)\subset\mathscr{C}^{\varsigma}_{\tau}E$
centered at $p\in\mathscr{C}^{\varsigma}_{\tau}E$. In particular, we define

$\mathcal{B}_{\tau}(p):=\\{y\in\mathscr{C}^{\varsigma}_{\tau}|\,y_{t}=p_{t}+z_{t},\,{\rm
with}\,\,z\in\mathscr{C}^{\varsigma}_{\tau}E,\,\,z_{0}=0,\,\,[y-p]_{\varsigma}\leq
1\\}.$

We begin with the invariance. From the first condition in (2.19), it is
readily checked that for $y\in\mathcal{B}_{\tau}(P)$

$[\Gamma_{\tau}(y)-p]_{\varsigma}\leq
C(1+|p_{0}|+[p]_{\varsigma})(1+[y]_{\varsigma})\tau^{\varepsilon},$ (2.22)

where we have used that for $y\in\mathcal{B}_{\tau}(P)$ , $\|y\|_{\infty}\leq
1+|p_{0}|+[p]_{\varsigma}$ Choosing a parameter ${\tau_{1}}>0$ such that

$\tau_{1}\leq(2C(1+|p_{0}|+[p]_{\varsigma})^{-\frac{1}{\varepsilon}}$

it follows that
$\Gamma_{\tau_{1}}(\mathcal{B}_{\tau_{1}}(p))\subset\mathcal{B}_{\tau_{1}}(p)$and
we say that $\Gamma_{\tau_{1}}$ leaves the ball $\mathcal{B}_{\tau_{1}}(p)$
invariant.

Next, we prove that $\Gamma_{\tau}$ is a contraction on
$\mathcal{B}_{\tau}(p)$. From the second condition in (2.19), it follows that
for two elements $y,\tilde{y}\in\mathcal{B}_{\tau}(p)$ we have

$[\Gamma(y)-\Gamma(\tilde{y})]_{\varsigma}\leq
2(1+[p]_{\varsigma})C(1+|p_{0}|+[p]_{\varsigma})[y-\tilde{y}]_{\varsigma}\tau^{\varepsilon},$

where we have used that $y_{0}=\tilde{y}_{0}$, and

$[y]_{\varsigma}\vee[\tilde{y}]_{\varsigma}\leq 1+[p]_{\varsigma}\quad{\rm
and}\quad\|y\|_{\infty}\vee\|\tilde{y}\|_{\infty}\leq
1+|p_{0}|+[p]_{\varsigma}.$

Again, choosing a parameter $\tau_{2}>0$ such that

$\tau_{2}\leq\left(4(1+[p]_{\varsigma})C(1+|p_{0}|+[p]_{\varsigma})\right)^{-\frac{1}{\varepsilon}}.$

it follows that

$[\Gamma(y)-\Gamma(\tilde{y})]_{\varsigma;\tau_{2}}\leq\frac{1}{2}[y-\tilde{y}]_{\varsigma;\tau_{2}}.$

Since $\tau_{2}\leq\tau_{1}$, we conclude that the solution map
$\Gamma_{\tau_{2}}$ both is an invariant map and a contraction on the unit
ball $\mathcal{B}_{\tau_{2}}(p)$. It follows by Picard-Lindlöfs fixed point
theorem that a unique solution to (2.21) exists in
$\mathcal{B}_{\tau_{2}}(p)$.

∎

The next theorem shows that if the locally bounded function $C$ appearing in
the conditions on $\Theta$ in (2.19) of Theorem 12, is uniformly bounded, then
there exists a unique global solution to (2.21).

######  Theorem 13 (Global existence and uniqueness).

Let
$\Theta:[0,T]\times\mathscr{C}^{\varsigma}_{T}E\rightarrow\mathscr{C}^{\varsigma}_{T}E$
satisfy (2.19) for a positive, globally bounded function $C$, i.e. there
exists a constant $M>0$ such that $\sup_{x\in\mathbb{R}_{+}}C(x)\leq M$.
Furthermore, suppose $\Theta$ is time-additive, in the sense that
$\Theta_{t}=\Theta_{s}+\Theta_{s,t}$ for any $s\leq t\in[0,T]$. Then for any
$p\in\mathscr{C}^{\varsigma}_{T}E$ there exists a unique solution
$y\in\mathscr{C}^{\varsigma}_{T}E$ to the equation

$y_{t}=p_{t}+\Theta(y)_{t},\quad t\in[0,T].$

###### Proof.

By Theorem 12 we know that there exists a unique solution to (2.21) on an
interval $[0,\tau]$, where $\tau$ satisfies (2.20), and $C$ is replaced by the
bounding constant $M$, i.e.

$\tau\leq\left[4(1+[p]_{\varsigma})M\right]^{-\frac{1}{\varepsilon}}.$ (2.23)

By a slight modification of the proof in Theorem 12 it is readily checked that
the existence and uniqueness of

$y_{t}=p_{t}+\Theta_{a,t}(y),\quad t\in[a,a+\tau],$

holds on any interval $[a,a+\tau]\subset[0,T]$, i.e. the solution is
constructed in $\mathcal{B}_{[a,a+\tau]}(p)$.

Now, we want iterate solutions to (2.21) to the domain $[0,T]$, by "gluing
together" solutions on the integrals $[0,\tau],[\tau,2\tau]...\subset[0,T]$.
Using the time-additivity property of $\Theta$, note that for
$t\in[\tau,2\tau]$, we have

$y_{t}=p_{t}+\Theta_{t}(y)=p_{t}+\Theta_{a}(y)+\Theta_{a,t}(y).$

Thus, set $\tilde{p}_{t}=p_{t}+\Theta_{a}(y|_{[0,\tau]})$, where
$y|_{[0,\tau]}$ denotes the solution to (2.21) restricted to $[0,\tau]$. Note
that

$[\tilde{p}]_{\varsigma}=[p]_{\varsigma},$

since the Hölder seminorm is invariant to constants, and
$t\mapsto\Theta_{a}(y|_{[0,\tau]})$ is constant. Therefore, there exists a
unique solution to $\eqref{eq: abs equation}$ in
$\mathcal{B}_{[\tau,2\tau]}(\tilde{p})$ where $\tau$ is the same as in (2.23).
We can repeat this to all intervals $[k\tau,(k+1)\tau]\subset[0,T]$. At last,
invoking the scalability of Hölder norms (see [12, Exc. 4.24]), it follows
that there exists a unique solution to (2.21) in
$\mathscr{C}^{\varsigma}_{T}E$.

∎

##  3\. Averaged fields

We give here a quick overview of the concept of averaged fields and averaging
operators. We begin with the following definition:

######  Definition 14.

Let $\omega$ be a measurable path from $[0,T]$ to $\mathbb{R}^{d}$, and let
$g\in\mathcal{S}^{\prime}(\mathbb{R}^{d};\mathbb{R})$. We define the average
of $g$ against $\omega$ as the element of
$C^{0}\big{(}[0,T];\mathcal{S}^{\prime}(\mathbb{R}^{d};\mathbb{R})\big{)}$
defined for all $s\leq t\in[0,T]$ and all test functions $\phi$ by

$\big{\langle}\phi,T^{\omega}_{s,t}g\big{\rangle}=\int_{s}^{t}\langle\phi(\cdot-\omega_{r}),g\rangle\mathop{}\\!\mathrm{d}r.$

Introduced in the analysis of regularization by noise in [7], the concept of
averaged fields and averaging operators is by now a well studied topic. For
example, the recent analysis of Galeati and Gubinelli [15] provides a good
overview of the analytic properties in the context of regularization by noise.
See also [23, 16, 17] for further details on probabilistic and analytical
aspects of averaged fields and averaging operators, and their connection to
the concept of occupation measures. In the current article, we investigate
these operators from an infinite-dimensional perspective in order to apply
them in the context of (S)PDEs, and thus some extra considerations needs to be
taken into account. In addition, we include in section 5 a construction of the
averaged field associated to a fractional Lévy process. The regularizing
properties of Volterra-Lévy processes was recently investigated in [22], where
the authors constructed the averaged field using the concept of local times.
The construction given in the current article provides better regularity
properties of the resulting averaged field than those obtained in [22], but
comes at the cost of lost generality. More precisely, constructing the
averaged field associated to a measurable stochastic process
$\omega:\Omega\times[0,T]\rightarrow\mathbb{R}^{d}$ acting on a distribution
$b\in\mathcal{S}^{\prime}(\mathbb{R}^{d})$, the exceptional set
$\Omega^{\prime}\subset\Omega$ on which the function $T^{\omega}g$ is a
sufficiently regular field (say, Hölder in time and differentiable in space..)
depends on $b\in\mathcal{S}^{\prime}(\mathbb{R}^{d})$, i.e.
$\Omega^{\prime}=\Omega^{\prime}(b)$. Thus choosing one construction or the
other depends on the problem at hand, and which properties are important to
retain. We will not investigate these differences in more details in this
article, and will view the analysis of the SPDE from a purely deterministic
point of view. Let us give the following statement which can be viewed as a
short summary of some of the results appearing in [15] and [23]:

######  Proposition 15.

There exists a $\delta$-Hölder continuous path
$\omega:[0,T]\rightarrow\mathbb{R}$ such that for any given
$g\in\mathcal{C}^{\eta}$ with $\eta>3-\frac{1}{2\delta}$, the corresponding
averaged field $T^{\omega}g$ is contained in
$\mathcal{C}^{\gamma}_{T}\mathcal{C}^{\kappa}$ for some $\kappa\geq 3$ and
$\gamma>\frac{1}{2}$. Moreover, there exists a continuous
$\omega:[0,T]\rightarrow\mathbb{R}$ such that for any
$g\in\mathcal{S}^{\prime}(\mathbb{R})$, the averaged field $T^{\omega}g$ is
contained in $\mathcal{C}^{\gamma}_{T}\mathcal{C}^{\kappa}(w)$ for some weight
$w:\mathbb{R}\rightarrow\mathbb{R}_{+}$ and any $\gamma\in(\frac{1}{2},1)$ and
any $\kappa\in\mathbb{R}$.

###### Proof.

The first statement can be seen as a simple version of [15, Thm. 1]. The
second is proven in [23, Prop. 24]. ∎

######  Remark 16.

In fact, the statement of [15, Thm. 1] is much stronger; given a
$g\in\mathcal{C}^{\eta}$ with $\eta\in\mathbb{R}$, then for almost all
$\delta$-Hölder continuous paths $\omega:[0,T]\rightarrow\mathbb{R}$ such that
$\eta>3-\frac{1}{2\delta}$, the averaged field
$T^{\omega}g\in\mathcal{C}^{\gamma}_{T}\mathcal{C}^{3}$ for some
$\gamma>\frac{1}{2}$. Similarly, it is stated that almost all continuous paths
$\omega$ are infinitely regularizing in the sense that for any
$g\in\mathcal{C}^{\eta}$ with $\eta\in\mathbb{R}$,
$T^{\omega}g\in\mathcal{C}^{\gamma}\mathcal{C}^{\kappa}$ for any
$\kappa\in\mathbb{R}$. The "almost surely" statement here is given through the
concept of prevalence. We refrain from writing proposition 15 in the most
general way in order to avoid going into details regarding the concept of
prevalence here. We therefore refer the reader to [15] for more details on
this result and this concept.

######  Remark 17.

The statement in Proposition 15 can also be generalized to measurable paths
$\omega:[0,T]\rightarrow\mathbb{R}$. Indeed, in [22] the authors show that
there exists a class of measurable Volterra-Lévy processes, which provides a
regularizing effect, similar to that of Gaussian processes, and a statement
similar to that of Proposition 15 can be found there.

in all the following we will use notions of (weighted)-Besov space. For a
recap on weighted Lebesgue and Besov spaces, as long as a recap on standard
(fractional)-Schauder estimates see Appendix Sections A.1 and A.2.

We continue with some properties which will be useful in later analysis.

######  Proposition 18.

Let $w$ be an admissible weight, $\kappa\in\mathbb{R}$ and $1\leq
p,q\leq+\infty$. Let $f\in B^{\kappa}_{p,q}(w)$. For all $j\geq-1$ and all
$0\leq s\leq t\leq T$,

$\Delta_{j}T^{\omega}_{s,t}g=T^{\omega}_{s,t}(\Delta_{j}g),$

where $\Delta_{j}$ denotes the standard Paley-Littlewood block (see Appendix
A.2).

In particular for all $\varepsilon,\delta>0$, and for all $f\in
B^{\kappa}_{p,q}(w)$, and for $\mathcal{S}_{k}=\sum_{j={-1}^{k}}\Delta_{j}$,

$T^{\omega}_{s,t}(\mathcal{S}_{k}g)\underset{k\to\infty}{\to}T^{\omega}_{s,t}g\quad\text{in}\quad\mathcal{S}^{\prime}\quad\text{and
in}\quad B^{\kappa-\varepsilon}_{p,q}(\langle\cdot\rangle^{\delta}w).$

Suppose that $g$ in a measurable locally bounded function, then
$T^{\omega}_{s,t}f$ is also a measurable function one has

$T^{\omega}_{s,t}g(x)=\int_{s}^{t}g(x+\omega_{r})\mathop{}\\!\mathrm{d}r.$

###### Proof.

Let us remind that the topology on $\mathcal{S}$ is the one generated by the
family of semi-norm

$\mathcal{N}_{n}(\phi)=\sum_{|k|,|l|\leq
n}\sup_{x\in\mathbb{R}^{d}}|x^{k}\partial^{l}\phi(x)|,$

where $k$ and $l$ are multi-indices. Furthermore, for any distribution
$f\in\mathcal{S}^{\prime}$, there exists $n\geq 0$ and $C>0$ such that

$|\langle\phi,g\rangle|\leq C\mathcal{N}_{n}(\phi).$

Let $\omega$ be a measurable path from $[0,T]$ to $\mathbb{R}^{d}$, then since
$\phi$ is bounded, there is a constant $C_{1}>0$ such that

$\mathcal{N}_{n}\big{(}\phi(\cdot-w_{r})\big{)}\leq
C\mathcal{N}_{n}\big{(}\phi\big{)}.$

In particular

$\Delta_{j}T^{\omega}_{t}g(x)=\left\langle
K_{j}(x-\cdot),T^{\omega}_{s,t}f\right\rangle=\int_{0}^{t}\left\langle
K_{j}\big{(}x+\omega_{r})-\cdot\big{)},g\right\rangle\mathop{}\\!\mathrm{d}r=\int_{0}^{t}\Delta_{j}g(x+\omega_{r})\mathop{}\\!\mathrm{d}r.$

Thanks to the previous remark, we are allowed to perform any Fubini arguments,
and for all $\phi\in\mathcal{S}$,

$\big{\langle}\phi,\Delta_{j}T^{\omega}_{t}g\big{\rangle}=\int_{\mathbb{R}^{d}}\phi(x)\int_{0}^{t}\Delta_{j}g(x+\omega_{r})\mathop{}\\!\mathrm{d}r\mathop{}\\!\mathrm{d}x=\int_{0}^{t}\int_{\mathbb{R}^{d}}\phi(x-\omega_{r})\Delta_{j}g(x)\mathop{}\\!\mathrm{d}x\mathop{}\\!\mathrm{d}r,$

which ends the proof by using properties of weighted-Besov spaces. ∎

For the purpose of this section, we will fix a distribution
$g\in\mathcal{S}^{\prime}(\mathbb{R}^{d})$ and a measurable path
$\omega:[0,T]\rightarrow\mathbb{R}^{d}$ with the property that there exists an
averaged field
$T^{\omega}g:[0,T]\times\mathbb{R}^{d}\rightarrow\mathbb{R}^{d}$ defined as in
Definition 14 which is Hölder continuous in time and three times locally
differentiable in space. More specifically, we will assume that
$T^{\omega}g\in\mathcal{C}^{\gamma}_{T}\mathcal{C}^{\kappa}(w)$ for some
admissible weight function
$w:\mathbb{R}^{d}\rightarrow\mathbb{R}_{+}\setminus\\{0\\}$.

Our first goal is to show how the averaged field $T^{w}g$ can be seen as a
function from $[0,T]\times\mathcal{C}^{\beta}\rightarrow\mathcal{C}^{\beta}$
for some $\beta\geq 0$.

######  Proposition 19.

Let $w$ be an admissible weight (see Section A.1). For a measurable path
$\omega:[0,T]\rightarrow\mathbb{R}^{d}$ and
$g\in\mathcal{S}^{\prime}(\mathbb{R}^{d})$, suppose
$T^{\omega}g\in\mathcal{C}^{\gamma}_{T}\mathcal{C}^{\kappa}(w)$ for some
$\gamma>\frac{1}{2}$ and $\kappa\geq 3$. Then for all
$x,y\in\mathcal{C}^{\beta}$ with $\beta\in[0,1)$, we have that

$\displaystyle\|T^{\omega}_{s,t}g(x)\|_{\mathcal{C}^{\beta}}\vee\|\nabla
T^{\omega}_{s,t}g(x)\|_{\mathcal{C}^{\beta}}$
$\displaystyle\leq\sup_{|z|\leq\|x\|_{\mathcal{C}^{\beta}}}w^{-1}(z)\|T^{\omega}g\|_{\mathcal{C}^{\gamma}_{T}\mathcal{C}^{\kappa}(w)}|t-s|^{\gamma}$
$\displaystyle\|T^{\omega}_{s,t}g(x)-T^{\omega}_{s,t}g(y)\|_{\mathcal{C}^{\beta}}$
$\displaystyle\leq\sup_{|z|\leq\|x\|_{\mathcal{C}^{\beta}}\vee\|y\|_{\mathcal{C}^{\beta}}}w^{-1}(z)\|T^{\omega}g\|_{\mathcal{C}^{\gamma}_{T}\mathcal{C}^{\kappa}(w)}\|x-y\|_{\mathcal{C}^{\beta}}|t-s|^{\gamma}$
$\displaystyle\|\nabla T^{\omega}_{s,t}g(x)-\nabla
T^{\omega}_{s,t}g(y)\|_{\mathcal{C}^{\beta}}$
$\displaystyle\leq\sup_{|z|\leq\|x\|_{\mathcal{C}^{\beta}}\vee\|y\|_{\mathcal{C}^{\beta}}}w^{-1}(z)\|T^{\omega}g\|_{\mathcal{C}^{\gamma}_{T}\mathcal{C}^{\kappa}(w)}\|x-y\|_{\mathcal{C}^{\beta}}|t-s|^{\gamma}.$

###### Proof.

For $x\in\mathcal{C}^{\beta}$ note that for any $\xi\in\mathbb{R}^{d}$

$|T^{\omega}_{s,t}g(x(\xi))|\leq
w^{-1}(x(\xi))|w(x(\xi))T^{\omega}_{s,t}g(x(\xi))|\leq\sup_{|z|\leq\|x\|_{\mathcal{C}^{\beta}}}w^{-1}(z)\|T^{\omega}g\|_{\mathcal{C}^{\gamma}_{T}\mathcal{C}^{\kappa}}.$

By similar computations using that $T^{\omega}b$ is (weighted) differentiable
in space, it is straightforward to verify that

$\|T^{\omega}_{s,t}g(x)\|_{\mathcal{C}^{\beta}}\vee\|\nabla
T^{\omega}_{s,t}g(x)\|_{\mathcal{C}^{\beta}}\leq\sup_{|z|\leq\|x\|_{\mathcal{C}^{\beta}}}w^{-1}(z)\|T^{\omega}g\|_{\mathcal{C}^{\gamma}_{T}\mathcal{C}^{\kappa}(w)}|t-s|^{\gamma}.$

Now consider $x,y\in\mathcal{C}^{\beta}$. Again using the differentiability of
$T^{\omega}b$, and elementary rules of calculus we observe that for any
$\xi\in\mathbb{R}^{d}$

$|T^{\omega}_{s,t}g(x(\xi))-T^{\omega}_{s,t}g(y(\xi))|\leq\sup_{|z|\leq\|x\|_{\mathcal{C}^{\beta}}\vee\|y\|_{\mathcal{C}^{\beta}}}w^{-1}(z)\|T^{\omega}g\|_{\mathcal{C}^{\gamma}_{T}\mathcal{C}^{\kappa}(w)}\|x-y\|_{\mathcal{C}^{\beta}}|t-s|^{\gamma},$

where we have used that $|x(\xi)-y(\xi)|\leq\|x-y\|_{\mathcal{C}^{\beta}}$ for
all $\xi\in\mathbb{R}^{d}$. At last, for $\xi,\xi^{\prime}\in\mathbb{R}^{d}$,
observe that

$\displaystyle
T^{\omega}_{s,t}g(x(\xi))-T^{\omega}_{s,t}g(y(\xi))-T^{\omega}_{s,t}g(x(\xi^{\prime}))+T^{\omega}_{s,t}g(y(\xi^{\prime}))$
$\displaystyle=\int_{0}^{1}\nabla
T^{\omega}_{s,t}g\Big{(}l\big{(}x(\xi)-y(\xi)\big{)}+y(\xi)\Big{)}\big{(}x(\xi)-x(\xi^{\prime})-y(\xi)+y(\xi^{\prime})\big{)}\mathop{}\\!\mathrm{d}l$
$\displaystyle+\int_{0}^{1}\int_{0}^{1}D^{2}T^{\omega}_{s,t}g\Big{(}\Lambda(l,l^{\prime})\Big{)}\Big{(}l\big{(}x(\xi)-x(\xi^{\prime})\big{)}+(1-l)\big{(}y(\xi)-y(\xi^{\prime})\big{)}\Big{)}\otimes\big{(}x(\xi)-y(\xi)\big{)}\mathop{}\\!\mathrm{d}l^{\prime}\mathop{}\\!\mathrm{d}l,$

with

$\Lambda(l,l^{\prime})=l^{\prime}\Big{(}l\big{(}x(z)-x(z^{\prime})\big{)}+(1-l)\big{(}y(z)-y(z^{\prime})\big{)}\Big{)}+lx(z^{\prime})+(1-l)y(z^{\prime}).$

Since $T^{\omega}g$ is twice (weighted) differentiable, it follows that

$\|T^{\omega}_{s,t}g(x)-T^{\omega}_{s,t}g(y)\|_{\mathcal{C}^{\beta}}\leq\sup_{|z|\leq\|x\|_{\mathcal{C}^{\beta}}\vee\|y\|_{\mathcal{C}^{\beta}}}w^{-1}(z)\|T^{\omega}g\|_{\mathcal{C}^{\gamma}_{T}\mathcal{C}^{\beta}(w)}\|x-y\|_{\mathcal{C}^{\beta}}|t-s|^{\gamma}.$

A similar argument for $\nabla T^{\omega}b$, using that $\kappa\geq 3$,
reveals that

$\|\nabla T^{\omega}_{s,t}g(x)-\nabla
T^{\omega}_{s,t}g(y)\|_{\mathcal{C}^{\beta}}\leq\sup_{|z|\leq\|x\|_{\mathcal{C}^{\beta}}\vee\|y\|_{\mathcal{C}^{\beta}}}w^{-1}(z)\|T^{\omega}b\|_{\mathcal{C}^{\gamma}_{T}\mathcal{C}^{\beta}(w)}\|x-y\|_{\mathcal{C}^{\beta}}|t-s|^{\gamma},$

which concludes this proof. ∎

##  4\. Existence and uniqueness of the mSHE

We are now ready to formulate the multiplicative stochastic heat equation
using the abstract non-linear Young integral and the averaged field
$T^{\omega}g$ introduced in the previous section.

###  4.1. Standard multiplicative Stochastic Heat equation with additive noise

Recall that the mSHE with additive (time)-noise is given in its mild form by

$u_{t}=P_{t}u_{0}+\int_{0}^{t}P_{t-s}\xi
g(u_{s})\mathop{}\\!\mathrm{d}s+\omega_{t},\quad u_{0}\in\mathcal{C}^{\beta},$
(4.1)

where $\xi\in\mathcal{C}^{-\vartheta}$ for some
$0\leq\vartheta<\beta<2-\vartheta$, $P$ denotes the standard heat semi-group
acting on functions $u$ through convolution, and
$\omega:[0,T]\rightarrow\mathbb{R}$ is a measurable path. We will see in this
section that $g$ can be chosen to be distributional given that $\omega$ is
sufficiently irregular.

Similarly as is done for pathwise regularization by noise for SDEs (e.g. [7]),
we begin to consider (4.1) with a smooth function $g$, and we set
$\theta=u-\omega$, and then study the integral equation

$\theta_{t}=P_{t}u_{0}+\int_{0}^{t}P_{t-s}\xi
g(\theta_{s}+\omega_{s})\mathop{}\\!\mathrm{d}s.$ (4.2)

The integral term can be written on the form of a non-linear Young–Volterra
integral. To motivate this, we begin with the following observation: Define
the linear operator $S_{t}=P_{t}\xi$, whose action is defined by

$S_{t}f:=\int_{\mathbb{R}}P_{t}(x-y)\xi(y)f(y)\mathop{}\\!\mathrm{d}y.$ (4.3)

For $\beta>\vartheta$ the classical Schauder estimates for the heat equation
tells us that

$\|S_{t}f\|_{\mathcal{C}^{\beta}}=\|S_{t}f\|_{\mathcal{C}^{-\vartheta+2\frac{\beta+\vartheta}{2}}}\lesssim
t^{-\frac{\beta+\vartheta}{2}}\|\xi f\|_{\mathcal{C}^{-\vartheta}}\lesssim
t^{-\frac{\beta+\vartheta}{2}}\|\xi\|_{\mathcal{C}^{-\vartheta}}\|f\|_{\beta},$

where in the last estimate we have used that the product between the
distribution $\xi\in\mathcal{C}^{-\vartheta}$ and the function
$f\in\mathcal{C}^{\beta}$ since $\beta>\vartheta$. Thus we can view $S$ as a
bounded linear operator from $\mathcal{C}^{\beta}$ to itself for any
$\beta>\vartheta$. This motivates the next proposition:

######  Proposition 20.

If $\vartheta\geq 0$ and $\beta>\vartheta$, then the operator $S$ defined in
(4.3) is a linear operator on $\mathcal{C}^{\beta}$ and satisfies Hypothesis 9
with singularity $\rho=\frac{\beta+\vartheta}{2}$.

###### Proof.

This follows from the properties of the heat kernel proven in Corollary 45 in
Section A.3, and the fact that the para-product $\xi f$ satisfies the bound
$\|\xi
f\|_{\mathcal{C}^{-\vartheta}}\leq\|\xi\|_{\mathcal{C}^{-\vartheta}}\|f\|_{\mathcal{C}^{\beta}}$
due to the assumption that $\beta>\vartheta$. See [2] for more details on the
para-product in Besov spaces. ∎

The integral in (4.2) can therefore be written as

$\int_{0}^{t}P_{t-s}\xi
g(\theta_{s}+\omega_{s})\mathop{}\\!\mathrm{d}s=\int_{0}^{t}S_{t-s}g(\theta_{s}+\omega_{s})\mathop{}\\!\mathrm{d}s.$

Furthermore, the classical Volterra integral on the right hand side can be
written in terms of the averaged field $T^{\omega}g$, in the sense that

$\int_{0}^{t}S_{t-s}g(\theta_{s}+\omega_{s})\mathop{}\\!\mathrm{d}s=\int_{0}^{t}S_{t-s}T^{\omega}_{\mathop{}\\!\mathrm{d}s}g(\theta_{s}),$

where for a partition $\mathcal{P}$ of $[0,t]$ with infinitesimal mesh, we
define

$\int_{0}^{t}S_{t-s}T^{\omega}_{\mathop{}\\!\mathrm{d}s}g(\theta_{s}):=\lim_{|\mathcal{P}|\rightarrow
0}\sum_{[u,v]\in\mathcal{P}}S_{t-u}T^{\omega}_{u,v}g(\theta_{u}).$

It is not difficult to see (and we will rigorously prove it later) that when
$g$ is smooth, the above definition agrees with the classical Riemann
definition of the integral. However, the advantage with this formulation of
the integral is that $T^{\omega}g$ might still make sense as a function when
$g$ is only a distribution, and thus we will see that this formulation allows
for an extension of the concept of integration to distributional coefficients
$g$.

######  Proposition 21.

Consider a measurable path $\omega:[0,T]\rightarrow\mathbb{R}$ and
$g\in\mathcal{S}^{\prime}(\mathbb{R})$, and suppose
$T^{\omega}g\in\mathcal{C}^{\gamma}_{T}\mathcal{C}^{\kappa}(w)$ for some
$\gamma>\frac{1}{2}$ and $\kappa\geq 3$. For some $0<\beta<1$, suppose
$S:[0,T]\rightarrow\mathcal{L}(\mathcal{C}^{\beta})$ satisfies Hypothesis 9
for some $0\leq\rho<1$ such that $\gamma-\rho>1-\gamma$. Take
$\gamma-\rho>\varsigma>1-\gamma$ Then for any
$y\in\mathscr{C}^{\varsigma}_{T}\mathcal{C}^{\beta}$, the integral

$\Theta(y)_{t}:=\int_{0}^{t}S_{t-s}T^{\omega}_{\mathop{}\\!\mathrm{d}s}g(y_{s}):=\lim_{|\mathcal{P}|\rightarrow
0}\sum_{[u,v]\in\mathcal{P}}S_{t-u}T^{\omega}_{u,v}g(y_{u})$ (4.4)

exists as a non-linear Young-Volterra integral according to Lemma 10.

###### Proof.

Let $E:=\mathcal{C}^{\beta}$. Since
$T^{\omega}g\in\mathcal{C}^{\gamma}_{T}\mathcal{C}^{\kappa}(w)$ for some
$\gamma>\frac{1}{2}$ and $\kappa\geq 3$, it follows from Proposition 19 that
$T^{\omega}g$ can be seen as a function from $[0,T]\times E$ to $E$ that
satisfies (i)-(ii) in Lemma 10 with $H(x):=\sup_{|z|\leq
x}w^{-1}(z)\|T^{\omega}g\|_{\mathcal{C}^{\gamma}_{T}\mathcal{C}^{\kappa}(w)}$.
Lemma 10 then implies that (4.4) exists, and satisfies the bound in (2.3). ∎

As an immediate consequence of the above proposition and Proposition 11, we
obtain the following Proposition:

######  Proposition 22.

Under the same assumptions as in Proposition 21, for two paths
$x,y\in\mathscr{C}^{\varsigma}_{T}\mathcal{C}^{\beta}$ we have that

$[\Theta(y)-\Theta(x)]_{\varsigma}\lesssim_{P,\xi}\Big{\\{}\sup_{z\leq\|x\|_{\infty}\vee\|y\|_{\infty}}w^{-1}(z)\|T^{\omega}g\|_{\mathcal{C}^{\gamma}_{T}\mathcal{C}^{\kappa}}\Big{\\}}\left(\|x_{0}-y_{0}\|_{\mathcal{C}^{\beta}}+[x-y]_{\varsigma}\right).$

Moreover, let $\\{g_{n}\\}_{n\in\mathbb{N}}$ be a sequence of smooth functions
converging to $g\in\mathcal{S}^{\prime}(\mathbb{R})$ such that
$T^{\omega}g_{n}\rightarrow
T^{\omega}g\in\mathcal{C}^{\gamma}_{T}\mathcal{C}^{\kappa}(w)$. Then the
sequence of integral operators $\\{\Theta_{n}\\}_{n\in\mathbb{N}}$ built from
$T^{\omega}g_{n}$ converge to $\Theta$, in the sense that for any
$y\in\mathscr{C}^{\varsigma}_{T}\mathcal{C}^{\beta}$

$\lim_{n\rightarrow\infty}\|\Theta_{n}(y)-\Theta(y)\|_{\mathcal{C}^{\varsigma}_{T}\mathcal{C}^{\beta}}=0,$

for any $1-\gamma<\varsigma<\gamma-\rho$.

###### Proof.

A combination of Proposition 21 and Proposition 11 gives the first claim. For
the second, consider the field
$X^{n}_{s,t}(x)=\left(T^{\omega}_{s,t}g_{n}-T^{\omega}_{s,t}g\right)(x)$. Then
$X$ satisfies condition (i)-(ii) of Lemma 10 with $H(x)=\sup_{|z|\leq
x}w^{-1}(z)\|T^{\omega}g_{n}-T^{\omega}g\|_{\mathcal{C}^{\gamma}_{T}\mathcal{C}^{\kappa}(w)}$.
Thus by Proposition 11 it follows that the integral operator
$\Theta^{X^{n}}:\mathscr{C}^{\varsigma}\mathcal{C}^{\beta}\rightarrow\mathscr{C}^{\varsigma}\mathcal{C}^{\beta}$
built from $X^{n}$ satisfies for any
$y\in\mathscr{C}^{\varsigma}_{T}\mathcal{C}^{\beta}$

$\|\Theta^{X^{n}}(y)\|_{\mathcal{C}^{\varsigma}_{T}\mathcal{C}^{\beta}}\lesssim_{S,\xi}\sup_{|z|\leq\|y\|_{\infty}}w^{-1}(z)\|T^{\omega}g_{n}-T^{\omega}g\|_{\mathcal{C}^{\gamma}_{T}\mathcal{C}^{\kappa}(w)}(1+[y]_{\varsigma}).$

Thus taking the limits when $n\rightarrow\infty$, it follows that
$\|\Theta^{X^{n}}(y)\|_{\mathcal{C}^{\varsigma}_{T}\mathcal{C}^{\beta}}\rightarrow
0$ due to the assumption that
$\|T^{\omega}g_{n}-T^{\omega}g\|_{\mathcal{C}^{\gamma}_{T}\mathcal{C}^{\kappa}(w)}\rightarrow
0$ when $n\rightarrow\infty$. ∎

We are now ready to prove existence and uniqueness of solutions to (4.2) in
the non-linear Volterra-Young formulation, through an application of the
abstract existence and uniqueness results of equations on the form

$\theta_{t}=p_{t}+\Theta(\theta)_{t},\quad t\in[0,T],$

proven in Theorem 12. In combination with Proposition 21 and 22 we consider
$p_{t}=P_{t}\psi$ for some $\psi\in\mathcal{C}^{\beta}$, $\Theta$ is the
integral operator constructed in (4.4) with $S_{t}=P_{t}\xi$ for some
$\xi\in\mathcal{C}^{-\vartheta}$ with $\beta>\vartheta$.

######  Theorem 23.

Consider a measurable path $\omega:[0,T]\rightarrow\mathbb{R}$ and
$g\in\mathcal{S}^{\prime}(\mathbb{R})$, and suppose
$T^{\omega}g\in\mathcal{C}^{\gamma}_{T}\mathcal{C}^{\kappa}(w)$ for some
$\gamma>\frac{1}{2}$ and $\kappa\geq 3$.

Let $0<\vartheta<1$ and $\vartheta<\beta<2-\vartheta$. Let
$\xi\in\mathcal{C}^{-\vartheta}$ and $\rho=\frac{\beta-\vartheta}{2}$ and
suppose that $1-\gamma<\gamma-\rho$. Suppose $\psi\in\mathcal{C}^{\beta}$.
Then there exists a time $\tau\in(0,T]$ such that there exists a unique
$\theta\in\mathcal{B}_{\tau}(P\psi)\subset\mathscr{C}^{\varsigma}_{\tau}\mathcal{C}^{\beta}$
which solves the non-linear Young equation

$\theta_{t}=P_{t}\psi+\int_{0}^{t}P_{t-s}\xi
T^{\omega}_{\mathop{}\\!\mathrm{d}s}g(\theta_{s}),\quad t\in[0,\tau],$ (4.5)

for any $\gamma-\rho>\varsigma>1-\gamma$ where $S_{t}:=P_{t}\xi$ as defined
Proposition 20, and the integral is understood in the sense of proposition 21.
There exists a $C=C({P,\xi})>0$ such that the solution $\theta$ satisfies the
following bound

$[\theta]_{\varsigma;\tau}\leq C\left([P\psi]_{\varsigma,\tau}+2\sup_{|z|\leq
1+\|\psi\|_{\mathcal{C}^{\beta}}+[P\psi]_{\varsigma;\tau}}w^{-1}(z)\|T^{\omega}g\|_{\mathcal{C}^{\gamma}_{T}\mathcal{C}^{\kappa}}\right).$
(4.6)

Moreover, if $w\simeq 1$, then there exists a unique
$\theta\in\mathscr{C}^{\varsigma}_{T}\mathcal{C}^{\beta}$ which is a global
solution, in the sense that $\tau=T$.

###### Proof.

By Proposition 20, it follows that $S_{t}:=P_{t}\xi$ is a linear operator on
$\mathcal{C}^{\beta}$ since $\beta>\vartheta$. It then follows from
Proposition 21 that the non-linear Young integral

$\Theta(y)_{t}:=\int_{0}^{t}S_{t-s}T^{\omega}_{\mathop{}\\!\mathrm{d}s}g(y_{s}),$

exists as a map
$\Theta:[0,T]\times\mathscr{C}^{\varsigma}_{T}\mathcal{C}^{\beta}\rightarrow\mathscr{C}^{\varsigma}_{T}\mathcal{C}^{\beta}$,
and we have that for any $y\in\mathscr{C}^{\varsigma}_{T}\mathcal{C}^{\beta}$
with $\gamma-\rho>\varsigma>1-\gamma$ there exists a constant $C>0$ such that

$[\Theta(y)]_{\varsigma}\leq
C\sup_{|z|\leq\|y\|_{\infty}}w^{-1}(z)\|T^{\omega}g\|_{\mathcal{C}^{\gamma}_{T}\mathcal{C}^{\kappa}(w)}(1+[y]_{\varsigma})T^{\gamma-\rho-\varsigma}.$
(4.7)

Moreover by Proposition 22 we have that for two paths
$x,y\in\mathscr{C}^{\varsigma}_{T}\mathcal{C}^{\beta}$ there exists a constant
$C=C(S,\xi)>0$ such that

$[\Theta(x)-\Theta(y)]_{\varsigma}\leq
C\sup_{|z|\leq\|x\|_{\infty}\vee\|y\|_{\infty}}w^{-1}(z)\|T^{\omega}g\|_{\mathcal{C}^{\gamma}_{T}\mathcal{C}^{\kappa}(w)}\left(\|x_{0}-y_{0}\|_{\mathcal{C}^{\beta}}+[x-y]_{\varsigma}\right)T^{\gamma-\rho-\varsigma}.$
(4.8)

Thus by theorem 12, for any $\tau>0$ such that

$\tau\leq\left[4(1+[P\psi]_{\varsigma})C\sup_{|z|\leq
1+\|\psi\|_{\mathcal{C}^{\beta}}+[P\psi]_{\varsigma}}w^{-1}(z)\right]^{-\frac{1}{\gamma-\rho-\varsigma}},$

there exists a unique
$\theta\in\mathscr{C}^{\varsigma}([0,\tau];\mathcal{C}^{\beta}(\mathbb{R}))$
for $1-\gamma<\varsigma<\gamma-\rho$ which satisfies (4.5). Thus local
existence and uniqueness holds for (4.5).

We now move on to prove the bound in (4.6) First observe that

$[\theta]_{\varsigma;\tau}\leq[P\psi]_{\varsigma,\tau}+[\Theta(\theta)]_{\varsigma;\tau}.$

From Lemma 10, we know that there exists a $C=C(P,\xi)>0$ such that

$[\Theta(\theta)]_{\varsigma;\tau}\leq
C\sup_{|z|\leq\|\theta\|_{\infty}}w^{-1}(z)\|T^{\omega}g\|_{\mathcal{C}^{\gamma}_{T}\mathcal{C}^{\kappa}}(1+[\theta]_{\varsigma;\tau})\tau^{\gamma-(\rho+\varsigma)}$

Recall from Theorem 12 that $\tau>0$ is chosen such that $\theta$ is contained
in the unit ball centred at $S\psi$, i.e.
$\theta\in\mathcal{B}_{\tau}(P\psi)$, and thus
$[\theta]_{\varsigma;\tau}\leq[P\psi]_{\varsigma;\tau}+1$, which yields

$[\theta]_{\varsigma;\tau}\leq[P\psi]_{\varsigma,\tau}+2\sup_{|z|\leq
1+\|\psi\|_{\mathcal{C}^{\beta}}+[P\psi]_{\varsigma;\tau}}w^{-1}(z)\|T^{\omega}g\|_{\mathcal{C}^{\gamma}_{T}\mathcal{C}^{\kappa}}.$

At last, if $w\simeq 1$, then for any
$x,y\in\mathscr{C}^{\varsigma}_{T}\mathcal{C}^{\beta}$, there exists an $M>0$
such that

$C\sup_{|z|\leq\|x\|_{\infty}\vee\|y\|_{\infty}}w^{-1}(z)\leq M.$

where $C$ is the largest of the constants in (4.7) and (4.8). By Theorem 13 it
follows that a global solution exists, i.e. there exists a unique
$\Theta\in\mathscr{C}^{\varsigma}_{T}\mathcal{C}^{\beta}$ satisfying (4.5). ∎

With the above theorem at hand, we are ready to prove Theorem 2 and Corollary
3.

###### Proof of Theorem 2.

From Theorem 23, we know that there exists a $\tau>0$ such that there exists a
unique $\theta\in\mathscr{C}^{\varsigma}_{\tau}\mathcal{C}^{\beta}$ which
solves (4.5). Thus we say that there exists a unique local solution $u$ in the
sense of Definition 1. As proven in Theorem 23, there exists a global solution
if $T^{\omega}g\in\mathcal{C}^{\gamma}_{T}\mathcal{C}^{\kappa}(w)$ with
$w\simeq 1$. This concludes the proof of theorem 2. ∎

###### Proof of Corollary 3.

Let $\\{g_{n}\\}$ be a sequence of smooth functions converging to $g$ such
that $T^{\omega}g_{n}$ converges to $T^{\omega}g$ in
$\mathcal{C}^{\gamma}_{T}\mathcal{C}^{\kappa}$, and define the field
$Y^{n}_{s,t}(x)=(t-s)g_{n}(x+\omega_{s})$. It is readily checked that $Y^{n}$
satisfies (i)-(ii) of Lemma 10, and thus $\Theta^{Y^{n}}$ is an integral
operator from
$\mathscr{C}^{\varsigma}_{T}\mathcal{C}^{\beta}\rightarrow\mathscr{C}^{\varsigma}_{T}\mathcal{C}^{\beta}$.
Furthermore, it follows from Theorem 12 together with the same reasoning as in
the proof of Theorem 23 that there exists a unique solution $\theta^{n}$ to
the equation

$\theta^{n}_{t}=P_{t}\psi+\Theta^{Y^{n}}(\theta)_{t}.$ (4.9)

In particular,
$\|\theta^{n}\|_{\mathscr{C}^{\varsigma}_{\tau}\mathcal{C}^{\beta}}<\infty$
for all $n\in\mathbb{N}$. Note that then $\Theta^{Y^{n}}$ corresponds to the
classical Riemann integral on $\mathcal{C}^{\beta}$. Indeed, since $g_{n}$ is
smooth, the integral $\Theta^{Y^{n}}$ is given by

$\Theta^{Y^{n}}(y)_{t}:=\int_{0}^{t}S_{t-s}g(y_{s})\mathop{}\\!\mathrm{d}s,$

where we recall that $S_{t}=P_{t}\xi$. Furthermore, it is readily checked that
$\Theta^{Y^{n}}$ agrees with the integral operator $\Theta^{T^{\omega}g_{n}}$,
since $g_{n}$ is smooth,
$T^{\omega}_{s,t}g_{n}(x)=\int_{s}^{t}g(x+\omega_{r})\mathop{}\\!\mathrm{d}r$
is differentiable in time. Thus note that the difference between the two
approximating integrals

$\displaystyle\Theta^{Y^{n}}_{\mathcal{P}}(y)_{t}=\sum_{[u,v]\in\mathcal{P}[0,t]}S_{t-u}g_{n}(y_{u}+\omega_{u})(v-u)$
$\displaystyle\Theta^{T^{\omega}g_{n}}_{\mathcal{P}}(y)_{t}=\sum_{[u,v]\in\mathcal{P}[0,t]}S_{t-u}\int_{u}^{v}g_{n}(y_{u}+\omega_{r})\mathop{}\\!\mathrm{d}r$

can be estimated by

$\left\|\sum_{[u,v]\in\mathcal{P}}S_{t-u}\int_{u}^{v}g(y_{u}+\omega_{u})-g(y_{u}+\omega_{r})\mathop{}\\!\mathrm{d}r\right\|_{\mathcal{C}^{\beta}}\rightarrow
0$

when $|\mathcal{P}|\rightarrow 0$. Therefore
$\Theta^{Y^{n}}\equiv\Theta^{T^{\omega}g_{n}}$. Let
$\theta\in\mathscr{C}^{\varsigma}_{\tau}\mathcal{C}^{\beta}$ be the solution
to (4.5) constructed from $T^{\omega}g$ with integral operator $\Theta$. We
now investigate the difference between $\theta$ and $\theta^{n}$ as found in
(4.9). It is readily checked that

$[\theta^{n}-\theta]_{\varsigma}\leq[\Theta(\theta)-\Theta^{T^{\omega}g_{n}}(\theta_{n})]_{\varsigma}+[\Theta^{T^{\omega}g_{n}}(\theta)-\Theta^{Y^{n}}]_{\varsigma}.$

As already argued, the last term on the right hand side is equal to zero, and
we are therefore left to show that the first term on the right hand side
converge to zero. Invoking Proposition 11 (in particular (2.17)) and following
along the lines of the proof of 22, it is readily checked that there exists a
$C=C(P,\xi)>0$ such that

$[\theta^{n}-\theta]_{\varsigma;\tau}\leq
C\sup_{|z|\leq\|\theta\|_{\infty}\vee\|\theta^{n}\|_{\infty}}w^{-1}(z)(\|T^{\omega}g_{n}-T^{\omega}g\|_{\mathcal{C}^{\gamma}_{T}\mathcal{C}^{\kappa}(w)}(1+[\theta]_{\varsigma;\tau}\vee[\theta^{n}]_{\varsigma;\tau})+[\theta-\theta^{n}]_{\varsigma;\tau})\tau^{\gamma-\rho-\varsigma}$

We can now choose a parameter $\bar{\tau}>0$ sufficiently small, such that

$[\theta^{n}-\theta]_{\varsigma;\bar{\tau}}\leq
2\sup_{|z|\leq\|\theta\|_{\infty}\vee\|\theta^{n}\|_{\infty}}w^{-1}(z)(\|T^{\omega}g_{n}-T^{\omega}g\|_{\mathcal{C}^{\gamma}_{T}\mathcal{C}^{\kappa}(w)}(1+[\theta]_{\varsigma;\tau}\vee[\theta^{n}]_{\varsigma;\tau}).$

From (4.6) we know that there exists a constant $M>0$ such that

$\|\theta^{n}\|_{\infty}\vee\|\theta\|_{\infty}\vee[\theta^{n}]_{\varsigma;\tau}\vee[\theta]_{\varsigma;\tau}\leq
M,$

and it follows that

$[\theta^{n}-\theta]_{\varsigma;\bar{\tau}}\leq 2\sup_{|z|\leq
M}w^{-1}(z)\|T^{\omega}g_{n}-T^{\omega}g\|_{\mathcal{C}^{\gamma}_{T}\mathcal{C}^{\kappa}(w)}(1+M).$
(4.10)

The above inequality can, by similar arguments, be proven to hold for any sub-
interval $[a,a+\bar{\tau}]\subset[0,\tau]$ and thus we conclude by [12, Exc.
4.24] that there exists a constant $C=C(P,\xi,M,\bar{\tau},w)>0$ such that

$[\theta^{n}-\theta]_{\varsigma;\tau}\leq
C\|T^{\omega}g_{n}-T^{\omega}g\|_{\mathcal{C}^{\gamma}_{T}\mathcal{C}^{\kappa}(w)}.$

Taking the limit when $n\rightarrow\infty$ it follows that
$\theta^{n}\rightarrow\theta$ due to the assumption that
$T^{\omega}g_{n}\rightarrow T^{\omega}g$. If the solution is global, i.e.
$w\simeq 1$, then the same arguments as above holds, and the inequality in
(4.10) can be proven to hold on any sub-interval
$[a,a+\bar{\tau}]\subset[0,T]$ and then by the same arguments as above
$\theta^{n}\rightarrow\theta$ in
$\mathscr{C}^{\varsigma}_{T}\mathcal{C}^{\beta}$. ∎

###  4.2. The drifted multiplicative SHE

So far we have proven existence and uniqueness of solutions to equations on
the form

$u_{t}=P_{t}\psi+\int_{0}^{t}P_{t-s}\xi
g(u_{s})\mathop{}\\!\mathrm{d}s+\omega_{t},\quad
s\in[0,T],\,\,u_{0}=\psi\in\mathcal{C}^{\beta}(\mathbb{R}).$

That is, we have throughout the text assumed that
$\xi\in\mathcal{C}^{-\vartheta}$ with $\vartheta<\beta$ is a multiplicative
spatial noise. More generally, it is frequently considered a drift term in the
above equation as well, such that

$u_{t}=S_{t}\psi+\int_{0}^{t}P_{t-s}b(u_{s})\mathop{}\\!\mathrm{d}s+\int_{0}^{t}P_{t-s}\xi
g(u_{s})\mathop{}\\!\mathrm{d}s+\omega_{t},\quad
s\in[0,T],\,\,u_{0}=\psi\in\mathcal{C}^{\beta}(\mathbb{R}).$

Our results extends easily to this type of drifted equation as well, as long
as $T^{\omega}b\in\mathcal{C}^{\gamma}_{T}\mathcal{C}^{3}(w)$. Indeed, again
setting $\theta=u-\omega$, we consider the non-linear Young-Volterra equation

$\theta_{t}=P_{t}\psi+\int_{0}^{t}P_{t-s}T^{\omega}_{\mathop{}\\!\mathrm{d}s}b(\theta_{s})+\int_{0}^{t}P_{t-s}\xi
T^{\omega}_{\mathop{}\\!\mathrm{d}s}g(\theta_{s}).$ (4.11)

It is straightforward to construct the non-linear Young-Volterra integral of
the form
$\int_{0}^{t}P_{t-s}T^{\omega}_{\mathop{}\\!\mathrm{d}s}b(\theta_{s})$. Since
there is no multiplicative noise $\xi$, the linear operator $P_{t}$ on
$\mathcal{C}^{\beta}$ is not singular in its time argument. An application of
Lemma 10 would then allow to construct
$y\mapsto\Theta^{b}(y):=\int_{0}^{t}P_{t-s}T^{\omega}_{\mathop{}\\!\mathrm{d}s}b(y_{s})$
as an operator from $\mathscr{C}^{\varsigma}_{T}\mathcal{C}^{\beta}$ into
itself. The second integral is constructed as before as an operator from
$\mathscr{C}^{\varsigma}_{T}\mathcal{C}^{\beta}$ into itself; let us denote
this by $\Theta^{g}$. Then (4.11) can be written as the abstract equation

$\theta_{t}=P_{t}\psi+\Theta(\theta)_{t},$

where $\Theta(y):=\Theta^{b}(y)+\Theta^{g}(y)$ for
$y\in\mathscr{C}^{\varsigma}_{T}\mathcal{C}^{\beta}$. An application of
Theorem 12 then provides local existence and uniqueness. If both $T^{\omega}b$
and $T^{\omega}g$ are contained in some unweighted Besov spaces, global
existence and uniqueness also holds, as proven in 13.

##  5\. Averaged fields with Lévy noise

In this section we will construct the averaged field associated to a
fractional Levy process, and prove it’s regularizing properties. In contrast
to the works of [22], we will not consider the regularity of the local time
associated with a process in order to obtain the regularizing effect, but
estimate the regularity of the averaged field $T^{\omega}b$ directly based on
probabilistic techniques developed in [7]. This has the benefit that it
improves the regularity of the averaged field, as discussed in the beginning
of section 3.

###  5.1. Linear fractional Stable motion

In this section we introduce a class of measurable processes which have the
regularization property, and allows us to deal with our equations.

######  Definition 24.

Let $\alpha\in(0,2]$. We say that $L$ is a $d$-dimensional symmetric
$\alpha$-stable Lévy process if

  * (i)

$L_{0}=0$,

  * (ii)

$L$ has independent and stationary increments,

  * (iii)

$L$ is right cádlág and

  * (iv)

there exists a constant such that for all $\lambda\in\mathbb{R}^{d}$

$\mathbb{E}[\exp(iL_{t}\cdot\xi)]=e^{-c_{\alpha}|\xi|^{\alpha}t}.$

Following [36], we define a fractional process with respect to a stable Lévy
process as followed:

######  Definition 25.

Let $\alpha\in(0,2]$ and $L,\tilde{L}$ be two independent $d$-dimensional
$\alpha$-stable symmetric Lévy processes. Let
$H\in(0,1)\backslash\\{\alpha^{-1}\\}$. For all $t\in\mathbb{R}_{+}$, we
define the $\alpha$-linear fractional stable motion ($\alpha$-LFSM) of Hurst
parameter $H$ by

$L^{H}_{t}=\int_{0}^{t}(t-v)^{H-\frac{1}{\alpha}}\mathop{}\\!\mathrm{d}L_{t}+\int_{0}^{+\infty}(t+v)^{H-\frac{1}{\alpha}}-v^{H-\frac{1}{\alpha}}\mathop{}\\!\mathrm{d}\tilde{L}_{v}.$

We extend this definition to $H=\frac{1}{\alpha}$ by setting
$L^{\frac{1}{\alpha}}=L$.

The existence of the LFSM is proved in [36], Chapter 3. Note that when
$\alpha=2$, we recover the standard fractional Brownian motion (up to a
multiplicative constant), and its different representations. Here we have
chosen the moving average representation, in the fractional Brownian motion
case, one could also use the harmonizable representation or something else.
Note that when $\alpha<2$, those different representations are no longer
equivalent.

We gather here some properties about the LFSM which will become useful in the
subsequent analysis of the regularity of averaged fields associated to LFSM.
One can again consult [36], but also [29] for the first three points. The last
point is a direct consequence of the definition of $L^{H}$.

######  Proposition 26.

Let $\alpha\in(0,2]$ and let $H\in(0,1)\cup\\{\alpha^{-1}\\}$, and let $L^{H}$
be an $\alpha$-LFSM.

  * (i)

$L^{H}$ is almost surely measurable.

  * (ii)

$L^{H}$ is continuous if and only if $\alpha=2$ or $H>\frac{1}{\alpha}$.

  * (iii)

$L^{H}$ is $H$ self-similar and its increments are stationary.

  * (iv)

For all $t\geq 0$ define
$\mathcal{F}_{t}=\sigma\left(\\{\tilde{L}_{(\tilde{t}})_{\tilde{t}\in\mathbb{R}_{+}}\\}\cup\\{L_{s}\,:\,s\leq
t\\}\right)$. Then $L^{H}$ is $(\mathcal{F}_{t})_{t\geq 0}$ adapted.

  * (v)

For all $0\leq s\leq r$, we have

$L^{H}_{r}=L^{1,H}_{s,r}+L^{2,H}_{s,r},$

with

$L^{1,H}_{s,r}=\int_{s}^{r}(r-v)^{H-\frac{1}{\alpha}}\mathop{}\\!\mathrm{d}L_{v}$

and

$L^{2,H}_{s,r}=\int_{0}^{s}(r-v)^{H-\frac{1}{\alpha}}\mathop{}\\!\mathrm{d}L_{v}+\int_{0}^{+\infty}(r+v)^{H-\frac{1}{\alpha}}-v^{H-\frac{1}{\alpha}}\mathop{}\\!\mathrm{d}\tilde{L}_{v}.$

Hence, $L^{1,H}_{s,r}$ is independent of $\mathcal{F}_{r}$ while
$L^{2,H}_{s,r}$ is measurable with respect to $\mathcal{F}_{r}$, and for all
$x\in\mathbb{R}^{d}$

$\mathbb{E}[e^{i\xi\cdot
L^{1,H}_{s,r}}]=e^{-c_{\alpha}|\xi|^{\alpha}(s-r)^{\alpha H}}.$

Note that we can (and we will) reformulate the last point by saying that for
all $g\in\mathcal{S}$,

$\mathbb{E}[g(x+L^{1,H}_{s,r})]=P^{\frac{\alpha}{2}}_{c_{\alpha}(r-s)^{\alpha
H}}g(x)$

Finally, let us recall a useful result about martingales in Lebesgue spaces
due to Pinelis [35] and adapted in weighted spaces (see Appendix A.1).

######  Proposition 27.

Let $p\geq 2$, let $w$ be an admissible weight, and let $(M_{n})_{n}$ be a
$(\Omega,\mathcal{F},(\mathcal{F}_{n})_{n},\mathbb{P})$ martingale with value
in $L^{p}(\mathbb{R}^{d},w;\mathbb{R})$. Suppose that there exists a constant
$c>0$ such that for all $n\geq 0$, $\|M_{n+1}-M_{n}\|_{L^{p}(w)}\leq c$. Then
there exists two constants $C_{1}>0$ and $C_{2}>0$ such that for all $x\geq
0$,

$\mathbb{P}\big{(}\|M_{N}-\mathbb{E}[M_{N}]\|_{L^{P}(w)}\geq x\big{)}\leq
C_{1}e^{-\frac{C_{2}x^{2}}{Nc^{2}}}.$

###  5.2. Averaging operator

In this section we give another proof of the regularizing effect of the
fractional Brownian motion. This proof is similar to the one in [7], but
provides bounds in general weighted Besov spaces directly. For other proofs of
the same kind of results, one can consult [8], [15] and [28]. Other proofs may
rely on Burkolder-Davis-Gundy inquality in UMD spaces. For information about
UMD spaces, one can consult [26].

######  Lemma 28.

Let $w$ be an admissible weight. Let $\alpha\in(0,2]$ and let
$H\in(0,1)\cup\\{\alpha^{-1}\\}$. Let $\nu\in(0,1)$. Then there exists three
constants $K,C_{1},C_{2}>0$ such that for all $j\geq-1$, all
$g\in\mathcal{S}^{\prime}(\mathbb{R}^{d};\mathbb{R})$ all $s\leq t$ and all
$x>K$

$\mathbb{P}\left(\frac{\|\int_{s}^{t}\Delta_{j}g(\cdot+L^{H}_{r})\mathop{}\\!\mathrm{d}r\|_{L^{p}(w)}}{|t-s|^{1-\frac{\nu}{2}}2^{-j\frac{\nu}{2H}}\|\Delta_{j}g\|_{L^{p}(w)}}\geq
x\right)\leq C_{1}e^{-C_{2}x^{2}}.$

###### Proof.

Let $0\leq s\leq t$ and let $N\in\mathbb{N}$ to be fixed from now. Let us
define $t_{n}=\frac{n}{N}(t-s)+s$, and $\mathcal{G}_{n}=\mathcal{F}_{t_{n}}$.
Let us define

$M_{n}(x)=\mathbb{E}\left[\int_{s}^{t}\Delta_{j}g(x+L^{H}_{r})\mathop{}\\!\mathrm{d}r\bigg{|}\mathcal{F}_{t_{n}}\right].$

Hence, $(M_{n})_{n}$ is a martingale with respect to $(\mathcal{G}_{n})_{n}$
and with value in $L^{p}(\mathbb{R}^{d},w;\mathbb{R})$ for all $p\geq 2$, and
furthermore. Note also that

$M_{N}=\int_{s}^{t}\Delta_{j}g(\cdot+L^{H}_{r})\mathop{}\\!\mathrm{d}r.$

Furthermore, we have for all $0\leq n\leq N-1$

$M_{n+1}-M_{n}=\int_{t_{n}}^{t_{n+1}}\Delta_{j}g(\cdot+L^{H}_{r})\mathop{}\\!\mathrm{d}r+A_{n+1}-A_{n},$

with

$A_{n}=\mathbb{E}\left[\int_{t_{n}}^{t}\Delta_{j}g(\cdot+L^{H}_{r})\mathop{}\\!\mathrm{d}r\bigg{|}\mathcal{F}_{t_{n}}\right].$

It is also readily checked that for all $n=0,\ldots,N$ we have

$\left\|\int_{t_{n}}^{t_{n+1}}\Delta_{j}g(\cdot+L^{H}_{r})\mathop{}\\!\mathrm{d}r\right\|_{L^{p}(w)}\leq\frac{|t-s|}{N}\|\Delta_{j}g\|_{L^{p}(w)}.$

Furthermore, using the decomposition of Proposition 26, one get for all
$n\in\\{0,\cdots,N\\}$,

$\displaystyle A_{n}(x)=$
$\displaystyle\int_{t_{n}}^{t}\mathbb{E}\left[\Delta_{j}g\left(x+L^{1,H}_{t_{n},r}+L^{2,H}_{t_{n},r}\right)\bigg{|}\mathcal{F}_{t_{n}}\right]\mathop{}\\!\mathrm{d}r$
$\displaystyle=$
$\displaystyle\int_{t_{n}}^{t}P^{\frac{\alpha}{2}}_{(r-t_{n})^{\alpha
H}}\Delta_{j}g(x+L^{2,H}_{t_{n},r})\mathop{}\\!\mathrm{d}r,$

Hence, thanks to Proposition 43, the following bound holds:

$\|A_{n}\|_{L^{p}(w)}\lesssim\int_{t_{n}}^{t}e^{-c2^{\alpha
j}(r-t_{n})^{\alpha H}}\|\Delta_{j}g\|_{L^{p}(w)}\lesssim
2^{-\frac{j}{H}}\|\Delta_{j}g\|_{L^{p}(w)}\int_{0}^{+\infty}e^{-cr^{\alpha
H}}\mathop{}\\!\mathrm{d}r.$

Finally, we have the bound

$\|M_{n+1}-M_{n}\|_{L^{p}(w)}\lesssim\left(\frac{t-s}{N}+2^{-\frac{j}{H}}\right)\|\Delta_{j}g\|_{L^{p}(w)}.$
(5.1)

Note also that we have the straightforward bound
$\|M_{n+1}-M_{n}\|_{L^{p}(w)}\lesssim(t-s)\|\Delta_{j}g\|_{L^{p}(w)}$.
Therefore, by interpolation, this gives

$\|M_{n+1}-M_{n}\|_{L^{p}(w)}\lesssim(t-s)^{1-\nu}\left(\frac{t-s}{N}+2^{-\frac{j}{H}}\right)^{\nu}\|\Delta_{j}g\|_{L^{p}(w)}.$
(5.2)

It is readily checked that

$\mathbb{P}\left(\left\|M_{N}\right\|_{L^{p}(w)}\geq
x|t-s|^{1-\frac{\nu}{2}}2^{-j\frac{\nu}{2H}}\|\Delta_{j}g\|_{L^{p}(w)}\right)\\\
\leq\mathbb{P}\left(\left\|M_{N}-\mathbb{E}[M_{N}]\right\|_{L^{p}(w)}\geq\frac{x}{2}|t-s|^{1-\frac{\nu}{2}}2^{-j\frac{\nu}{2H}}\|\Delta_{j}g\|_{L^{p}(w)}\right)\\\
+\mathbb{P}\left(\left\|\mathbb{E}[M_{N}]\right\|_{L^{p}(w)}\geq\frac{x}{2}|t-s|^{1-\frac{\nu}{2}}2^{-j\frac{\nu}{2H}}\|g\|_{L^{p}(w)}\right).$

Furthermore, note that $\mathbb{E}[M_{N}]=\mathbb{E}[A_{0}]$, and
$\|A_{0}\|_{L^{p}(w)}\leq(t-s)\|\Delta_{j}g\|_{L^{p}(w)}$, where
$\|A_{0}\|_{L^{p}(w)}\lesssim 2^{-\frac{j}{H}}\|\Delta_{j}g\|_{L^{p}(w)}$.
Hence, there exists a constant $K>0$ such that

$\|\mathbb{E}[M_{N}]\|_{L^{p}(w)}\leq\frac{K}{2}|t-s|^{1-\frac{\nu}{2}}2^{-j\frac{\nu}{2H}}\|\Delta_{j}g\|_{L^{p}(w)}.$

Therefore, for $x>K$,

$\mathbb{P}\left(\left\|\mathbb{E}[M_{N}]\right\|_{L^{p}(w)}\geq\frac{x}{2}|t-s|^{1-\frac{\nu}{2}}2^{-j\frac{\nu}{2H}}\|\Delta_{j}g\|_{L^{p}(w)}\right)=0.$

Finally, this gives us, thanks to Proposition 27, there exists a constant
$\tilde{C}_{2}$ such that

$\mathbb{P}\left(\left\|\int_{s}^{t}\Delta_{j}g(\cdot+L^{H}_{r})\mathop{}\\!\mathrm{d}r\right\|_{L^{p}(w)}\geq
x|t-s|^{1-\frac{\nu}{2}}2^{-j\frac{\nu}{2H}}\|\Delta_{j}g\|_{L^{p}(w)}\right)\\\
\lesssim\exp\left(-\tilde{C}_{2}\frac{x^{2}|t-s|^{2-\nu}2^{-j\frac{\nu}{H}}\|\Delta_{j}g\|^{2}_{L^{p}(w)}}{N\left((t-s)^{1-\nu}\left(\frac{t-s}{N}+2^{-\frac{j}{H}}\right)^{\nu}\right)^{2}\|\Delta_{j}g\|^{2}_{L^{p}(w)}}\right).$

When optimizing in $N$, one obtains the desired result. ∎

######  Remark 29.

Thanks to the bound (5.2) and to UMD space standard argument (see for example
[26], one could rather use the Burkoldher-Davis-Gundy inequality, and get _for
all $1<p<+\infty$_, and all $m\geq 2$

$\mathbb{E}\left[\left\|\int_{s}^{t}\Delta_{j}g(\cdot+L^{H}_{r})\mathop{}\\!\mathrm{d}r\right\|^{m}_{L^{p}(w)}\right]\lesssim_{m}|t-s|^{\frac{m}{2}}2^{-\frac{m}{2H}j}\|\Delta_{j}g\|^{m}_{L^{p}(w)}.$

This would allow for a more standard proof of regularizing effect of LFSM
(using Kolmogorov continuity theorem), but prevents to have Gaussian tails.

######  Lemma 30.

Let $X$ be a non-negative random variable such that there exists
$K,C_{1},C_{2}>0$ such that for all $x>K$,

$\mathbb{P}(X\geq x)\leq C_{1}e^{-C_{2}x^{2}}.$

Then for all $m\geq 1$

$\mathbb{E}[X^{2m}]\leq K^{2m}+\frac{C_{1}}{2}\frac{m!}{C_{2}^{m}}.$

###### Proof.

A simple computation reveals that

$\displaystyle\mathbb{E}[X^{2m}]$
$\displaystyle=\int_{0}^{K}2mX^{2m-1}\mathbb{P}(X\geq
x)\mathop{}\\!\mathrm{d}x+\int_{K}^{+\infty}2mX^{2m-1}\mathbb{P}(X\geq
x)\mathop{}\\!\mathrm{d}x$ $\displaystyle\leq
K^{2m}+C_{1}\int_{0}^{+\infty}e^{-C_{2}x^{2}}\mathop{}\\!\mathrm{d}x$
$\displaystyle=K^{2m}+C_{1}C_{2}^{-m}\int_{0}^{+\infty}e^{-x^{2}}\mathop{}\\!\mathrm{d}x$
$\displaystyle=K^{2m}+\frac{C_{1}}{2}\frac{m!}{C_{2}^{m}}.$

∎

Finally, we have all the tools to prove the following theorem of regularity of
the averaging operator with respect the LFSM. We will use a version of the
Garsia-Rodemich-Rumsey inequality from Friz and Victoir ([11] -Theorem 1.1 p.
571).

######  Theorem 31.

Let $w$ be an admissible weight. Let $\alpha\in(0,2]$. Let
$H\in(0,1)\cup\\{\alpha^{-1}\\}$. Let $\kappa\in\mathbb{R}$ and $\nu\in[0,1]$.
Let $2\leq p\leq\infty$ and let $1\leq q\leq+\infty$. There exists a constant
$C>0$ such that for any $g\in B^{\kappa}_{p,q}(w)$ a positive random variable
$F$ exists with

$\mathbb{E}[e^{CF^{2}}]<+\infty$

, and such that for any $0\leq s\leq t\leq T$

$\|T^{L^{H}}_{s,t}g\|_{B^{\kappa+\frac{\nu}{2H}}_{p,q}(w)}\leq
F|t-s|^{1-\frac{\nu}{2}-\frac{\varepsilon}{2}}\|g\|_{B^{\kappa}_{p,q}(w)}\quad\text{if}\quad
q<+\infty$

and

$\|T^{L^{H}}_{s,t}g\|_{B^{\kappa+\frac{\nu}{2H}-\delta}_{p,\infty}(w)}\leq
F|t-s|^{1-\frac{\nu}{2}-\frac{\varepsilon}{2}}\|g\|_{B^{\kappa}_{p,\infty}(w)}.$

###### Proof.

First consider $1\leq q<\infty$. Thanks to Lemma 28 and Lemma 30, we know that
there exists a constant $c_{q}>0$ such that for all $0\leq s<t\leq T$

$\mathbb{E}\left[\left(\frac{\left\|\Delta_{j}T^{L^{H}}_{s,t}g\right\|_{L^{p}(w)}}{|t-s|^{1-\frac{\nu}{2}}2^{-j\frac{\nu}{2H}}\left\|\Delta_{j}g\right\|_{L^{p}(w)}}\right)^{q}\right]\leq
c_{q}$

and that for all $m\geq 1$,

$\mathbb{E}\left[\left\|\Delta_{j}T^{L^{H}}_{s,t}g\right\|^{2m}_{L^{p}(w)}\right]\lesssim\left(K^{2m}+\frac{m!}{C_{2}^{m}}\right)|t-s|^{(2-\nu)m}2^{-j\frac{\nu
m}{H}}\left\|\Delta_{j}g\right\|^{2m}_{L^{p}(w)}.$

Let us take $C<C_{2}$. We have

$\mathbb{E}\left[\exp\left(C\left(\frac{\|T^{L^{H}}_{s,t}g\|_{B^{\kappa+\frac{\nu}{2H}}_{p,q}(w)}}{|t-s|^{1-\frac{\nu}{2}}\|g\|_{B^{\kappa}_{p,q}(w)}}\right)^{2}\right)\right]=A+B,$

with

$A=\sum_{2m\leq
q}\frac{C^{m}}{m!|t-s|^{(2-\nu)m}\|g\|^{2m}_{B^{\kappa}_{p,q}(w)}}\mathbb{E}\left[\left(\sum_{j\geq
1}2^{j\left(\kappa+\frac{\nu}{2H}\right)q}\left\|\Delta_{j}T^{L^{H}}_{s,t}g\right\|^{q}_{L^{p}(w)}\|\right)^{\frac{2m}{q}}\right]$

and

$B=\sum_{2m>q}\frac{C^{m}}{m!}\mathbb{E}\left[\left(\sum_{j\geq{-1}}\frac{2^{jq\kappa}\|\Delta_{j}g\|^{q}_{L^{P}(w)}}{\sum_{j\geq{-1}}2^{jq\kappa}\|\Delta_{j}g\|^{q}_{L^{P}(w)}}\left(\frac{\|\Delta_{j}T^{L^{H}}_{s,t}g\|_{L^{p}(w)}}{|t-s|^{1-\frac{\nu}{2}}2^{-j\frac{\nu}{2}}\|\Delta_{j}g\|_{L^{P}(w)}}\right)^{q}\right)^{\frac{2m}{q}}\right].$

For $A$, we use Jensen inequality in the concave case, and we have thanks to
Lemma 30

$\displaystyle A\leq$ $\displaystyle\sum_{2m\leq
q}\frac{C^{m}}{m!|t-s|^{(2-\nu)m}\|g\|^{2m}_{B^{\kappa}_{p,q}(w)}}\left(\sum_{j\geq
1}2^{j\left(\kappa+\frac{\nu}{2H}\right)q}\mathbb{E}\left[\left\|\Delta_{j}T^{L^{H}}_{s,t}g\right\|^{q}_{L^{p}(w)}\|\right]\right)^{\frac{2m}{q}}$
$\displaystyle\lesssim$
$\displaystyle\sum_{2m<q}\frac{C^{m}\left(c_{q}^{\frac{2}{q}}\right)^{m}}{m!}$
$\displaystyle\lesssim$ $\displaystyle e^{Cc_{q}^{\frac{2}{q}}}.$

For $B$, we use Jensen inequality in the convex case, and we have again thanks
to Lemma 30

$\displaystyle B=$
$\displaystyle\sum_{2m>q}\frac{C^{m}}{m!}\mathbb{E}\left[\left(\sum_{j\geq{-1}}\frac{2^{jq\kappa}\|\Delta_{j}g\|^{q}_{L^{P}(w)}}{\sum_{j\geq{-1}}2^{jq\kappa}\|\Delta_{j}g\|^{q}_{L^{P}(w)}}\left(\frac{\|\Delta_{j}T^{L^{H}}_{s,t}g\|_{L^{p}(w)}}{|t-s|^{1-\frac{\nu}{2}}2^{-j\frac{\nu}{2}}\|\Delta_{j}g\|_{L^{P}(w)}}\right)^{q}\right)^{\frac{2m}{q}}\right]$
$\displaystyle\leq$
$\displaystyle\sum_{2m>q}\frac{C^{m}}{m!}\sum_{j\geq{-1}}\frac{2^{jq\kappa}\|\Delta_{j}g\|^{q}_{L^{P}(w)}}{\sum_{j\geq{-1}}2^{jq\kappa}\|\Delta_{j}g\|^{q}_{L^{P}(w)}}\mathbb{E}\left[\left(\frac{\|\Delta_{j}T^{L^{H}}_{s,t}g\|_{L^{p}(w)}}{|t-s|^{1-\frac{\nu}{2}}2^{-j\frac{\nu}{2}}\|\Delta_{j}g\|_{L^{P}(w)}}\right)^{2m}\right]$
$\displaystyle\lesssim$
$\displaystyle\sum_{2m>q}\frac{C^{m}}{m!}\left(K^{2m}+\frac{m!}{C_{2}^{m}}\right)$
$\displaystyle\lesssim$ $\displaystyle
e^{CK^{2}}+\frac{1}{1-\frac{C}{C_{2}}}.$

Hence, if we define

$F=\int_{0}^{T}\int_{0}^{T}\exp\left(C\left(\frac{\|T^{L^{H}}_{s,t}g\|_{B^{\kappa+\frac{\nu}{2H}}_{p,q}(w)}}{|t-s|^{1-\frac{\nu}{2}}\|g\|_{B^{\kappa}_{p,q}(w)}}\right)^{2}\right)\mathop{}\\!\mathrm{d}s\mathop{}\\!\mathrm{d}t,$

we are exactly in the scope of the Garsia-Rodemich-Rumsey inequality with
$\psi(x)=e^{Cx^{2}}$ and $p(u)=u^{1-\frac{\nu}{2}}$, which leads to the wanted
result when $q<+\infty$. When $q=\infty$ one needs to deal with the supremum
in $q$, and thus needs to lose a bit in $q$. We leave it to the reader since
its a direct adaptation of the previous proof. ∎

With the above construction of the averaged field associated with the
fractional Lévy process, we are ready to prove Theorem 4.

###### Proof of Theorem 4.

As in Section 4, let us first consider a distribution
$\xi\in\mathcal{C}^{-\vartheta}$ and an initial condition
$\psi\in\mathcal{C}^{\beta}$ with $0<\vartheta<1$ and
$\beta=\vartheta+\varepsilon_{1}$. The singularity
$\rho=\frac{\vartheta+\beta}{2}$ is one of the limiting thresholds in all the
previous computations. Hence, the best choice of $2-\vartheta>\beta>\vartheta$
for the space of the initial condition is for a small $\varepsilon_{1}>0$.
Note also that in Theorem 31 one want to take $\nu$ as close as possible to 1,
since $\frac{\nu}{2H}$ is somehow the index of regularization of the LFSM.
Note also that for some $\varepsilon_{2}>0$ small enough, one has

$\gamma=1-\frac{\nu}{2}-\varepsilon_{2}.$

The condition which allows the nonlinear Young–Volterra calculus to work is

$\frac{1}{2}<\gamma-\rho$

here it gives us

$\vartheta<1-\nu-\frac{\varepsilon_{1}}{2}-\varepsilon_{2},$

which gives for some $\varepsilon_{3}>0$

$\nu=1-\vartheta-\frac{\varepsilon_{1}}{2}-\varepsilon_{2}-\varepsilon_{3}.$

Hence, for an admissible weight $w$ and a function $g\in\mathcal{C}^{\kappa}$
with $\kappa>3-\frac{1-\vartheta}{2H}$, there exists $\varepsilon_{1}>0$,
$\varepsilon_{2}>0$ and $\varepsilon_{3}>0$ such that all the previous
condition are satisfied and such that almost surely

$T^{L^{H}}\in\mathcal{C}^{\gamma}_{T}\mathcal{C}^{3}(w).$

Applying Theorem 2 and 3 we have the desired result. ∎

###### Proof of Corollary 5.

The proof is straightforward when one recall that the space white noise $\xi$
is almost surely in $\mathcal{C}^{-\vartheta}$ for any
$\vartheta>\frac{d}{2}$. One can then apply Theorem 4. ∎

##  6\. Conclusion

We have proven that a certain class of measurable paths
$\omega:[0,T]\rightarrow\mathbb{R}$ provides strong regularizing effects on
the multiplicative stochastic heat equation of the form of (1.6). In
particular, we prove that there exists measurable paths $\omega$ such that
local existence and uniqueness of such equations holds even when the non-
linear function $g$ is only a Schwartz distribution and
$\xi\in\mathcal{C}^{-\vartheta}$ for some $\vartheta<1$, thus allowing for
very rough spatial noise in the one dimensional setting. To this end, we apply
the concept of non-linear Young integration, and extends this to the infinite
dimensional setting with Volterra operators. This sheds new light on the
application of the "pathwise regularization by noise" techniques developed in
[7] to the context of SPDEs, and we believe that this program can be taken
further in several directions in the future, and we provide some thoughts on
such developments here.

In the current article we restricted our analysis to the spatial space white
noise on $\mathbb{T}$ (Corollary 5). Our techniques could be extended to
$\mathbb{T}^{d}$, but with the same techniques, one could not allow for white
spatial noise $\xi$. Indeed, recall that if
$\xi:\mathbb{T}^{d}\rightarrow\mathbb{R}$ then
$\xi\in\mathcal{C}^{-\frac{d}{2}-\varepsilon}$ for any $\varepsilon>0$ and
thus even in $d=2$ the noise is too rough, since the best we can deal with in
the Young setting developed above is when $\xi\in\mathcal{C}^{-\vartheta}$
with $\vartheta<1$. However, there is a possibility that techniques from the
theory of paracontrolled calculus as developed in [18] could be applied here
to make sense of the product, and thus a generalization could then be
possible. In that connection, one would possibly need "second order correction
terms" associated to the averaged field $T^{\omega}g$ which at this point is
unclear (at least to us) how one should construct.

Another possible direction would be to allow multiplicative space-time noise,
i.e. consider $\xi$ as a distribution on $[0,T]\times\mathbb{R}^{d}$ (or
$\mathbb{T}^{d}$). When $\xi$ is depending on time, one can no longer use the
non-linear Young integral in the same way as developed here. The situation
looks like the one encountered when considering regularization by noise for
ODEs with multiplicative (time dependent) noise of the form

$y_{t}=y_{0}+\int_{0}^{t}b(y_{s}+\omega_{s})\mathop{}\\!\mathrm{d}\beta_{s}+\omega_{t},\qquad
y_{0}\in\mathbb{R}^{d}.$ (6.1)

Existence and uniqueness of the above equation when $\beta$ is a fractional
Brownian motion with $H>\frac{1}{2}$ was recently established in [17], even
when $b$ is a distribution. The key idea in this result was to consider the
average operator

$\Gamma_{s,t}^{\omega}b(x):=\int_{s}^{t}b(x+\omega_{r})\mathop{}\\!\mathrm{d}\beta_{r}$

and use a recently developed probabilistic lemma by Hairer and Li [20] to show
that the regularity of $\Gamma^{\omega}b$ is linked to the regularity of
$T^{\omega}b$. By considering an averaging operator given on the form on a
Banach space

$\Pi_{s,t}^{\omega}b(x)=\int_{s}^{t}b(x+\omega_{s})\mathop{}\\!\mathrm{d}\xi_{s},$
(6.2)

where $x\in E$ and $\xi_{t}\in E$ is a time-colored spatially-white noise, for
some Banach space $E$. If one can extend the lemmas of Hairer and Li to the
infinite dimensional setting, showing the connection between the regularity of
$\Pi^{\omega}b$ and $T^{\omega}b$ (when considering $T^{\omega}b$ as an
infinite dimensional averaged field, as in Proposition 19) there is a
possibility that one could prove regularization by noise for stochastic heat
equations with space-time noise on the form

$x_{t}=P_{t}\psi+\int_{0}^{t}P_{t-s}g(x_{s})\mathop{}\\!\mathrm{d}\xi_{s}+\omega_{t},$

by using similar techniques as developed in the current article. We leave a
deeper investigation into these possibilities open for future work.

##  Appendix A Basic concepts of Besov spaces and properties of the heat
kernel

We gather here some material about Besov spaces, heat kernel estimates and
embedding in those spaces. This section is strongly inspired by [2] and [37].
See also [31] for the weighted besov norms. For the sake of the
comprehensions, we give elementary proofs of the main points for developing
the theory. This is the purpose of subsection A.1 and A.2. For the sake of the
Volterra sewing lemma (see Section 2.1), we need a few non-standard estimates
on the heat kernel action on Besov spaces. This is the purpose of subsection
A.3. In Section B we prove the elementary Cauchy-Lipschitz theorem for
multiplicative SHE without additive perturbation.

###  A.1. Weighted Lebesgue spaces

In order to work in a general setting, we will define the weighted Besov
spaces. To do so, let us define the class of admissible weight, following
Triebel chapter 6 [37]. In [31] one can find a more general definition for
weights, which somehow allows the same kind of estimates.

######  Definition 32.

We say that $w\in C^{\infty}(\mathbb{R}^{d};\mathbb{R}_{+}\backslash\\{0\\})$
is an admissible weight function if

  * (i)

For all $\kappa\in\mathbb{N}^{d}$, there exists a positive constant
$c_{\kappa}$ such that

$\forall x\in\mathbb{R}^{d},\quad\partial^{\kappa}w(x)\leq c_{\kappa}w(x).$

  * (ii)

There exists $\lambda^{\prime}\geq 0$ and $c>0$ such that

$w(x)\leq cw(y)(1+|x-y|^{2})^{\frac{\lambda^{\prime}}{2}}.$ (A.1)

Furthermore for any admissible weight we define the weighted $L^{p}$ spaces as
the following :

$L^{p}(\mathbb{R}^{d};\mathbb{R}|w)=\\{f:\mathbb{R}^{d}\to\mathbb{R}\,:\,\|f\|_{L^{p}(\mathbb{R}^{d},w)}=\|wf\|_{L^{p}(\mathbb{R}^{d};\mathbb{R})}<+\infty\\}.$

A direct consequence of the definition is that the product of two admissible
weights is also an admissible weight. Furthermore, standard polynomials
weights are of course admissible, as proved in the following Proposition.
Finally, Hölder inequality in weighted Lebesgue spaces is straightforward.

######  Proposition 33.

Let $\lambda\in\mathbb{R}$, and let us define for all $x\in\mathbb{R}^{d}$,
$\langle x\rangle=(1+|x|^{2})^{\frac{1}{2}}$. Then
$\langle\cdot\rangle^{\lambda}$ is an admissible weight with
$\lambda^{\prime}=|\lambda|$.

###### Proof.

Let us remark that for all $x,y\in\mathbb{R}^{d}$

$(1+y\cdot(x-y))^{2}\leq 2\left(1+|y|^{2}|x-y|^{2}\right)\leq
2(1+|y|^{2}|x-y|^{2})^{2}.$

Hence,

$1+y\cdot(x-y)\leq\sqrt{2}(1+|y|^{2}|x-y|^{2}),$

then

$\displaystyle\langle x\rangle^{2}=$ $\displaystyle
1+|x-y|^{2}+|y|^{2}+2y\cdot(x-y)$ $\displaystyle\leq$
$\displaystyle|x-y|^{2}+|y|^{2}+2\sqrt{2}\left(1+|y|^{2}|x-y|^{2}\right)$
$\displaystyle\leq$ $\displaystyle
2\sqrt{2}\left(1+|x-y|^{2}+|y|^{2}+|y|^{2}|x-y|^{2}\right)$ $\displaystyle=$
$\displaystyle 2\sqrt{2}\langle y\rangle^{2}\langle x-y\rangle^{2}.$

and it therefore follows that

$\langle x\rangle\leq 2^{\frac{3}{4}}\langle y\rangle\langle x-y\rangle.$

If $\lambda\geq 0$,

$\langle x\rangle^{\lambda}\leq 2^{\frac{\lambda 3}{4}}\langle
y\rangle^{\lambda}\langle x-y\rangle^{\lambda},$

and

$\langle x\rangle^{-\lambda}=\langle x-y\rangle^{\lambda}\left(\langle
x\rangle\langle y-x\rangle\right)^{-\lambda}\leq 2^{\frac{3\lambda}{4}}\langle
y\rangle^{-\lambda}\langle x-y\rangle^{\lambda},$

which concludes the proof. ∎

######  Lemma 34.

Let $w$ be an admissible weight and $\lambda^{\prime}$ defined as in Equation
(A.1). Then for $1\leq p,q,r\leq+\infty$ with
$\frac{1}{p}+\frac{1}{q}=1+\frac{1}{r}$ and any measurable functions $f$ and
$g$,

$\|f*g\|_{L^{r}(w)}\leq\|f\|_{L^{p}(\langle\cdot\rangle^{\lambda^{\prime}})}\|g\|_{L^{p}(w)}.$

###### Proof.

The result is a direct consequence of the definition of admissible weights and
of standard Young inequality. Indeed,

$\displaystyle\|f*g\|_{L^{r}(w)}=$
$\displaystyle\left(\int_{\mathbb{R}^{d}}\left|\int_{\mathbb{R}^{d}}f(y)g(x-y)\mathop{}\\!\mathrm{d}y\right|^{r}w(x)^{r}\mathop{}\\!\mathrm{d}x\right)^{\frac{1}{r}}$
$\displaystyle\lesssim$
$\displaystyle\left(\int_{\mathbb{R}^{d}}\left|\int_{\mathbb{R}^{d}}\langle
x-y\rangle^{\lambda^{\prime}}f(x-y)w(y)g(y)\mathop{}\\!\mathrm{d}y\right|^{r}w(x)^{r}\mathop{}\\!\mathrm{d}x\right)^{\frac{1}{r}}$
$\displaystyle=$
$\displaystyle\|(\langle\cdot\rangle^{\lambda^{\prime}}f)*(wg)\|_{L^{r}}$
$\displaystyle\leq$
$\displaystyle\|\langle\cdot\rangle^{\lambda^{\prime}}f\|_{L^{p}}\|wg\|_{L^{q}}$
$\displaystyle=$
$\displaystyle\|f\|_{L^{p}(\langle\cdot\rangle^{\lambda^{\prime}})}\|g\|_{L^{q}(w)}$

which is the desired result. ∎

Furthermore, as usual in order to define Besov spaces we will work with
functions with compactly supported Fourier transform. In order to deal with
such functions, let us prove a Bernstein-type lemma in weighted $L^{p}$ space:

######  Lemma 35.

Let $w$ be an admissible weight. Let
$\mathcal{C}=\\{\xi\in\mathbb{R}^{d}\,:\,c_{1}\leq|\xi|\leq c_{2}\\}$ be an
annulus and $\mathcal{B}$ be a ball. There exists a constant $C>0$ such that
for all $1\leq p\leq p^{\prime}\leq+\infty$, $n\geq 0$, $a\geq 1$, and for any
function $f\in L^{p}(w)$, we have

  * (i)

If $\mathrm{supp}\hat{f}\subset a\mathcal{B}$ then for any
$k\in\mathbb{N}^{d}$

$\|D^{n}f\|=\sup_{|k|=n}\|\partial^{k}f\|_{L^{p^{\prime}}(w)}\leq
C^{n+1}a^{n+d\left(\frac{1}{p}-\frac{1}{p^{\prime}}\right)}\|f\|_{L^{p}(w)}.$

  * (ii)

If $\mathrm{supp}\hat{f}\subset a\mathcal{A}$ then

$\frac{1}{C^{n+1}}a^{n}\|f\|_{L^{P}(w)}\leq\|D^{n}f\|_{L^{p}(w)}\leq
C^{n+1}a^{n}\|f\|_{L^{p}(w)}.$

Here $\hat{f}$ is the Fourier transform of $f$.

###### Proof.

Let $\hat{K}$ be a function such that $\hat{K}\equiv 1$ on $\mathcal{B}$ and
$\mathrm{supp}(\hat{K})$ is compactly supported and let us define
$K_{a}=a^{d}K(a\cdot)$. If $\mathrm{supp}\hat{f}\subset a\mathcal{B}$, then
$f=K_{a}*f$, and
$\partial^{k}f=(\partial^{k}K_{a})*f=a^{|k|}((\partial^{k}K)_{a})*f$, where
$(\partial^{k}K)_{a}=a^{d}\partial^{k}K(a\cdot)$. Hence, by the previous
weighted Young inequality, for $1\leq p\leq p^{\prime}\leq+\infty$ and
$\frac{1}{r}=1-\left(\frac{1}{p}-\frac{1}{p^{\prime}}\right)$,

$\|\partial^{k}f\|_{L^{p^{\prime}}(w)}=a^{|k|}\|(\partial^{k}K)_{a}*f\|_{L^{p^{\prime}}(w)}\\\
\leq
a^{|k|}\|(\partial^{k}K)_{a}\|_{L^{r}(\langle\cdot\rangle^{\lambda^{\prime}})}\|f\|_{L^{p}(w)}.$

Furthermore, since $a\geq 1$ and $\lambda^{\prime}\geq 0$, one has $\langle
a^{-1}x\rangle^{\lambda^{\prime}}\leq\langle x\rangle^{\lambda^{\prime}}$, and

$\displaystyle\|(\partial^{k}K)_{a}\|_{L^{r}(\langle\cdot\rangle^{\lambda^{\prime}})}^{r}=$
$\displaystyle a^{rd}\int_{\mathbb{R}^{d}}|\partial^{k}K(ax)|^{r}\langle
x\rangle^{r\lambda^{\prime}}\mathop{}\\!\mathrm{d}x$ $\displaystyle=$
$\displaystyle a^{(r-1)d}\int_{\mathbb{R}^{d}}|\partial^{k}K(x)|^{r}\langle
a^{-1}x\rangle^{r\lambda^{\prime}}\mathop{}\\!\mathrm{d}x$ $\displaystyle\leq$
$\displaystyle a^{(r-1)d}\int_{\mathbb{R}^{d}}|\partial^{k}K(x)|^{r}\langle
x\rangle^{r\lambda^{\prime}}\mathop{}\\!\mathrm{d}x.$

It follows that

$\|(\partial^{k}K)_{a}\|_{L^{r}(\langle\cdot\rangle^{\lambda^{\prime}})}\lesssim
a^{\left(1-\frac{1}{r}\right)}\lesssim
a^{d\left(\frac{1}{p}-\frac{1}{p^{\prime}}\right)}.$

Gathering the above considerations proves (i).

The second inequality of (ii) is just a sub-case of (i). For the first
inequality of the (ii), consider a smooth function $L$ such that
$\mathrm{supp}\hat{L}$ is included in an annulus and such that $\hat{L}\equiv
1$ on $\mathcal{A}$. Following [2] Lemma 2.1 and (1.23) page 25, there exists
some real numbers $(A_{k})$ such that

$|\xi|^{2n}=\sum_{|k|=n}A_{k}(-i\xi)^{k}(i\xi)^{k},$

with
$(\xi_{1},\cdots,\xi_{d})^{(k_{1},\cdots,k_{d})}=\xi_{1}^{k_{1}}\cdots\xi_{d}^{k_{d}}$.
Hence, we have

$\sum_{|k|=n}A_{k}\frac{(-i\xi)^{k}}{|\xi|^{2n}}\hat{L}(a^{-1}\xi)\widehat{\partial^{k}f}(\xi)=L(a^{-1}\xi)\hat{f}(\xi)\sum_{|k|=n}A_{k}\frac{(-i\xi)^{k}(i\xi)^{k}}{|\xi|^{2n}}=\hat{f}(\xi).$

For $k\in\mathbb{N}^{d}$ with $|k|=n$ let us define

$L^{k}(x)=A_{k}\int_{\mathbb{R}^{d}}(-i\xi)^{k}|\xi|^{-2n}\hat{L}(\xi)e^{i\xi\cdot
x}\mathop{}\\!\mathrm{d}x$

and we have $f=a^{-n}\sum_{|k|=n}L^{k}_{a}*\partial^{k}f$. One can use the
weighted Young inequality to obtain that

$\|f\|_{L^{p}(w)}\lesssim
a^{-|n|}\sum_{|k|=n}\|L^{k}_{a}\|_{L^{1}(\langle\cdot\rangle^{\lambda^{\prime}})}\|\partial^{k}f\|_{L^{p}(w)}.$

Furthermore, since $\langle a^{-1}x\rangle^{\lambda^{\prime}}\leq\langle
x\rangle^{\lambda^{\prime}}$, it follows that
$\|L^{k}_{a}\|_{L^{1}(\langle\cdot\rangle^{\lambda^{\prime}})}\leq\|L^{k}\|_{L^{1}(\langle\cdot\rangle^{\lambda^{\prime}})}$.
Finally, we get

$\|f\|_{L^{p}(w)}\lesssim a^{-|n|}\|D^{n}f\|_{L^{p}(w)},$

which proves our claim. ∎

###  A.2. Weighted Besov Spaces and standards estimates

######  Definition 36.

Let
$\mathcal{A}=\\{\lambda\in\mathbb{R}^{d}\,:\,\frac{3}{4}\leq|\lambda|\leq\frac{8}{3}\\}$.
There exists two radial function $\chi$ and $\varphi$ such that
$\mathrm{supp}(\chi)=B(0,\frac{3}{4})$,
$\mathrm{supp}(\varphi)\subset\mathcal{A}$,

$\forall\lambda\in\mathbb{R}^{d}\,\chi(\lambda)+\sum_{j\geq
0}\varphi(2^{-j}\lambda)=1,$

and for $j\geq 1$,

$\mathrm{supp}(\chi)\cap\mathrm{supp}(\varphi(2^{-j}\cdot))=\emptyset$

and for $|j-j^{\prime}|\geq 2$,

$\mathrm{supp}(\varphi(2^{-j}\cdot))\cap\mathrm{supp}(\varphi(2^{-j^{\prime}}\cdot))=\emptyset.$

For all $f\in\mathcal{S}^{\prime}$ and $j\geq 0$, we define the in-homogeneous
Paley-Littlewood blocks by

$\Delta_{-1}f=\mathcal{F}^{-1}\big{(}\chi\hat{f}\big{)},\quad\mathrm{and}\quad\Delta_{j}f=\mathcal{F}^{-1}(\varphi(2^{-j}\cdot)\hat{f}),$

where $\hat{f}$ denotes the Fourier transform of $f$ and $\mathcal{F}^{-1}$
the inverse Fourier transform.

Note that the Paley-Litllewood blocks define a nice appproximation of the
unity. We refer to [2] proposition 2.12 for a proof.

######  Proposition 37.

For all $f\in\mathcal{S}^{\prime}$, let us define for all $j\geq-1$
$\mathcal{S}_{j}f=\sum_{j^{\prime}\leq j-1}\Delta_{j}f$. Then

$f=\lim_{j\to\infty}\mathcal{S}_{j}f\quad\text{in}\quad\mathcal{S}^{\prime}.$

######  Definition 38.

Let $1\leq p,q\leq+\infty$, and let $\kappa\in\mathbb{R}$ and $w$ be an
admissible weight. For a distribution
$f\in\mathcal{S}^{\prime}(\mathbb{R}^{d};\mathbb{R})$ we define the (in-
homogeneous) weighted Besov norm by

$\|f\|_{B^{\kappa}_{p,q}(w)}=\Big{\|}\big{(}2^{\kappa
j}\|\Delta_{j}f\|_{L^{p}(\mathbb{R}^{d},w)}\big{)}_{j\geq-1}\Big{\|}_{\ell^{q}(\mathbb{N}\cup\\{-1\\})}$

where
$\|f\|_{L^{p}(\mathbb{R}^{d},w)}=\left(\int_{\mathbb{R}^{d}}f(x)^{p}w(x)^{p}\mathop{}\\!\mathrm{d}x\right)^{\frac{1}{p}}$.
When $w\equiv 1$, we only write $B^{\kappa}_{p,q}$.

We gather here some basic properties of (weighted) Besov spaces.

######  Proposition 39.

Let $w$ be an admissible weight.

  * (i)

The space

$B^{s}_{p,q}(w)=\\{f\in\mathcal{S}^{\prime}(\mathbb{R}^{d};\mathbb{R})\,:\,\|f\|_{B^{\kappa}_{p,q}(w)}<+\infty\\}$

does not depend on the choice of $\varphi$ and $\chi$.

  * (ii)

The two following quantities $\|f\|_{B^{\kappa}_{p,q}(w)}$ and
$\|wf\|_{B^{\kappa}_{p,q}}$ are equivalent norms on $B^{\kappa}_{p,q}(w)$.

  * (iii)

For all $n\geq 0$,
$\|D^{n}f\|_{B^{\kappa}_{p,q}(w)}\lesssim\|f\|_{B^{\kappa+n}_{p,q}(w)}$.

  * (iv)

Let $1\leq p\leq p^{\prime}\leq+\infty$ and $1\leq q\leq
q^{\prime}\leq+\infty$, then for all $\varepsilon>0$,

$\|f\|_{B^{\kappa-d\left(\frac{1}{p}-\frac{1}{p^{\prime}}\right)}_{p^{\prime},q}(w)}\lesssim\|f\|_{B^{\kappa}_{p,q}(w)}\lesssim\|f\|_{B^{\kappa}_{p^{\prime},q}\left(\langle\cdot\rangle^{d\left(\frac{1}{p}-\frac{1}{p^{\prime}}\right)+\varepsilon}w\right)}$

and

$\|f\|_{B^{\kappa}_{p,\infty}(w)}\lesssim\|f\|_{B^{\kappa}_{p,q}(w)}\lesssim\|f\|_{B^{\kappa-\varepsilon}_{p,q^{\prime}}(w)}.$

  * (v)

For all $\varepsilon,\delta>0$ and all $\kappa\in\mathbb{R}$ and all $1\leq
p,q\leq+\infty$,

$B^{\kappa}_{p,q}(w)\quad\text{is compactly embedded in}\quad
B^{\kappa-\varepsilon}_{p,q}(\langle\cdot\rangle^{-\delta}w)$

  * (vi)

Suppose that $\kappa>0$ and $\kappa\notin\mathbb{N}$ and for
$f:\mathbb{R}^{d}\mapsto\mathbb{R}$ let us define

$\|f\|_{\mathcal{C}^{\kappa}(w)}=\sum_{|k|\leq[\kappa]}\sup_{x\in\mathbb{R}^{d}}|w(x)\partial^{k}f(x)|+\sum_{|k|=[\kappa]}\sup_{0<|h|\leq
1}\sup_{x\in\mathbb{R}^{d}}\frac{w(x)|\partial^{k}f(x+h)-\partial^{k}f(x)|}{|h|^{\kappa-[\kappa]}}.$

Then

$\mathcal{C}^{\kappa}(w)=\\{f\,:\,\|f\|_{\mathcal{C}^{\kappa}(w)}<+\infty\\}=B^{\kappa}_{\infty,\infty}(w)$

and furthermore $\|\cdot\|_{\mathcal{C}^{\kappa}(w)}$ and
$\|\cdot\|_{B^{\kappa}_{\infty,\infty}(w)}$ are equivalent norms on this
space.

###### Proof.

We only prove the weighted inequality in the fourth point, and we refer [37]
and the references therein for the other ones. The first and the third
inequalities are direct consequences of Lemma 35. For the second one, let us
take $1\leq p<p^{\prime}<+\infty$. We have, thanks to Jensen inequality, for
any $\varepsilon>0$

$\displaystyle\|\Delta_{j}f\|_{L^{p}(w)}^{p^{\prime}}\lesssim_{d,\varepsilon}$
$\displaystyle\int_{\mathbb{R}^{d}}|\Delta_{j}f(x)|^{p^{\prime}}w(x)^{p^{\prime}}\langle
x\rangle^{\frac{(d+\varepsilon)p^{\prime}}{p}}\langle
x\rangle^{-\frac{(d+\varepsilon)p^{\prime}}{p^{\prime}}}\mathop{}\\!\mathrm{d}x$
$\displaystyle=$
$\displaystyle\int_{\mathbb{R}^{d}}|\Delta_{j}f(x)|^{p^{\prime}}\left(w(x)\langle
x\rangle^{(d+\varepsilon)\left(\frac{1}{p}-\frac{1}{p^{\prime}}\right)}\right)^{p^{\prime}}\mathop{}\\!\mathrm{d}x.$

The constant in the previous inequality does not depends on $p^{\prime}$. This
gives

$\|\Delta_{j}f\|_{L^{p}(w)}\lesssim_{d,\varepsilon}\|\Delta_{j}f\|_{L^{p^{\prime}}\left(\langle\cdot\rangle^{(d+\varepsilon)\left(\frac{1}{p}-\frac{1}{p^{\prime}}\right)}w\right)}.$

∎

Finally, in order to deal with product of elements in Besov space, we give the
following result, which can be proved thanks to standard techniques (see [2]
Lemma 2.69 and 2.84 and [31] Theorem 3.17 and Corollary 3.19 and 3.21). Note
that it mostly relies on Hölder and Young inequality, and therefore is
available in the context of weighted spaces.

######  Proposition 40.

[Corollary 2.86 in [2] and Corollary 3.19 in [31]] Let $w$ be an admissible
weight, $\kappa_{2}\leq\kappa_{1}$ with $\kappa_{1}\geq 0$ and suppose that
$\kappa_{1}+\kappa_{2}>0$, and let $1\leq p,p_{1},p_{2},q\leq+\infty$ such
that $\frac{1}{p}=\frac{1}{p_{1}}+\frac{1}{p_{2}}$. Let
$\varepsilon,\delta>0$. Then for all
$f\in\mathcal{B}^{\kappa_{1}}_{p_{1},q}(w)$ and all
$g\in\mathcal{B}^{\kappa_{2}}_{p_{2},q}(w)$,

  * (i)

$\left(\mathcal{S}_{j}f\mathcal{S}_{j}g\right)_{j\geq 0}$ converges in
$B^{\kappa_{2}-\varepsilon}_{p,q}(\langle\cdot\rangle^{-\delta}w)$ to a limit
in $B^{\kappa_{2}}_{p,q}(w)$.

  * (ii)

We have

$\left\|\lim_{j}\mathcal{S}_{j}f\mathcal{S}_{j}g\right\|_{B^{\kappa_{2}}_{p,q}(w)}\lesssim\left\|f\right\|_{B^{\kappa_{1}}_{p,q}(w)}\left\|g\right\|_{B^{\kappa_{2}}_{p,q}(w)}$

  * (iii)

When $\kappa_{2}\geq 0$, $\lim_{j}\mathcal{S}_{j}f\mathcal{S}_{j}g=fg$ the
standard point wise product.

######  Remark 41.

The previous proposition shows that the limit does not depend on the choice of
the blocks, and therefore it extends canonically the notion of product of
functions to a product of distributions, as soon as $\kappa_{1}+\kappa_{2}>0$,
with $\kappa_{1}\geq 0$. For this reason, we will denote by
$fg=\lim_{j}\mathcal{S}_{j}f\mathcal{S}_{j}g$, and this is a bi-linear
functional from $B^{\kappa_{1}}_{p_{1},q}(w)\times
B^{\kappa_{2}}_{p_{2},q}(w)$ to $B^{\kappa_{2}}_{p,q}(w)$.

###  A.3. Heat kernel estimates

In order to deal with heat kernel estimates on weighted Besov spaces, we first
need some heat kernel estimates for function whose Fourier transform has a
support in an annulus. In order to deal with non-Gaussian noise, we need to
consider heat semi-group for fractional Laplacian. For a full study of the
fractional Laplacian, we refer to [27]. Here we just define the fractional
Laplacian for smooth functions.

######  Definition 42.

Let $\alpha\in(0,2]$. For any function
$f\in\mathcal{S}(\mathbb{R}^{d};\mathbb{R})$ we define the fractional Laplace
operator $\Delta^{\frac{\alpha}{2}}=-(-\Delta)^{\frac{\alpha}{2}}$ by

$\Delta^{\frac{\alpha}{2}}f=\mathcal{F}^{-1}(-|\cdot|^{\alpha}\hat{f}(\cdot)).$

Furthermore, we define the semi-group associated to
$\Delta^{\frac{\alpha}{2}}$

$P^{\frac{\alpha}{2}}_{t}f=\mathcal{F}^{-1}\left(e^{-|\cdot|^{\alpha}t}\hat{f}(\cdot)\right).$

Finally we extend this definition to the whole space $\mathcal{S}^{\prime}$ by
the standard procedure.

Note that when $\alpha=2$, the previous definition gives the standard Laplace
operator. When this is the case we simply write $P$ instead of $P^{1}$.

######  Proposition 43.

Let $w$ be an admissible weight and $\lambda^{\prime}$ be defined as in (A.1).
Let $\alpha\in(0,2]$. Let
$\mathcal{A}=\\{\xi\in\mathbb{R}^{d}\,:\,c_{1}\leq|\xi|\leq c_{2}\\}$ be an
annulus, let $a\geq 1$ and let $f:\mathbb{R}^{d}\to\mathbb{R}^{d}$ be a
function such that $supp\hat{f}\subset a\mathcal{A}$. Let $0<s\leq t$. There
exists a constant $c>0$ such that for all $p\geq 1$,

$\|P^{\frac{\alpha}{2}}_{t}f\|_{L^{p}(w)}\lesssim
e^{-cta^{\alpha}}\|f\|_{L^{p}(w)},$

and for all $\rho\geq 0$,

$\|(P_{t}-P_{s})f\|_{L^{p}(w)}\lesssim
a^{-\alpha\rho}\left|\frac{1}{s^{\rho}}-\frac{1}{t^{\rho}}\right|\|f\|_{L^{p}(w)}.$

Finally, for all $s\leq u\leq\tau^{\prime}\leq\tau$, and for all $\rho>0$,

$\Big{\|}\big{(}(P_{\tau-s}-P_{\tau-u})-(P_{\tau^{\prime}-s}-P_{\tau^{\prime}-u})\big{)}f\Big{\|}_{L^{p}(w)}\\\
\lesssim
a^{-\alpha\rho}\left(\frac{1}{(\tau^{\prime}-u)^{\rho}}-\frac{1}{(\tau^{\prime}-s)^{\rho}}-\frac{1}{(\tau-u)^{\rho}}+\frac{1}{(\tau-s)^{\rho}}\right)\|f\|_{L^{p}(w)}$

###### Proof.

We follow the proof of [2], Lemma 2.4. The first affirmation is the result of
this lemma but in the context of weighted spaces and of fractional operator.
We also refer to [31] Lemma 2.10 to a proof. Since the proof of the second and
third points are similar to the proof of the first one, we will not detail the
first point. For the second and third one, let us define

$E(s,t;\xi):=e^{-|\xi|^{\alpha}t}-e^{-|\xi|^{\alpha}s}.$

Note that in that case, we have

$(P_{t}-P_{s})f(x)=\int_{\mathbb{R}^{d}}E(s,t;\xi)\hat{f}(\xi)e^{i\xi
x}\mathop{}\\!\mathrm{d}x.$

Thanks to the hypothesis, there also exists a smooth function
$\varphi:\mathbb{R}^{d}\to\mathbb{R}^{d}$ such that $\varphi\equiv 1$ on
$\mathcal{A}$ and
$\mathrm{supp}\varphi\subset\\{\xi\in\mathbb{R}^{d}\,:\,\frac{c_{1}}{2}\leq|\xi|\leq
2c_{2}\\}$. Let us define for all $s\leq t$

$K(s,t;x)=\int E(s,t;\xi)\varphi(\xi)e^{i\xi x}d\xi,$

and $K_{a}(s,t,x)=a^{d}K(s,t,ax)$. Hence we have

$\displaystyle(P_{t}-P_{s})f(x)=$
$\displaystyle\int_{\mathbb{R}^{d}}E(s,t;\xi)\hat{f}(\xi)e^{i\xi\cdot x}d\xi$
$\displaystyle=$
$\displaystyle\int_{\mathbb{R}^{d}}E(s,t;\xi)\varphi(a^{-1}\xi)\hat{f}(\xi)e^{i\xi\cdot
x}d\xi$ $\displaystyle=$ $\displaystyle
a^{d}\int_{\mathbb{R}^{d}}E(s,t;a\xi)\varphi(\xi)\hat{f}(\xi)e^{i\xi\cdot
ax}d\xi.$

And since $E(s,t,a\xi)=E(a^{\alpha}s,a^{\alpha}t;\xi)$, one has

$(P_{t}-P_{s})*f(x)=K_{a}(a^{\alpha}s,a^{\alpha}t;\cdot)*f(x).$

Finally, thanks to Young inequality, one has

$\|(P_{t}-P_{s})*f\|_{L^{p}(w)}\leq\|K_{a}(a^{\alpha}s,a^{\alpha}t;\cdot)\|_{L^{1}(\langle\cdot\rangle^{\lambda^{\prime}})}\|f\|_{L^{p}(w)}.$

Note also that since $\lambda^{\prime}\geq 0$ and $a\geq 1$, we have $\langle
a^{-1}x\rangle^{\lambda^{\prime}}\leq\langle x\rangle^{\lambda^{\prime}}$, and

$\displaystyle\|K_{a}(a^{\alpha}s,a^{\alpha}t;\cdot)\|_{L^{1}(\langle\cdot\rangle^{\lambda^{\prime}})}=$
$\displaystyle a^{d}\int_{\mathbb{R}^{d}}K_{a^{\alpha}s,a^{\alpha}t,ax}\langle
x\rangle^{\lambda^{\prime}}\mathop{}\\!\mathrm{d}x$ $\displaystyle=$
$\displaystyle\int_{\mathbb{R}^{d}}K_{a^{\alpha}s,a^{\alpha}t,ax}\langle
a^{-1}x\rangle^{\lambda^{\prime}}\mathop{}\\!\mathrm{d}x$ $\displaystyle=$
$\displaystyle\leq
a^{d}\int_{\mathbb{R}^{d}}K_{a^{\alpha}s,a^{\alpha}t,x}\langle
x\rangle^{\lambda^{\prime}}\mathop{}\\!\mathrm{d}x$ $\displaystyle=$
$\displaystyle\|K(a^{\alpha}s,a^{\alpha}t;\cdot)\|_{L^{1}(\langle\cdot\rangle^{\lambda^{\prime}})}$

Hence, it is enough to prove the proposition for $a=1$ and for all
$0<s^{\prime}\leq t^{\prime}$, and then specify $s^{\prime}=a^{\alpha}s$ and
$t^{\prime}=a^{\alpha}t$.

Let $M\in\mathbb{N}$ such that $2M>d+\lambda^{\prime}$. In order to prove the
proposition, it is then enough to bound
$\big{|}(1+|x|^{2})^{M}K(s,t;x)\big{|}$ by the wanted quantity
$\frac{1}{s^{\rho}}-\frac{1}{t^{\rho}}$. We have

$\displaystyle\left(1+|x|^{2}\right)^{M}K(s,t;x)=$
$\displaystyle\int_{\mathbb{R}^{d}}\left((1-\Delta)^{M}e^{ix\cdot}\right)(\xi)\varphi(\xi)E(s,t;\xi)d\xi$
$\displaystyle=$
$\displaystyle\int_{\mathbb{R}^{d}}e^{ix\xi}(1-\Delta)^{M}\big{(}\varphi(\cdot)E(s,t;\cdot)\big{)}(\xi)d\xi$

And thanks to the Faà di Bruno formula, there exists constants
$(c_{\nu,\kappa})$ where $\nu$ and $\kappa$ are multi-indices such that

$\left(1+|x|^{2}\right)^{M}K(s,t;x)=\sum_{|\nu|+|\kappa|\leq
2M}c_{\nu,\kappa}\int_{\mathbb{R}^{d}}e^{i\xi
x}\partial^{\nu}\varphi(\xi)\partial^{\kappa}E(s,t;\xi)d\xi.$

Hence, in order to prove the proposition, one only has to bound all the
derivatives up to order $2M$ of $E(s,t;\cdot)$ for $\xi\in\mathcal{A}$ by the
wanted quantity $\frac{1}{s^{\rho}}-\frac{1}{t^{\rho}}$.

Note that the same strategy could be used in the case where we have the
rectangular increment of the semi-group if we replace $E(s,t;\xi)$ by

$F(s,u,\tau^{\prime},\tau;\xi):=\left(e^{-|\xi|^{\alpha}(\tau-s)}-e^{-|\xi|^{\alpha}(\tau-u)}\right)-\left(e^{-|\xi|^{\alpha}(\tau^{\prime}-s)}-e^{-|\xi|^{\alpha}(\tau^{\prime}-u)}\right).$

The same partial conclusion holds, one only has to control all the derivatives
of $F(s,u,\tau^{\prime},\tau;\cdot)$ up to order $2M$ for $\xi\in\mathcal{A}$.

Furthermore, observe that

$E(s,t;\xi)=\int_{s}^{t}-|\xi|^{\alpha}e^{-r|\xi|^{\alpha}}\mathop{}\\!\mathrm{d}r.$

Hence, by a direct induction and since $\xi\in\mathcal{A}$, for every
multiindexe $k=(k_{1}\cdots,k_{d})\in\mathbb{N}^{d}$, there exists a
polynomial

$P_{k,\xi}(t)=\sum_{l=0}^{|k|}a_{l}^{k}(\xi)t^{l},$

where for all $l\in\\{0,\cdots,n\\},$ $\xi\to a_{l}^{k}(\xi)$ are non-negative
smooth functions on $\mathcal{A}$, and for all $\xi\in\mathcal{A}$,
$a_{n}^{k}(\xi)\neq 0$ and such that

$\partial^{k}\big{(}|\cdot|^{2}e^{-r|\cdot|^{\alpha}}\big{)}(\xi)=P_{k,\xi}(r)e^{-r|\xi|^{\alpha}}.$

Furthermore, since $c_{1}\leq|\xi|\leq c_{2}$, there exists a constant $c>0$
(depending on $\mathcal{A}$) such that

$\big{|}\partial^{k}\big{(}|\cdot|^{\alpha}e^{-r|\cdot|^{\alpha}}\big{)}(\xi)\big{|}\lesssim\tilde{P}_{k,\xi}(r)e^{-r|\xi|^{2}}\lesssim_{k,\mathcal{A}}e^{-cr}\lesssim
r^{-\rho-1}$

for any $\rho\geq-1$. Hence

$|\partial^{k}E(s,t;\cdot)|\lesssim\int_{s}^{t}r^{-\rho}\mathop{}\\!\mathrm{d}r\lesssim\frac{1}{s^{\rho}}-\frac{1}{t^{\rho}}.$

With the previous discussion, this gives the wanted result, when we replace
$t$ by $a^{\alpha}t$ and $s$ by $a^{\alpha}s$. In order to deal with the last
estimates, one only has to remember that

$\displaystyle F(s,u,\tau^{\prime},\tau;\xi)=$
$\displaystyle-\int_{u}^{s}|\xi|^{\alpha}\left(e^{-|\xi|^{\alpha}(\tau-r)}-e^{-|\xi|^{\alpha}(\tau^{\prime}-r)}\right)\mathop{}\\!\mathrm{d}r$
$\displaystyle=$
$\displaystyle\int_{s}^{u}\int_{\tau^{\prime}-r}^{\tau-r}|\xi|^{2\alpha}e^{-|\xi|^{\alpha}v}\mathop{}\\!\mathrm{d}v\mathop{}\\!\mathrm{d}r$

The same argument as before gives the bound

$|\partial^{k}F(s,u,\tau^{\prime},\tau;\cdot)|\lesssim\int_{s}^{u}\int_{\tau^{\prime}-r}^{\tau-r}e^{-cv}\mathop{}\\!\mathrm{d}v\mathop{}\\!\mathrm{d}r.$

Finally, for any $\rho\geq 0$,

$\displaystyle|\partial^{k}F(s,u,\tau^{\prime},\tau;\cdot)|\lesssim$
$\displaystyle\int_{s}^{u}\int_{\tau^{\prime}-r}^{\tau-r}v^{-(\rho+2)}\mathop{}\\!\mathrm{d}v\mathop{}\\!\mathrm{d}r$
$\displaystyle\lesssim$
$\displaystyle\frac{1}{(\tau^{\prime}-u)^{\rho}}-\frac{1}{(\tau^{\prime}-s)^{\rho}}-\frac{1}{(\tau-u)^{\rho}}+\frac{1}{(\tau-s)^{\rho}}$

Again, this give the wanted result when recalling that one must replace
$\tau$,$\tau^{\prime}$,$u$ and $s$ by $a^{\alpha}\tau$,
$a^{\alpha}\tau^{\prime}$, $a^{\alpha}u$ and $a^{\alpha}s$. ∎

Let us gives the following useful and straightforward corollary for the action
of the fractional heat semi-group on weighted Besov spaces. Let us first
remind a rather elementary be useful lemma ([10], Lemma 4.4). For the sake of
the reader, we provide a full proof of it.

######  Lemma 44.

Let $\rho\geq 0$ and $\theta\in[0,1]$. There is a constant $c>0$ such that for
any $0<s\leq t$,