# The stochastic heat equation with multiplicative
Lévy noise: Existence, Moments, and Intermittency

Quentin Berger  LPSM, Sorbonne Université
UMR 8001 Campus Pierre et Marie Curie
Boîte courrier 158
4 Place Jussieu
75252 Paris Cedex 05
France<EMAIL_ADDRESS>,  Carsten Chong  Department of
Statistics
Columbia University
1255 Amsterdam Avenue
New York, NY 10027
USA<EMAIL_ADDRESS>and  Hubert Lacoin  IMPA
Institudo de Matemática Pura e Aplicada
Estrada Dona Castorina 110
Rio de Janeiro CEP-22460-320
Brazil<EMAIL_ADDRESS>
###### Abstract.

We study the stochastic heat equation (SHE) $\partial_{t}u=\frac{1}{2}\Delta
u+\beta u\xi$ driven by a multiplicative Lévy noise $\xi$ with positive jumps
and amplitude $\beta>0$, in arbitrary dimension $d\geq 1$. We prove the
existence of solutions under an optimal condition if $d=1,2$ and a close-to-
optimal condition if $d\geq 3$. Under an assumption that is general enough to
include stable noises, we further prove that the solution is unique. By
establishing tight moment bounds on the multiple Lévy integrals arising in the
chaos decomposition of $u$, we further show that the solution has finite $p$th
moments for $p>0$ whenever the noise does. Finally, for any $p>0$, we derive
upper and lower bounds on the moment Lyapunov exponents of order $p$ of the
solution, which are asymptotically sharp in the limit as $\beta\to 0$. One of
our most striking findings is that the solution to the SHE exhibits a property
called _strong intermittency_ (which implies moment intermittency of all
orders $p>1$ and pathwise mass concentration of the solution), for any non-
trivial Lévy measure, at any disorder intensity ${\beta}>0$, in any dimension
$d\geq 1$.

###### Key words and phrases:

continuum directed polymer, decoupling inequalities, Lyapunov exponents, non-
integer moments, parabolic Anderson model, partition function, size-biased
measure, stable noise

###### 2020 Mathematics Subject Classification:

Primary: 60H15, 82D60, 37H15; Secondary: 60K37, 60G51

###### Contents

  1. 1 Introduction
  2. 2 Main results
  3. 3 Extensions and consequences of our results
  4. 4 Preliminary tools
  5. 5 Qualitative properties of Lyapunov exponents
  6. 6 Moments of order $p>1$
  7. 7 Proof of Theorem 2.1 and upper bounds in Theorems 2.11 and 2.12
  8. 8 Existence and uniqueness of solutions to the SHE
  9. 9 Moments of order $p\in(0,1)$ and lower bounds in Theorems 2.11 and 2.12
  10. A Technical results

##  1\. Introduction

We consider the stochastic partial differential equation

$\partial_{t}u=\tfrac{1}{2}\Delta u+\beta u\xi\,,\qquad u(0,\cdot)=u_{0},$
(1.1)

where $\xi$ is a space-time Lévy noise, ${\beta}>0$ is an intensity parameter
and $u_{0}$ is some initial condition. In most parts of the paper, we assume
that $\xi$ is spectrally positive (i.e., only has positive jumps) without a
Gaussian part; extensions to the general case will be discussed in Section 3.3
below. The equation (1.1) is usually referred to as the _stochastic heat
equation (SHE)_ with multiplicative noise or the _parabolic Anderson model
(PAM)_ ; see [17, 28] for early works on the subject and [30, 34, 63, 69, 70,
71] for a selection of more recent contributions in the case where $\xi$ is
Gaussian. We opt for the SHE denomination since the expression parabolic
Anderson model is also often used in the literature to designate the equation

$\partial_{t}u=\tfrac{1}{2}\Delta u+\beta uV,$ (1.2)

where the multiplicative noise term $V$ does not depend on time (see [60, 62]
for examples of continuous models and [52, 54, 72] for lattice models). In
this case, Equation (1.2) is the real-valued analogue of the Schrödinger
equation associated with the Anderson Hamiltonian $H=\frac{1}{2}\Delta u+\beta
V$. A lattice version of this Schrödinger equation appears in one of the
original papers concerning Anderson localization [6, Equation (1)]; see also
[54, Equation (0.2)]. The study of the localization properties of $H$ (which
bears some connection to (1.2)) has been and remains an important area of
research in physics. We refer the reader to the reviews [80, 90] and the
references therein.

Compared to (1.2), Equation (1.1) corresponds to a related but different
problem: the diffusion of particles in a random medium that varies at small
time scales. This equation has been widely studied in the literature, firstly
and mostly in the case where $\xi$ is a space-time Gaussian white noise [17].
In this case, the equation can be related, via the Cole–Hopf transform, to the
KPZ equation, a model for the progression of growth fronts that has attracted
a lot of attention in the literature; see [5, 19, 61, 68] and the two review
papers [45, 86]. Furthermore, in the case of a Dirac initial condition, the
solution to (1.1) has been used to describe the scaling limit of a one-
dimensional directed polymer in a random environment [3, 4], called the
_continuum directed polymer model_. With Lévy noise, the existence of
solutions to (1.1) has only been proved under some additional integrability
assumptions on the Lévy measure; see [13, 36, 89] as well as Sections 1.2 and
1.3 below for a more detailed review. Recently, [14] further showed that
directed polymers in certain heavy-tailed environments have scaling limits
that, as we shall show below, agree with solutions to (1.1) with
${\alpha}$-stable $\xi$, providing a physical motivation for studying the SHE
with Lévy noise.

Another central point of interest in past studies of the SHE is the phenomenon
of localization of solutions, or _intermittency_ [28, 54, 96]. One striking
manisfestation is the appearance of sharp peaks at large times $t$ in the
random field $(u(t,x))_{x\in\mathbb{R}^{d}}$, where $u$ is the solution to
(1.1) with initial condition $u_{0}\equiv 1$. These peaks result from mass
concentration of $u(t,\cdot)$ on a fraction of space that decreases
exponentially in $t$. A common way to quantify this concentration property
(see Section 1.4 for more details) is to prove that for some $p<p^{\prime}$,
the ratio of moments $\mathbb{E}[u(t,x)^{p^{\prime}}]/\mathbb{E}[u(t,x)^{p}]$
grows exponentially in time, and to compute the growth rate. For the SHE with
Gaussian noise, this type of moment intermittency was studied in depth by, for
example, [17, 30, 33, 47, 69]. We also refer to [70, 71] for a path-by-path
analysis of the intermittency peaks. For the SHE with Lévy noise, the issue of
moment intermittency was investigated only much more recently in [38]. In
[39], it was further proved that unusually large peaks (on a logarithmic
scale) already appear in the solution to the SHE with an _additive_ Lévy
noise.

Before we describe our main results in Section 2, we provide a detailed
technical introduction to Equation (1.1), which aims at being as self-
contained as possible.

###  1.1. Lévy noise with positive jumps

Given a measure $\lambda$ on $(0,\infty)$ that satisfies
${\lambda}([1,\infty))<\infty$ and
$\int_{(0,1)}z^{2}\,\lambda(\mathrm{d}z)<\infty$, we provide a constructive
definition of a pure-jump Lévy noise with intensity ${\lambda}$. General Lévy
noises, with a Gaussian part or with negative jumps, will be considered in
Section 3.3. Let ${\omega}$ be a Poisson point process on
$\mathbb{R}\times\mathbb{R}^{d}\times(0,\infty)$ with intensity
$\nu:=\mathrm{d}t\otimes\mathrm{d}x\otimes\lambda(\mathrm{d}z)$. The law of
${\omega}$ is denoted by $\mathbb{P}$. For convenience, we sometimes split
${\omega}$ into two processes ${\omega}_{<}$ and ${\omega}_{\geq}$ that
correspond to the restriction of ${\omega}$ to
$\mathbb{R}\times\mathbb{R}^{d}\times(0,1)$ and
$\mathbb{R}\times\mathbb{R}^{d}\times[1,\infty)$, respectively. We let
$\mathbb{P}_{<}$ and $\mathbb{P}_{\geq}$ denote the associated probabilities,
so that $\mathbb{P}=\mathbb{P}_{<}\otimes\mathbb{P}_{\geq}$. For $a\in(0,1]$,
consider the measure

$\xi_{{\omega}}^{a}:=\sum_{(t,x,z)\in{\omega}}z\mathbf{1}_{\\{z\geq
a\\}}\delta_{(t,x)}-\kappa_{a}\mathcal{L}\,,$ (1.3)

where $\mathcal{L}$ is the Lebesgue measure on
$\mathbb{R}\times\mathbb{R}^{d}$ and
$\kappa_{a}=\int_{[a,1)}z\,\lambda(\mathrm{d}z)$. We also set

$\xi_{{\omega}_{<}}^{a}:=\sum_{(t,x,z)\in{\omega}}z\mathbf{1}_{\\{z\in[a,1)\\}}\delta_{(t,x)}-\kappa_{a}\mathcal{L}\qquad\text{and}\qquad\xi_{{\omega}_{\geq}}:=\sum_{(t,x,z)\in{\omega}}z\mathbf{1}_{\\{z\geq
1\\}}\delta_{(t,x)}\,.$ (1.4)

Under the assumption $\int_{(0,1)}z^{2}\,{\lambda}(\mathrm{d}z)<\infty$, the
measure $\xi_{{\omega}}^{a}$ converges a.s., in the local Sobolev space
$H_{\mathrm{loc}}^{-s}(\mathbb{R}\times\mathbb{R}^{d})$ for any $s>(1+d)/2$,
towards a limit $\xi_{{\omega}}\in
H_{\mathrm{loc}}^{-s}(\mathbb{R}\times\mathbb{R}^{d})$ (this is a standard
result, we refer to [13, Appendix A] for a proof of this exact statement). For
$k\geq 1$ and $u\in\mathbb{R}$, the Sobolev space $H^{u}(\mathbb{R}^{k})$ is
the Hilbert space of all Schwartz distributions
$\varphi\in{\mathcal{S}}^{\prime}(\mathbb{R}^{k})$ for which

$\lVert\varphi\rVert_{H^{u}(\mathbb{R}^{k})}:=\bigg{(}\int_{\mathbb{R}^{k}}(1+|\xi|^{2})^{u}|\hat{\varphi}(\xi)|^{2}\,\mathrm{d}\xi\bigg{)}^{\frac{1}{2}}<\infty,$

where $\hat{\varphi}$ denotes the Fourier transform of $\varphi$. When
$\varphi\in C^{\infty}_{c}(\mathbb{R}^{k})$ (i.e., $\varphi$ is smooth and
compactly supported), then $\hat{\varphi}$ is defined by
$\hat{\varphi}(\xi):=\int_{\mathbb{R}^{k}}e^{i\xi\cdot
x}\varphi(x)\,\mathrm{d}x.$ The local Sobolev space
$H^{u}_{\mathrm{loc}}(\mathbb{R}^{k})$ is then defined by

$H^{u}_{\mathrm{loc}}(\mathbb{R}^{k}):=\Big{\\{}\varphi\in{\mathcal{S}}^{\prime}(\mathbb{R}^{k})\
:\ \forall\rho\in C^{\infty}_{c}(\mathbb{R}^{k}),\ \rho\varphi\in
H^{u}(\mathbb{R}^{k})\Big{\\}}\,,$

equipped with the topology induced by the family of seminorms
$\\{\varphi\mapsto\|\rho\varphi\|_{H^{u}(\mathbb{R}^{k})}:\rho\in
C^{\infty}_{c}(\mathbb{R}^{k})\\}$. The random distribution $\xi_{{\omega}}$
is called a _Lévy (space-time white) noise_ with intensity or Lévy measure
$\lambda$. In the the case where $\lambda(\mathrm{d}x)=\alpha
x^{-(1+\alpha)}\,\mathrm{d}x$ for some $\alpha\in(0,2)$, $\xi_{\omega}$ is
referred to as an _$\alpha$ -stable noise_. We refer the reader to [7] and
[46] for more background on Lévy noises. In the remainder, we let
$\mathcal{F}:=(\mathcal{F}_{t})_{t\geq 0}$ be the completed natural filtration
associated with $\xi_{{\omega}}$, that is,

$\mathcal{F}_{t}:=\overline{\sigma\Big{(}{\omega}\cap([0,t]\times\mathbb{R}^{d}\times(0,\infty))\Big{)}}.$
(1.5)

######  Remark 1.1.

Note that the Poisson process ${\omega}$ is measurable with respect to
$\sigma(\xi_{{\omega}})$. Hence the solution $u$ to (1.1) that we construct in
this paper (which is defined in terms of ${\omega}$) is a function of the
noise $\xi$. Our choice to index by ${\omega}$ instead of $\xi$ is only for
convenience because many quantities in our proofs are easier to express in
terms of ${\omega}$ than in terms of $\xi$; in particular, this is not because
we need an extension of the probability space in our construction. We mention
this explicitly because there are martingale constructions which are similar
to the one used in this paper and that do require an extension of the
probability space: this is, for instance, the case in Kahane’s construction of
Gaussian multiplicative chaos associated with a kernel of $\sigma$-positive
type (see [66] for the seminal paper or [88, Section 2.1] for a review).

###  1.2. Mild solutions to SHE

A random field $(u(t,x))_{t>0,\ x\in\mathbb{R}^{d}}$ is called a _mild
solution_ to the SHE (1.1) with initial condition $u_{0}$ if for all $t>0$ and
$x\in\mathbb{R}^{d}$,

$u(t,x)=\int_{\mathbb{R}^{d}}\rho(t,x-y)u_{0}(\mathrm{d}y)+\beta\int^{t}_{0}\int_{\mathbb{R}^{d}}\rho(t-s,x-y)u(s,y)\,\xi(\mathrm{d}s,\mathrm{d}y)\qquad\text{a.s.}$
(1.6)

The statement includes the requirement that the integrals be well-defined and
finite. In (1.6),

$\rho(t,x):=(2\pi t)^{-\frac{d}{2}}e^{-\frac{\|x\|^{2}}{2t}}$

is the $d$-dimensional heat kernel and $\lVert\cdot\rVert$ denotes the
Euclidean norm on $\mathbb{R}^{d}$.

######  Remark 1.2.

If $\int_{(0,1)}z\,{\lambda}(\mathrm{d}z)=\infty$, the noise $\xi$ does not
have a locally finite total variation. In this case, the stochastic integral
on the right-hand side of (1.6) cannot be defined as a path-by-path Lebesgue
integral (i.e., an integral with respect to a random measure defined on
$[0,t]\times\mathbb{R}^{d}$) and must be interpreted in Itô’s sense.

In [89], it was shown that (1.1) has a unique mild solution if the Lévy
measure ${\lambda}$ satisfies

$\exists\,q\in(1,1+\tfrac{2}{d})\ :\
\int_{(0,\infty)}z^{q}\,{\lambda}(\mathrm{d}z)<\infty\,.$ (1.7)

While this condition covers many examples, it is not fully satisfying, in
particular because it rules out ${\alpha}$-stable noises for all
${\alpha}\in(0,2)$. Subsequent papers that considered the case of
$\alpha$-stable noise (e.g., [9, 13, 36, 37]) neither established uniqueness
nor the finiteness of moments of the solution to (1.1). At this point, let us
also mention [81, 83], who investigated an SHE with stable noise but with a
non-Lipschitz nonlinearity.

###  1.3. The truncation approach to SHE

Prior to this article, the most general existence condition for the solution
to the SHE (1.1) was obtained, for a Lévy noise with positive jumps, in [13],
but using a (possibly) weaker notion of solution than (1.6). The approach in
[13] consists in solving the equation

$\partial_{t}u^{a}=\tfrac{1}{2}\Delta u^{a}+\beta
u^{a}\xi^{a}_{\omega}\,,\qquad u^{a}(0,\cdot)=u_{0},$ (1.8)

with $\xi^{a}_{{\omega}}$ from (1.3) and then taking the limit of $u^{a}$ as
$a\downarrow 0$.

######  Remark 1.3.

In [13], it is not proved that the obtained limit satisfies (1.6). In fact,
the main focus of [13] is not the SHE, but rather constructing a continuum
path measure on $\mathbb{R}^{d}$: _the continuum directed polymer in Lévy
noise_. This is further discussed in Section 3.1 below.

In order to review the results of [13], let us start by introducing some
important notations and quantities. For $0<s<t<\infty$,
$x,y\in\mathbb{R}^{d}$, let
${\mathfrak{X}}_{k}(s,t):=\\{(t_{1},\ldots,t_{k})\in\mathbb{R}^{k}:s<t_{1}<\cdots<t_{k}<t\\}$
denote the $k$-dimensional simplex delimited by $s$ and $t$ and define, for
$\boldsymbol{t}\in{\mathfrak{X}}_{k}(s,t)$ and
$\boldsymbol{x}\in(\mathbb{R}^{d})^{k}$,

$\rho_{s,x;t,y}(\boldsymbol{t},\boldsymbol{x})=\prod_{i=1}^{k+1}\rho(\Delta
t_{i},\Delta x_{i})\,,$ (1.9)

where

$\Delta t_{i}:=t_{i}-t_{i-1}\qquad\text{and}\qquad\Delta
x_{i}:=x_{i}-x_{i-1}\,,$ (1.10)

with the convention that $t_{0}:=s$, $x_{0}:=x$ and $t_{k+1}:=t$,
$x_{k+1}:=y$. Given $a>0$ we let $|\xi_{{\omega}}^{a}|$ and
$\xi_{{\omega}}^{a,+}$ denote the total variation and the positive part of
$\xi_{{\omega}}^{a}$ (considered as a measure), that is,

$|\xi_{{\omega}}^{a}|=\xi_{{\omega}}^{a}+2\kappa_{a}\mathcal{L}\qquad\text{and}\qquad\xi_{{\omega}}^{a,+}=\xi_{{\omega}}^{a}+\kappa_{a}\mathcal{L}.$
(1.11)

We define the point-to-point partition function associated with (1.8) and
truncated Lévy noise $\xi_{{\omega}}^{a}$ as

${\mathcal{Z}}_{{\beta}}^{{\omega},a}(s,x;t,y):=\rho(t-s,y-x)+\sum_{k=1}^{\infty}{\beta}^{k}\int_{{\mathfrak{X}}_{k}(s,t)\times(\mathbb{R}^{d})^{k}}\rho_{s,x;t,y}(\boldsymbol{t},\boldsymbol{x})\prod_{i=1}^{k}\xi_{{\omega}}^{a}(\mathrm{d}t_{i},\mathrm{d}x_{i}),$
(1.12)

under the assumption that the sum of integrals is absolutely convergent, that
is,

$\sum_{k=0}^{\infty}{\beta}^{k}\int_{{\mathfrak{X}}_{k}(s,t)\times(\mathbb{R}^{d})^{k}}\rho_{s,x;t,y}(\boldsymbol{t},\boldsymbol{x})\prod_{i=1}^{k}|\xi_{{\omega}}^{a}|(\mathrm{d}t_{i},\mathrm{d}x_{i})<\infty.$
(1.13)

It is proved in [13, Prop. 2.5 and 2.6] that (1.13) holds if and only if

$\int_{[1,\infty)}(\log z)^{\frac{d}{2}}\,\lambda(\mathrm{d}z)<\infty.$ (1.14)

The case where $x=0$ and $s=0$ is of particular interest to us, hence we
introduce the notational convention

${\mathcal{Z}}^{{\omega},a}_{\beta}(x;t,y):={\mathcal{Z}}^{{\omega},a}_{\beta}(0,x;t,y),\qquad{\mathcal{Z}}^{{\omega},a}_{\beta}(t,x):={\mathcal{Z}}^{{\omega},a}_{\beta}(0,0;t,x),$
(1.15)

which applies similarly to other quantities such as
${\mathfrak{X}}_{k}(t):={\mathfrak{X}}_{k}(0,t)$ and
$\rho_{t,x}(\boldsymbol{t},\boldsymbol{x}):=\rho_{0,0;t,x}(\boldsymbol{t},\boldsymbol{x})$.
When the condition (1.14) holds,
${\mathcal{Z}}^{{\omega},a}_{\beta}(\cdot,\cdot)$ is a mild solution solution
to the SHE with noise $\xi^{a}_{\omega}$ and initial condition $\delta_{0}$.
Indeed, from (1.13), the integrals are absolutely convergent, so (1.12) can be
rewritten as

${\mathcal{Z}}_{{\beta}}^{{\omega},a}(t,x)=\rho(t,x)+{\beta}\int_{0}^{t}\int_{\mathbb{R}^{d}}\rho(t-s,x-y){\mathcal{Z}}_{{\beta}}^{{\omega},a}(s,y)\,\xi_{{\omega}}^{a}(\mathrm{d}s,\mathrm{d}y)\,.$
(1.16)

In the same manner, ${\mathcal{Z}}_{{\beta}}^{{\omega},a}(y;\cdot,\cdot)$ is a
mild solution to the SHE with initial condition $\delta_{y}$ and noise
$\xi^{a}_{\omega}$, and given some uniformly bounded
$u_{0}:\mathbb{R}^{d}\to[0,\infty)$, the random field $u^{a}$ defined by

$u^{a}(t,x):=\int_{\mathbb{R}^{d}}u_{0}(y){\mathcal{Z}}_{{\beta}}^{{\omega},a}(y;t,x)\,\mathrm{d}y$
(1.17)

is a mild solution to the SHE with initial condition $u_{0}$ and noise
$\xi_{\omega}^{a}$; see [13, Prop. 2.19].

######  Remark 1.4.

Note that by translation invariance, we have

$\begin{split}\Big{(}{\mathcal{Z}}^{{\omega},a}_{\beta}(s,x;t,y)\Big{)}_{y\in\mathbb{R}^{d}}&\stackrel{{\scriptstyle(d)}}{{=}}\Big{(}{\mathcal{Z}}^{{\omega},a}_{\beta}(t-s,y-x)\Big{)}_{y\in\mathbb{R}^{d}}\stackrel{{\scriptstyle(d)}}{{=}}\bigg{(}e^{-\frac{\|x-y\|^{2}+\|y\|^{2}}{2(t-s)}}{\mathcal{Z}}^{{\omega},a}_{\beta}(t-s,y)\bigg{)}_{y\in\mathbb{R}^{d}}.\end{split}$
(1.18)

For this reason, we present most results only for the case $s=0$ and $x=0$,
without loss of generality.

The next step is to investigate existence and “relevance” of the limit
${\mathcal{Z}}^{{\omega},a}_{\beta}(t,x)$ as $a\downarrow 0$. If (1.14) holds,
integrating the “Lebesgue part” of $\xi_{{\omega}}^{a}$ (see [13, Prop. 2.15]
for details), we obtain the alternative expression

${\mathcal{Z}}_{{\beta}}^{{\omega},a}(t,x)\\\
=e^{-\beta\kappa_{a}t}\left(\rho(t,x)+\sum_{k=1}^{\infty}{\beta}^{k}\int_{{\mathfrak{X}}_{k}(t)\times(\mathbb{R}^{d})^{k}}\rho_{t,x}(\boldsymbol{t},\boldsymbol{x})\prod_{i=1}^{k}\xi_{{\omega}}^{a,+}(\mathrm{d}t_{i},\mathrm{d}x_{i})\right),$
(1.19)

which entails the positivity of ${\mathcal{Z}}_{{\beta}}^{{\omega},a}(t,x)$.
Still under (1.14), considering the reverse filtration
$\mathcal{G}:=(\mathcal{G}_{a})_{a\in(0,1)}$ defined by

$\mathcal{G}_{a}:=\sigma(\xi^{a}_{\omega})=\sigma({\omega}\cap(\mathbb{R}\times\mathbb{R}^{d}\times[a,\infty))),$
(1.20)

one can observe that for almost every realization of ${\omega}_{\geq}$,
$({\mathcal{Z}}_{\beta}^{{\omega},a}(t,x))_{a\in(0,1]}$ is a non-negative
càdlàg time-reversed $\mathbb{P}_{<}$-martingale with respect to $\mathcal{G}$
(cf. [13, Lemma 3.5]) and thus admits a limit as $a\downarrow 0$. To determine
whether $\lim_{a\to 0}{\mathcal{Z}}_{\beta}^{{\omega},a}(t,x)$ is a good
candidate for being a solution to (1.6), a first step is to determine whether
this limit is degenerate or not. The answer depends on the intensity measure
${\lambda}$ and the following result summarizes the main findings of [13, Thm.
2.7 & Prop. 2.10–2.15].

######  Theorem A.

Assume that ${\lambda}$ satisfies the condition in (1.14).

  * (i)

If, in addition,

$\begin{cases}\displaystyle\int_{(0,1)}z^{2}\,\lambda(\mathrm{d}z)<\infty\
\quad\ &\text{if }d=1,\\\ \exists\,p\in\big{(}1,1+\frac{2}{d}\big{)}\ :\
\displaystyle\int_{(0,1)}z^{p}\,\lambda(\mathrm{d}z)<\infty&\text{if }d\geq
2,\end{cases}$ (1.21)

then for every $0<s<t<\infty$ and $x,y\in\mathbb{R}^{d}$, we have

${\mathcal{Z}}_{{\beta}}^{{\omega}}(s,x;t,y):=\lim_{a\to
0}{\mathcal{Z}}_{{\beta}}^{{\omega},a}(s,x;t,y)>0\qquad\text{$\mathbb{P}$-a.s.}$
(1.22)

Moreover, the convergence holds in $L^{1}(\mathbb{P}_{<})$ for
$\mathbb{P}_{\geq}$-a.e. realization of ${\omega}_{\geq}$, that is,

$\lim_{a\to
0}\mathbb{E}_{<}\Big{[}\Big{|}{\mathcal{Z}}_{\beta}^{{\omega}}(s,x;t,y)-{\mathcal{Z}}_{\beta}^{{\omega},a}(s,x;t,y)\Big{|}\Big{]}=0\qquad\text{$\mathbb{P}_{\geq}$-a.s.}$

  * (ii)

If, on the other hand, we have

$\begin{cases}\displaystyle\int_{(0,1)}z^{2}\,\lambda(\mathrm{d}z)=\infty\
\quad\ &\text{if }d=1,\\\ \displaystyle\int_{(0,1)}z^{2}\lvert\log
z\rvert\,\lambda(\mathrm{d}z)=\infty\ \quad\ &\text{if }d=2,\\\
\displaystyle\int_{(0,1)}z^{1+\frac{2}{d}}\,\lambda(\mathrm{d}z)=\infty&\text{if
}d\geq 3,\end{cases}$ (1.23)

then $\mathbb{P}$-a.s., we have $\lim_{a\to
0}{\mathcal{Z}}_{\beta}^{{\omega},a}(s,x;t,y)=0$.

One can also consider a free-end (or point-to-line) version of the partition
function with truncated Lévy noise, given by

${\mathcal{Z}}_{{\beta}}^{{\omega},a}(s,x;t,\ast):=\int_{\mathbb{R}^{d}}{\mathcal{Z}}^{{\omega},a}_{\beta}(s,x;t,y)\,\mathrm{d}y.$
(1.24)

Theorem A applies to ${\mathcal{Z}}_{{\beta}}^{{\omega},a}(s,x;t,\ast)$
mutatis mutandis. Following the convention (1.15), we use the notations
${\mathcal{Z}}^{{\omega}}_{{\beta}}(s,x;t,\ast)$,
${\mathcal{Z}}^{{\omega}}_{{\beta}}(x;t,\ast)$ and
${\mathcal{Z}}^{{\omega}}_{{\beta}}(t,\ast)$ for the limits we obtain as $a$
tends to zero (as a consequence of the reverse martingale property).
Furthermore, under the assumptions of Theorem 2.1, [13, Prop. 2.20] showed
that $\lim_{a\to 0}u^{a}(t,x)$ exists a.s. for $u^{a}$ defined in (1.17),
without proving that the limit is a solution to (1.6). Since moments of the
measure ${\lambda}$ play an important role in our assumptions, we introduce a
notation for partial moments of the measure ${\lambda}$ by setting

$\mu_{a,b}(p):=\int_{[a,b)}z^{p}\,\lambda(\mathrm{d}z)\,,$ (1.25)

for $0\leq a\leq b\leq\infty$. We simply write $\mu:=\mu_{1,\infty}(1)$ for
the first moment restricted to $[1,\infty)$.

######  Remark 1.5.

In the case of a finite mean $\mu$, given $t>0$ and $x\in\mathbb{R}$, the
convergence of both ${\mathcal{Z}}_{\beta}^{{\omega},a}(t,x)$ and
${\mathcal{Z}}_{\beta}^{{\omega},a}(t,\ast)$ as $a\to 0$ also holds in
$L^{1}(\mathbb{P})$.

If $\mu<\infty$, it is convenient to consider the normalized partition
functions, defined as

$\bar{\mathcal{Z}}^{{\omega},a}_{\beta}(t,x):=e^{-\beta\mu
t}{\mathcal{Z}}^{{\omega},a}_{\beta}(t,x),\qquad\bar{\mathcal{Z}}^{{\omega},a}_{\beta}(t,\ast):=e^{-\beta\mu
t}{\mathcal{Z}}^{{\omega},a}_{\beta}(t,\ast).$ (1.26)

In the same manner, we can center the noise by setting

$\bar{\xi}^{a}_{{\omega}}:=\xi^{a}_{{\omega}}-\mu\mathcal{L},$ (1.27)

which has the effect that integrals with respect to $\bar{\xi}^{a}_{{\omega}}$
have mean zero. Note that we have

$\bar{\mathcal{Z}}^{{\omega},a}_{\beta}(t,x)=\rho(t,x)+\sum_{k=1}^{\infty}{\beta}^{k}\int_{{\mathfrak{X}}_{k}(t)\times(\mathbb{R}^{d})^{k}}\rho_{t,x}(\boldsymbol{t},\boldsymbol{x})\prod_{i=1}^{k}\bar{\xi}_{{\omega}}^{a}(\mathrm{d}t_{i},\mathrm{d}x_{i})$
(1.28)

and a similar identity for $\bar{\mathcal{Z}}^{{\omega},a}_{\beta}(t,\ast)$.

###  1.4. Intermittency and related notions

Consider a random field $(v(t,x))_{t>0,x\in\mathbb{R}^{d}}$ where $d\geq 0$;
when $d=0$ this means that $(v(t))_{t>0}$ is a stochastic process only indexed
by $t$. Assuming that the _moment Lyapunov exponents_

${\gamma}(p)=\gamma(v,p):=\lim_{t\to\infty}\frac{1}{t}\log\mathbb{E}\Big{[}\lvert
v(t,x)\rvert^{p}\Big{]}$ (1.29)

exist on the extended real line for all $p\in(0,\infty)$ and are independent
of $x\in\mathbb{R}^{d}$, we let

$I=I(v):=\\{p\geq 0\ :\ {\gamma}(v,p)<\infty\\},$ (1.30)

with the convention ${\gamma}(0)=0$. Further assuming
$\lvert{\gamma}(1)\rvert<\infty$ in the following, we define the _normalized
moment Lyapunov exponents_ of $v$ by

$\bar{\gamma}(p):=\gamma(p)-p\,\gamma(1).$ (1.31)

Clearly, $\bar{\gamma}(1)=0$. Following the terminology of [28, 54, 96], if
$p>1$ and $\lvert{\gamma}(1)\rvert<\infty$, we say that $v$ exhibits _(moment)
intermittency of order $p$_ if $\bar{\gamma}(p)\in(0,\infty)$. We say that $v$
exhibits _full intermittency_ if $v$ is intermittent of all orders
$p\in(1,\infty)\cap I$.

Let us also introduce some new terminology when we consider moments of order
$p\in(0,1)$: we say that $v$ exhibits _strong intermittency_ (in analogy with
“very strong disorder” used in the directed polymer context, see the
discussion below) if $\bar{\gamma}(p)\in(-\infty,0)$ for some $p\in(0,1)$. As
$p\mapsto\bar{\gamma}(p)$ is convex on $I$ (cf. Proposition 2.9 (i) below),
strong intermittency implies $\bar{\gamma}(p)<0$ for all $p\in(0,1)$ but also
that $\bar{\gamma}(p)>0$ for all $p>1$. Thus, strong intermittency implies
full intermittency. As Section 3.2 below reveals, strong intermittency, plus
some ergodic properties in $x$, yields a _geometric characterization_ of
intermittency: we have a mass concentration of the paths of $v$ at large
times, characterized by the appearance of exponentially large peaks on islands
covering only an exponentially small fraction of space (a more quantitative
study of this phenomenon was undertaken in [70, 71], and we also refer to [53]
for a similar work concerning the parabolic Anderson model on
$\mathbb{Z}^{d}$).

#### Intermittency for the SHE

Let us first discuss the known intermittency result in a semi-discrete setting
(this is the setup where the most is known). For the SHE on $\mathbb{Z}^{d}$
with either a Gaussian or a finite-variance Lévy noise, the results of [1, 2,
28] and [16] show that

  * •

if $d=1,2$, then strong intermittency holds for every $\beta>0$;

  * •

if $d\geq 3$, then strong intermittency holds if and only if
$\beta>\bar{\beta}_{c}$ for some $\bar{\beta}_{c}>0$.

For the SHE on $\mathbb{R}$ driven by a multiplicative Gaussian white noise
$\xi$, the analysis of intermittency has a long history: if $\xi$ has variance
$\beta^{2}$ and $u_{0}\equiv 1$, the authors of [17] derived the formula

${\gamma}(p)={\gamma}(u,p)=\frac{p(p^{2}-1)}{24}\beta^{4}$ (1.32)

for $p\in\mathbb{N}$ and proved it for $p=2$. Later, a proof of (1.32) was
given in [30] and [78] for all integers $p\in\mathbb{N}$ and all real numbers
$p\geq 2$, respectively. The formula (1.32) for all $p>0$ (and in particular,
strong intermittency) was only established recently in [47, 56] using
integrable probability methods. For $d\geq 2$, there is no notion of solution
to the SHE with a Gaussian space-time white noise (cf. Section 3.3). For the
SHE on $\mathbb{R}^{d}$ driven by a Gaussian noise that is white in time and
colored in space with, say, a compactly supported correlation function, one
has a similar picture to the discrete-space setting: strong intermittency
always holds if $d=1,2$ and only for large $\beta$ if $d\geq 3$; see [32, 75].
The situation may be different if the noise has long-range spatial
correlation, as shown in [32, 51, 75].

For the solution to the Lévy-driven SHE, under the assumption (1.7), [38]
established intermittency of order $p$ in the following cases: for all
$p\in(1,3)$ if $d=1$; for $p$ close enough to (but smaller than)
$1+\frac{2}{d}$ if $d\geq 2$; for any fixed $p\in(1,1+\frac{2}{d})$ in any
dimension $d\geq 2$ if $\beta>\beta_{p,d}$ for some $\beta_{p,d}>0$. Whether
strong intermittency (or full intermittency in dimensions $d\geq 2$) holds in
the Lévy case or not, has been an open problem so far.

#### Very strong disorder for the directed polymer model

To complete the former discussion, let us mention some results that have been
proved for the directed polymer in a random environment (DPRE), whose
partition function formally corresponds to the solution to an SHE in discrete
space and time. In the context of the directed polymer model, the notion
equivalent to strong intermittency is that of very strong disorder (see for
instance [4, 27, 74]). For cultural reasons (DPRE is a statistical mechanics
model), very strong disorder is a property of the _free energy_ , that is, of
the asymptotic behavior of $\mathbb{E}[\log Z_{\beta}(N,\ast)]$, rather than a
property of the moments of order $p\in(0,1)$ of the partition function; but
this is not relevant for the present discussion. Very strong disorder has been
proved to hold for directed polymers in a various settings:

  * •

When $d=1$ [43] and $d=2$ [74] for any $\beta>0$;

  * •

When $d\geq 3$, if the environment has a power-law distribution with exponent
$\alpha\in(1,1+\frac{2}{d}]$, for any $\beta>0$, in [93] (this roughly
corresponds to $\alpha$-stable noise in the SHE context).

The method used in [74] does not rely much on the discrete nature on the model
and has been adapted to prove analogous results when either space or time are
continuous: Let us mention the case of the SHE on $\mathbb{Z}^{d}$ with
Gaussian white noise [16], the SHE on $\mathbb{R}^{d}$ with a Gaussian noise
which is white in time but colored in space [75, Theorem 1.2], or directed
polymers in a Poisson environment [40, Remark 3.4.3] (the partition function
of which corresponds to that of the SHE with a spatially convoluted Lévy
noise). In fact, the techniques can also be adapted to the SHE with Lévy
noise, and the content of Section 9.2 in the present paper is strongly
inspired by the proofs in [74] (for $d=1$) and [93] (for the case of heavy-
tailed noise).

On the other hand there are other situations where the directed polymer model
does not display very strong disorder:

  * •

When $d\geq 3$, if the environment has bounded second moment and $\beta$ is
small [22, 64];

  * •

When $d\geq 3$, if the environment has a power-law distribution with exponent
$\alpha>1+\frac{2}{d}$ and $\beta$ is sufficiently small [93].

In both cases, it has been shown that the directed polymer displays weak
disorder, a property which implies but goes beyond the absence of very strong
disorder. Weak disorder implies, for instance, that the sample paths drawn
from the polymer measure have diffusive behavior [44].

For this reason, the question of whether strong intermittency holds for the
SHE with Lévy space-time white noise is particularly intriguing when $d\geq 3$
and $\beta$ is small.

##  2\. Main results

Our main results can be summarized as follows:

$\bullet$ Firstly, we considerably reduce the gap conditions (1.21) and (1.23)
leave regarding the non-degeneracy of ${\mathcal{Z}}^{{\omega}}_{\beta}(t,x)$
if $d\geq 2$. We obtain, in Theorem 2.1 and Proposition 2.2, a necessary and
sufficient condition in dimension $d=2$ and a close-to-optimal condition in
dimensions $d\geq 3$. We further prove the $L^{p}$-convergence of
${\mathcal{Z}}^{{\omega},a}_{\beta}(t,x)$ to
${\mathcal{Z}}^{{\omega}}_{\beta}(t,x)$ for all $p\in(1,1+\frac{2}{d})$ as
soon as the noise $\xi_{\omega}$ has a finite $p$th moment (i.e.,
$\mu_{1,\infty}(p)<\infty$).

$\bullet$ Secondly, in Theorem 2.5, we show that the limit
${\mathcal{Z}}^{{\omega}}_{\beta}(t,x)$, when non-degenerate, is indeed a mild
solution to the SHE with initial condition $\delta_{0}$. Theorem 2.7 further
establishes the uniqueness of solutions under conditions that are general
enough to include the case of ${\alpha}$-stable noise.

$\bullet$ Thirdly, after establishing the existence of the moment Lyapunov
exponents of ${\mathcal{Z}}^{{\omega}}_{\beta}(t,\ast)$ and
${\mathcal{Z}}^{{\omega}}_{\beta}(t,x)$ in Proposition 2.8, we show our most
striking result in Theorem 2.10: the solution to the SHE with a multiplicative
Lévy noise and $\delta_{0}$-initial condition exhibits strong
intermittency—and thus, in particular, full intermittency—for any non-trivial
environment with finite expectation, that is, for any ${\beta}>0$, in any
dimension $d\geq 1$, for any non-trivial intensity measure ${\lambda}$ with
$\mu<\infty$. In Theorems 2.11 and 2.12, we further complement this result by
deriving sharp asymptotic estimates on the moment Lyapunov exponents as
$\beta$ tends to $0$.

The proof of the aforementioned results relies on two main methodological
achievements:

$\bullet$ By combining decoupling inequalities with an iterative partition of
${\mathfrak{X}}_{k}(t)\times(0,\infty)^{k}$ (the integration domain of
$(\boldsymbol{t},\boldsymbol{z})$), we establish, in Propositions 6.1 and 6.3,
sharp moment bounds of order $p>1$ on the multiple Lévy integrals that arise
in the series representation (1.28) of
$\bar{\mathcal{Z}}^{{\omega},a}_{\beta}(t,x)$. From these, we will then derive
the upper bounds for $p>1$ and, by convexity, the lower bounds for $p<1$ in
Theorems 2.11 and 2.12. The proof of Theorems 2.1, 2.5 and 2.7 will also rely
on (variants of) these moment bounds.

$\bullet$ By combining a change-of-measure technique with a coarse-graining
approach, summarized in Lemma 9.1, we obtain moment upper bounds of order
$p\in(0,1)$ for ${\mathcal{Z}}^{{\omega}}_{\beta}(t,\ast)$ which are then used
to show Theorem 2.10 as well as the lower bounds in Theorems 2.11 and 2.12.
Moreover, we apply a simpler version of this method to prove Proposition 2.2.
While the coarse-graining and change-of-measure approach is derived from that
used in [74], which itself was inspired from earlier work on disordered
pinning [50, 58, 91], its implementation to prove Proposition 9.5 (which is
the most important part of the proof of Theorem 2.10) and Proposition 2.2
relies on important novel ideas, which we discuss in more details at the
beginning of Section 9).

###  2.1. Optimal conditions for the non-degeneracy of
${\mathcal{Z}}^{{\omega}}_{\beta}(t,x)$

When $d\geq 2$, Theorem A displays a gap between the sufficient condition
(1.21) and the necessary condition (1.23) for the non-degeneracy of
${\mathcal{Z}}^{{\omega}}_{\beta}(s,x;t,y)$. Our first result reduces this gap
when $d\geq 3$ and identifies the necessary and sufficient condition in
dimension $d=2$. Our new sufficient condition reads as follows:

$\begin{cases}\displaystyle\int_{(0,1)}z^{2}\,\lambda(\mathrm{d}z)<\infty\
\quad\ &\text{if }d=1\,,\\\
\displaystyle\int_{(0,1)}z^{1+\frac{2}{d}}\lvert\log
z\rvert\,\lambda(\mathrm{d}z)<\infty&\text{if }d\geq 2\,.\end{cases}$ (2.1)

For later reference, we also included the case $d=1$, which is identical to
(1.21).

######  Theorem 2.1 (Non-degeneracy).

Let $d\geq 1$ and assume that (1.14) and (2.1) hold. Then, for any $t$ and
$x$, we have

${\mathcal{Z}}_{\beta}^{{\omega}}(t,x):=\lim_{a\to
0}{\mathcal{Z}}_{\beta}^{{\omega},a}(t,x)>0\qquad\text{$\mathbb{P}$-a.s.}$
(2.2)

In addition, for any $p\in[1,1+\frac{2}{d})$, we have

$\lim_{a\to
0}\mathbb{E}_{<}\Big{[}\Big{|}{\mathcal{Z}}_{\beta}^{{\omega},a}(t,x)-{\mathcal{Z}}_{\beta}^{{\omega}}(t,x)\Big{|}^{p}\Big{]}=0\qquad\text{$\mathbb{P}_{\geq}$-a.s.}$
(2.3)

If furthermore $\mu_{1,\infty}(p)<\infty$, then

$\lim_{a\to
0}\mathbb{E}\Big{[}\Big{|}{\mathcal{Z}}_{\beta}^{{\omega},a}(t,x)-{\mathcal{Z}}_{\beta}^{{\omega}}(t,x)\Big{|}^{p}\Big{]}=0.$
(2.4)

The same convergence results hold for the free-end partition function
${\mathcal{Z}}^{\omega}_{\beta}(t,\ast)$.

Together with (1.23), this shows that (2.1) is necessary and sufficient for
the non-degeneracy of ${\mathcal{Z}}_{\beta}^{{\omega}}(t,x)$ and
${\mathcal{Z}}^{\omega}_{\beta}(t,\ast)$ in dimension $d=2$. If $d\geq 3$, the
following result improves upon the necessary condition in (1.23).

######  Proposition 2.2 (Degeneracy).

Suppose that (1.14) holds and that $d\geq 3$. If for some ${\varepsilon}>0$,
we have that

$\int_{(0,e^{-e})}\,\frac{z^{1+\frac{2}{d}}\lvert\log z\rvert}{(\log\lvert\log
z\rvert)^{5+\frac{4}{d}+{\varepsilon}}}\,{\lambda}(\mathrm{d}z)=\infty\,,$
(2.5)

then for all $t>0$ and $x\in\mathbb{R}^{d}$, we have $\lim_{a\to
0}{\mathcal{Z}}_{\beta}^{{\omega},a}(t,x)=\lim_{a\to
0}{\mathcal{Z}}_{\beta}^{{\omega},a}(t,\ast)=0$ $\mathbb{P}$-a.s.

While there is still a small gap between (2.1) and (2.5) if $d\geq 3$, we
believe that (2.1) is necessary and sufficient for non-degeneracy.

######  Remark 2.3.

There is a small abuse of language when we say that we establish an almost
necessary and sufficient condition for existence of solutions to the SHE,
since Proposition 2.2 does not a priori exclude the existence of solutions
that are not given by limits of solutions with truncated noise.

######  Remark 2.4.

Besides replacing (1.21) by the weaker condition (1.23) and the inclusion of
$L^{p}$-convergence, Theorem 2.1 contains a third important improvement.
Contrary to Theorem A, the proof of (2.3), which is sufficient to establish
non-triviality of the limit, does not rely on the positivity of
${\mathcal{Z}}_{\beta}^{{\omega},a}(t,x)$ and can thus be directly adapted to
the case of signed noise, an observation that we will further elaborate on in
Section 3.3. On the other-hand, our proof of Proposition 2.2 strongly relies
on having only positive jumps.

###  2.2. Existence and uniqueness of solutions to SHE

Our next result shows that the point-to-point partition function
${\mathcal{Z}}^{\omega}_{\beta}(t,x)$ solves the SHE with Lévy noise
$\xi_{{\omega}}$ and initial condition $\delta_{0}$.

######  Theorem 2.5 (Existence).

Suppose that (1.14) and (2.1) hold.

  1. (i)

The point-to-point partition function
${\mathcal{Z}}_{{\beta}}^{{\omega}}(t,x)$ is a mild solution to the SHE (1.6)
with initial condition $\delta_{0}$, that is, for every
$(t,x)\in(0,\infty)\times\mathbb{R}^{d}$, the stochastic integral below is
well defined and the following identity holds a.s.:

${\mathcal{Z}}_{{\beta}}^{{\omega}}(t,x)=\rho(t,x)+{\beta}\int_{0}^{t}\int_{\mathbb{R}^{d}}\rho(t-s,x-y){\mathcal{Z}}^{\omega}_{{\beta}}(y,s)\,\xi_{{\omega}}(\mathrm{d}s,\mathrm{d}y)\,.$
(2.6)

  2. (ii)

If $u_{0}$ is a locally finite signed measure on $\mathbb{R}^{d}$ such that

$\limsup_{r\to\infty}r^{-2}\log\Big{(}\lvert
u_{0}\rvert([-r,r]^{d})\Big{)}<\frac{1}{2T}$ (2.7)

for some $T>0$, then

$v(t,x):=\int_{\mathbb{R}^{d}}{\mathcal{Z}}_{{\beta}}^{{\omega}}(y;t,x)\,u_{0}(\mathrm{d}y)$
(2.8)

is well defined and finite for all $(t,x)\in[0,T]\times\mathbb{R}^{d}$ and is
a solution to the SHE (1.6) with initial condition $u_{0}$, on
$[0,T]\times\mathbb{R}^{d}$.

######  Remark 2.6.

The proof presented in this paper actually implies that

${\mathcal{Z}}^{\omega}_{\beta}(t,x)=\rho(t,x)+\sum_{k=1}^{\infty}{\beta}^{k}\int_{{\mathfrak{X}}_{k}(t)\times(\mathbb{R}^{d})^{k}}\rho_{t,x}(\boldsymbol{t},\boldsymbol{x})\prod_{i=1}^{k}\xi_{{\omega}}(\mathrm{d}t_{i},\mathrm{d}x_{i}),$
(2.9)

where the right-hand side is well defined as a convergent sum of iterated
stochastic integrals (see Section 4.3 for the corresponding framework).

Under the stronger condition (1.21), we prove that the solution found above is
unique.

######  Theorem 2.7 (Uniqueness).

We assume that ${\lambda}$ satisfies (1.14) and (2.1) and that $u_{0}$ and $T$
satisfy (2.7). We let $(v(t,x))_{t\in[0,T],\ x\in\mathbb{R}^{d}}$ be the
solution to the SHE defined in (2.8). Then there exists $\theta>1$ such that
for all $p\in(1,1+\tfrac{2}{d})$, $t\in(0,T]$ and $x\in\mathbb{R}^{d}$, we
have

$\begin{cases}\mathbb{E}_{<}\Big{[}\lvert
v(t,x)\rvert^{p}\Big{]}<\infty\qquad&\text{$\mathbb{P}_{\geq}$-a.s.,}\\\
\displaystyle\int_{(0,t)\times\mathbb{R}^{d}}\rho(\theta(t-s),x)^{p}\mathbb{E}_{<}\Big{[}\lvert
v(s,x)\rvert^{p}\Big{]}\,\mathrm{d}s\,\mathrm{d}x<\infty\qquad&\text{$\mathbb{P}_{\geq}$-a.s.}\end{cases}$
(2.10)

Furthermore, if ${\lambda}$ satisfies (1.21), $v$ is the unique—up to
modifications—predictable random field that is a mild solution to the SHE
(1.6) and satisfies (2.10).

###  2.3. Lyapunov exponents and intermittency

Having established the finiteness of moments in Theorem 2.1, we now
investigate the growth rate of these moments as $t\to\infty$ and their
dependence on $\beta$. Our first task is to establish the existence of the
(moment) Lyapunov exponents.

######  Proposition 2.8.

Assume that (1.14) and (2.1) hold.

  1. (i)

If $p\in(0,1+\tfrac{2}{d})$ and $\mu_{1,\infty}(p)<\infty$, then the moment
Lyapunov exponent

$\gamma_{\beta}(p):={\gamma}({\mathcal{Z}}_{{\beta}}^{{\omega}}(t,\ast),p)=\lim_{t\to\infty}\frac{1}{t}\log\mathbb{E}\Big{[}{\mathcal{Z}}_{{\beta}}^{{\omega}}(t,\ast)^{p}\Big{]}$
(2.11)

exists and is finite.

  2. (ii)

The exponents $\gamma_{\beta}(p)$ also capture the growth of the point-to-
point partition function, in the sense that

$\lim_{t\to\infty}\frac{1}{t}\log\mathbb{E}\Big{[}{\mathcal{Z}}_{{\beta}}^{{\omega}}(t,0)^{p}\Big{]}=\gamma_{\beta}(p)\,.$
(2.12)

By [38, Theorem 3.1], we have
$\mathbb{E}[{\mathcal{Z}}_{{\beta}}^{{\omega}}(t,\ast)^{1+2/d}]=\mathbb{E}[{\mathcal{Z}}_{\beta}^{{\omega}}(t,0)^{1+2/d}]=\infty$
for any $t>0$ and non-trivial ${\lambda}$. We also have
$\mathbb{E}[{\mathcal{Z}}_{{\beta}}^{{\omega}}(t,\ast)^{p}]=\infty$ if
$\mu_{1,\infty}(p)=\infty$. Hence, $\gamma_{\beta}(p)$ is well defined
(possibly infinite) for every $p\geq 0$. Letting

$p_{\max}=p_{\max}({\lambda}):=\sup\Big{\\{}p\in[0,1+\tfrac{2}{d})\ :\
\mu_{1,\infty}(p)<\infty\Big{\\}}\,,$ (2.13)

we obtain from the above observation that the set

$I_{{\lambda}}:=I({\mathcal{Z}}_{{\beta}}^{{\omega}}(t,\ast))=\\{p\geq 0\ :\
\gamma_{\beta}(p)<\infty\\}$ (2.14)

is either $[0,p_{\max})$ or $[0,p_{\max}]$, the latter occurring if
$p_{\max}<1+\tfrac{2}{d}$ and $\mu_{1,\infty}(p_{\max})<\infty$.

The next result lists a few (partly classical) qualitative properties of
${\gamma}_{\beta}(p)$. If $\mu<\infty$ (in which case $1\in I_{{\lambda}}$),
we write

$\bar{\gamma}_{\beta}(p):=\bar{\gamma}({\mathcal{Z}}_{{\beta}}^{{\omega}}(t,\ast),p)=\gamma_{{\beta}}(p)-p\gamma_{\beta}(1)=\gamma_{\beta}(p)-p\mu={\gamma}(\bar{\mathcal{Z}}_{{\beta}}^{{\omega}}(t,\ast),p).$

Note that by Jensen’s inequality, we have $\bar{\gamma}_{\beta}(p)\geq 0$ for
$p>1$ and $\bar{\gamma}_{\beta}(p)\leq 0$ for $p<1$.

######  Proposition 2.9.

Under the assumptions of Proposition 2.8, we have the following:

  1. (i)

The map $p\mapsto{\gamma}_{\beta}(p)$ is convex on $I_{{\lambda}}$.

  2. (ii)

If $\mu<\infty$ and $p\in I_{{\lambda}}$, the map
${\beta}\mapsto\bar{\gamma}_{\beta}(p)$ is non-decreasing if $p\geq 1$ and
non-increasing if $p\in(0,1]$.

  3. (iii)

If $p_{\max}>1$, then $p\mapsto{\gamma}_{\beta}(p)/p$ is strictly increasing
on $\\{p\in I_{{\lambda}}:\gamma_{{\beta}}(p)-p\gamma_{\beta}(1)>0\\}$.

Our next goal is to understand under what conditions we have strong
intermittency. Based on the results reviewed in Section 1.4, one may
conjecture that the solution to the SHE on $\mathbb{R}^{d}$ driven by a Lévy
noise (that is white in time _and_ space) exhibits a similar behavior: strong
intermittency in dimensions $d=1,2$ and strong intermittency only for large
$\beta$ if $d\geq 3$. This turns out to be a fallacy: the SHE with a non-
trivial Lévy noise _always_ exhibits strong intermittency, irrespective of
$\beta$, $\lambda$, $d$ or $p$.

######  Theorem 2.10 (Strong intermittency).

Let $d\geq 1$. If ${\lambda}\not\equiv 0$, $\mu<\infty$ and $\beta>0$, we have
$\bar{\gamma}_{\beta}(p)<0$ for $p\in(0,1)$ and $\bar{\gamma}_{\beta}(p)>0$
for $p\in I_{{\lambda}}\cap(1,\infty)$.

To shed more light on this peculiar result, we further investigate the
detailed behavior of $\bar{\gamma}_{\beta}(p)$ in the small $\beta$ limit. Let
us start with the case of light-tailed noise at infinity.

######  Theorem 2.11 (Light-tailed noise).

Assume that (2.1) holds.

  * (i)

If $d=1$ and $\mu_{1,\infty}(2)<\infty$, then for every $p\in
I_{{\lambda}}\setminus\\{1\\}$, there exists $C_{p}\in(0,\infty)$ such that
for every $\beta\in(0,1]$, we have

$(C_{p})^{-1}\beta^{4}\leq\lvert\bar{\gamma}_{\beta}(p)\rvert\leq
C_{p}\beta^{4}.$ (2.15)

In the special case $p=2$, we have
$\bar{\gamma}_{\beta}(2)=\frac{1}{4}\mu_{0,\infty}(2)^{2}{\beta}^{4}$.

  * (ii)

If $d\geq 2$ and $\mu_{1,\infty}(1+\frac{2}{d})<\infty$, then for every
$p\in\big{(}0,1+\frac{2}{d}\big{)}\setminus\\{1\\}$, we have

$\lim_{\beta\to
0}\frac{\log|\log|\bar{\gamma}_{\beta}(p)||}{|\log\beta|}=1+\tfrac{2}{d}.$
(2.16)

If $d=1$ and $d=2$, these results match the asymptotics for the free energy of
the directed polymer in a random environment (see [84] for $d=1$ and [11] for
$d=2$). In dimensions $d\geq 3$, however, one has in the directed polymer
setting that $\bar{\gamma}_{{\beta}}(p)=0$ for sufficiently small ${\beta}$,
see [22] and [93, Prop. 4.1]. Note that (2.16) implies that
$|\bar{\gamma}_{\beta}(p)|$ vanishes very fast as $\beta$ goes to zero. As a
consequence, there is not a lot of margin to play with to prove that
$\lvert\bar{\gamma}_{{\beta}}(p)\rvert>0$ when $\beta$ is small, and the proof
indeed relies on a fine tuning of parameters. The behavior of
$\bar{\gamma}_{\beta}(p)$ is different if ${\lambda}$ displays an heavier tail
at infinity. In this case, $\bar{\gamma}_{\beta}(p)$ rather has a power-law
behavior close to $\beta=0$. Let us introduce

$\nu_{p}:=1-\tfrac{d}{2}(p-1).$ (2.17)

######  Theorem 2.12 (Heavy-tailed noise).

Assume that (2.1) holds and that $\mu<\infty$.

  * (i)

If $\mu_{1,\infty}(q)<\infty$ for some $q\in(1,\min(2,1+\frac{2}{d}))$, then
for every $p\in(0,q]\setminus\\{1\\}$, there exists $C_{p,q}\in(0,\infty)$
such that for every $\beta\in(0,1]$,

$\lvert\bar{\gamma}_{\beta}(p)\rvert\leq C_{p,q}\beta^{\frac{q}{\nu_{q}}}\,,$
(2.18)

which implies

$\liminf_{\beta\to
0}\frac{\log\lvert\bar{\gamma}_{\beta}(p)\rvert}{\log\beta}\geq\frac{q}{\nu_{q}}.$
(2.19)

  * (ii)

If $\mu_{1,\infty}(q)=\infty$ for some $q\in(1,\min(2,1+\frac{2}{d}))$, then
for every $p\in(0,q)\setminus\\{1\\}$,

$\limsup_{\beta\to
0}\frac{\log\lvert\bar{\gamma}_{\beta}(p)\rvert}{\log\beta}\leq\frac{q}{\nu_{q}}.$
(2.20)

In particular, if

$\lim_{z\to\infty}\frac{\log{\lambda}([z,\infty))}{\log z}=-\alpha$ (2.21)

for some $1<\alpha<\min(2,1+\frac{2}{d})$ (which includes the $\alpha$-stable
case), then for all $p\in(0,{\alpha})\setminus\\{1\\}$

$\lim_{\beta\to
0}\frac{\log\lvert\bar{\gamma}_{\beta}(p)\rvert}{\log\beta}=\frac{\alpha}{\nu_{\alpha}}\,.$
(2.22)

###  2.4. Overview of the rest of the paper

The remainder of the paper is organized as follows. In Section 3, we discuss
some extensions of our results and some related conjectures concerning
geometric localization. Section 4 introduces a few important tools that are
needed throughout the paper. In Section 5, we prove all basic properties
concerning the moment Lyapunov exponents, that is, Propositions 2.8 and 2.9.
Section 6 is devoted to proving our first main technical achievement:
Propositions 6.1 and 6.3, which contain upper estimates on moments of order
$p>1$ of ${\mathcal{Z}}^{{\omega}}_{\beta}(t,x)$. In Section 7, we then use
these moment bounds to derive Theorem 2.1 and the first halves of Theorems
2.11 and 2.12. The existence and uniqueness of solutions (i.e., Theorems 2.5
and 2.7) are addressed in Section 8. Finally, Section 9 gathers the proofs
that rely on a coarse-graining/change-of-measure approach summarized in Lemma
9.1. In particular, we show the remaining halves of Theorems 2.11 and 2.12 as
well as Proposition 2.2. The Appendix contains proofs of some of the results
introduced in Section 4.

##  3\. Extensions and consequences of our results

###  3.1. Continuum directed polymer model

As mentioned in Remark 1.3, the main purpose of introducing
$\mathcal{Z}^{{\omega},a}_{\beta}(t,\ast)$ in [13] is not to study the SHE but
rather to define a random probability measure on the space of continuous
functions

$C_{0}([0,T]):=\\{\theta:[0,T]\to\mathbb{R}^{d}\ :\ \theta\text{ continuous
and }\theta(0)=0\\},$

called the continuum directed polymer in a Lévy environment. For $T>0$ the
directed polymer in the truncated environment $\xi^{a}_{{\omega}}$ is the
probability measure $\mathbf{Q}^{{\omega},a}_{T,\beta}$ on $C_{0}([0,T])$
defined via the integral of bounded Borel measurable functions as follows:

$\mathbf{Q}^{{\omega},a}_{T,\beta}(f)=\frac{e^{-\beta\kappa_{a}t}}{{\mathcal{Z}}^{{\omega},a}_{\beta}(T,\ast)}\left(\mathbf{Q}\Big{[}f((B_{t})_{t\in[0,T]})\Big{]}+\sum_{k=1}^{\infty}{\beta}^{k}\int_{{\mathfrak{X}}_{k}(T)\times(\mathbb{R}^{d})^{k}}\rho(\boldsymbol{t},\boldsymbol{x},f)\prod_{i=1}^{k}\xi_{{\omega}}^{a,+}(\mathrm{d}t_{i},\mathrm{d}x_{i})\right),$
(3.1)

where
$\rho(\boldsymbol{t},\boldsymbol{x},f):=\mathbf{Q}[f((B_{t})_{t\in[0,T]})\ |\
\forall i\in\llbracket k\rrbracket:B_{t_{i}}=x_{i}]$ and $\llbracket
k\rrbracket:=\\{1,\dots,k\\}$. Here, $\mathbf{Q}$ denotes the distribution of
a standard $d$-dimensional Brownian motion and, with some light abuse of
notation, $\mathbf{Q}[\cdot\ |\ \forall i\in\llbracket
k\rrbracket:B_{t_{i}}=x_{i}]$ is the law of the concatenation of Brownian
bridges obtained by conditioning a Brownian motion on the null event
$\\{\forall i\in\llbracket k\rrbracket:B_{t_{i}}=x_{i}\\}$. The main result in
[13] is to prove that $\mathbf{Q}^{{\omega},a}_{T,\beta}$ converges in
distribution to a limit $\mathbf{Q}^{{\omega}}_{T,\beta}$ under the
assumptions (1.14) and (1.21). By the findings of the present paper, we can
replace the assumption (1.21) by (2.1) in the results of [13].

######  Theorem 3.1.

If the measure ${\lambda}$ satisfies (1.14) and (2.1), then there exists a
measure $\mathbf{Q}^{{\omega}}_{T,\beta}$ on $C_{0}([0,T])$ such that almost
surely, for every bounded continuous function $f$ in $C_{0}([0,T])$, we have

$\lim_{a\to
0}\mathbf{Q}^{{\omega},a}_{T,\beta}(f)=\mathbf{Q}^{{\omega}}_{T,\beta}(f).$
(3.2)

###### Proof.

The convergence of $\mathbf{Q}^{{\omega},a}_{T,\beta}(f)$ for a fixed $f$ is a
consequence of Theorem 2.1. Indeed, $\mathbf{Q}^{{\omega},a}_{T,\beta}(f)$ can
be written as a quotient
${\mathcal{Z}}^{{\omega},a}_{\beta}(T,f)/{\mathcal{Z}}^{{\omega},a}_{\beta}(T,\ast)$.
Theorem 2.1 asserts that the denominator converges to a positive limit and the
martingale argument below (1.20) entails the almost-sure convergence of
${\mathcal{Z}}^{{\omega},a}_{\beta}(T,f)$. The fact that convergence holds
simultaneously for all continuous bounded functions is a consequence of
tightness. The proof given in [13, Section 4.8] shows that the family of non-
normalized measures
$\\{{\mathcal{Z}}^{{\omega},a}_{\beta}(T,\ast)\times\mathbf{Q}^{{\omega},a}_{T,\beta}:a\in(0,1]\\}$
is tight, with no other assumption than (1.14). The tightness of
$\\{\mathbf{Q}^{{\omega},a}_{T,\beta}:a\in(0,1]\\}$ then follows from the
almost-sure positivity of ${\mathcal{Z}}^{{\omega},a}_{\beta}(T,\ast)$; cf.
(2.2). ∎

###  3.2. Geometric localization of the solution to the SHE

#### The case $u_{0}\equiv 1$.

The intermittency result given in Theorem 2.10 (especially the part concerning
strong intermittency) has direct implications on the distribution of mass for
the solution to (1.6) with initial condition $u_{0}\equiv 1$. To illustrate
this, let us present an argument from [17, §2.4], adapted to the case of non-
integer moments—the argument in [17, §2.4] is about mass concentration for the
square of the solution.

If $\mu<\infty$ and (2.1) holds, then the solution to (1.6) with $u_{0}\equiv
1$ is given by

$U(t,x):=\int_{\mathbb{R}^{d}}{\mathcal{Z}}^{{\omega}}_{\beta}(y;t,x)\,\mathrm{d}y.$

Let us define the normalized solution $\bar{U}(t,x):=e^{-\mu t}U(t,x)$. By
symmetry, we have that
$\bar{U}(t,x)\stackrel{{\scriptstyle(d)}}{{=}}\bar{\mathcal{Z}}^{{\omega}}_{\beta}(t,\ast)$
for any fixed $x$. For a fixed value of $t$, let us assume that the field
$\bar{U}(t,x)_{x\in\mathbb{R}^{d}}$ is ergodic (in the case of a Gaussian
noise, this was proved in [31]): we then get

$\lim_{R\to\infty}\frac{\int_{\|x\|\leq
R}\bar{U}(t,x)\,\mathrm{d}x}{\sigma_{d}R^{d}}=\mathbb{E}\Big{[}\bar{\mathcal{Z}}^{{\omega}}_{\beta}(t,\ast)\Big{]}=1\,,$
(3.3)

where $\sigma_{d}=\pi^{d/2}{\Gamma}(\tfrac{d}{2}+1)$ is the volume of the unit
ball in $\mathbb{R}^{d}$. On the other hand, if $p\in(0,1)$ and
$\bar{\gamma}_{\beta}(p)<0$ (which is ensured by Theorem 2.10), then for any
$\alpha\in(0,\frac{\bar{\gamma}_{\beta}(p)}{p-1})$ we have

$\begin{split}\lim_{R\to\infty}\frac{\int_{\|x\|\leq
R}\bar{U}(t,x)\mathbf{1}_{\\{\bar{U}(t,x)\leq e^{\alpha
t}\\}}\,\mathrm{d}x}{\sigma_{d}R^{d}}&=\mathbb{E}\Big{[}\bar{\mathcal{Z}}^{{\omega}}_{\beta}(t,\ast)\mathbf{1}_{\\{\bar{\mathcal{Z}}^{{\omega}}_{\beta}(t,\ast)\leq
e^{\alpha t}\\}}\Big{]}\\\ &\leq e^{(1-p)\alpha
t}\mathbb{E}[\bar{\mathcal{Z}}^{{\omega}}_{\beta}(t,\ast)^{p}]\leq
e^{[(1-p)\alpha+\bar{\gamma}_{\beta}(p)]t},\end{split}$ (3.4)

where the last inequality relies on super-multiplicativity, see Lemma 5.1
below. Setting $\delta:=\alpha(p-1)-\bar{\gamma}_{\beta}(p)>0$ and combining
(3.3) and (3.4), we obtain

$\lim_{R\to\infty}\frac{\int_{\|x\|\leq
R}\bar{U}(t,x)\mathbf{1}_{\\{\bar{U}(t,x)\geq e^{\alpha
t}\\}}\,\mathrm{d}x}{\int_{\|x\|\leq R}\bar{U}(t,x)\,\mathrm{d}x}\geq
1-e^{-\delta t}\,.$ (3.5)

Note that we also have

$\lim_{R\to\infty}\frac{\int_{\|x\|\leq R}\mathbf{1}_{\\{\bar{U}(t,x)\geq
e^{\alpha
t}\\}}\,\mathrm{d}x}{\sigma_{d}R^{d}}=\mathbb{P}\Big{(}\bar{\mathcal{Z}}^{{\omega}}_{\beta}(t,\ast)\geq
e^{\alpha t}\Big{)}\leq e^{-\alpha t}\,,$ (3.6)

thanks to Markov’s inequality. Hence, the two identities (3.5) and (3.6) show
that in the large $t$ limit, the mass of the solution concentrates on a very
small portion of space.

#### The case $u_{0}=\delta_{0}$.

Although this is much more difficult to prove rigorously, we believe that
intermittency in Theorem 2.10 also has implications on the localization of the
solution $\mathcal{Z}^{{\omega},a}_{\beta}(t,x)$ to the SHE with $\delta_{0}$
initial condition. Let us consider the probability measure on $\mathbb{R}^{d}$
given by

$P^{{\omega}}_{\beta,t}(\mathrm{d}x)=\frac{\mathcal{Z}^{{\omega},a}_{\beta}(t,x)}{\mathcal{Z}^{{\omega},a}_{\beta}(t,\ast)}\,\mathrm{d}x.$

When $\beta=0$, $P^{{\omega}}_{\beta,t}$ is simply $\rho(t,x)\,\mathrm{d}x$.
In this case, for large $t$, its mass is roughly homogeneously spread out on
the centered Euclidean ball of radius $t^{1/2}$. By contrast, if $\beta>0$,
the large time behavior of $P^{{\omega}}_{\beta,t}(\mathrm{d}x)$ is
conjectured to be different: the mass of $P^{{\omega}}_{\beta,t}$ should
typically be concentrated on a set of bounded volume (the volume here should
not depend on $t$).

######  Conjecture 1.

Given ${\varepsilon}>0$ and $\beta>0$, there exist two constants
$k=k({\varepsilon},\beta)\in\mathbb{N}$ and $R=R({\varepsilon},\beta)>0$ such
that for all $t\geq 0$,

$\mathbb{P}\Bigg{(}\exists z_{1},z_{2},\dots,z_{k}\in\mathbb{R}^{d}\ :\
P^{{\omega}}_{\beta,t}\bigg{(}\bigcup_{i=1}^{k}B(z_{i},R)\bigg{)}\geq
1-{\varepsilon}\Bigg{)}\geq 1-{\varepsilon},$

where $B(z,R):=\\{x\in\mathbb{R}^{d}:\lVert x-z\rVert<R\\}$.

A weaker version of the above conjecture is that localization holds in Cesàro
mean, that is, for ${\varepsilon}>0$ and $\beta>0$ fixed, there exist two
constants $k=k({\varepsilon},\beta)\in\mathbb{N}$ and
$R=R({\varepsilon},\beta)>0$ such that

$\liminf_{T\to\infty}\frac{1}{T}\int^{T}_{0}\max_{z_{1},\dots,z_{k}\in\mathbb{R}^{d}}P^{{\omega}}_{\beta,t}\bigg{(}\bigcup_{i=1}^{k}B(z_{i},R)\bigg{)}\,\mathrm{d}t\geq
1-{\varepsilon}.$ (3.7)

In fact, such statements have been rigorously proved for a discrete analogue
of the SHE: the partition function of the discrete polymer in
$\mathbb{Z}^{d}$. In this setup, the link between very strong disorder and
localization in the Cesàro sense was first explored in [26, 42]. More
recently, in [10], it was shown that for directed polymers, very strong
disorder implies (3.7); the result was extended to continuous space and
discrete time in [8]. The approach in [8, 10] is to prove that the measure
$\frac{1}{T}\int^{T}_{0}P^{{\omega}}_{\beta,t}\,\mathrm{d}t$ converges to a
limiting object, using a specific topology (introduced in [82]) on the space
of finite measures on $\mathbb{R}^{d}$. The corresponding localization result
has also been shown for a continuum directed polymer model (and SHE) with
spatially convoluted Gaussian white noise in [23], using a Gaussian
multiplicative chaos approach.

While many details of the proofs presented in [8, 10] use the discrete nature
of the polymer model, the general ideas the proof is based on do not seem to
rely on it. For this reason, we believe that while there are technical
challenges to be overcome, these proofs could in principle be adapted to the
continuum case, to show that (3.7) holds whenever $\bar{\gamma}_{\beta}(p)<0$
for $p\in(0,1)$ and hence, by Theorem 2.10, for any $\beta>0$ as soon as
$\mu<\infty$.

######  Remark 3.2.

Note that the localization result in [10] holds for every $\beta$ in
dimensions $1$ and $2$ but only for $\beta$ above a certain threshold in
dimensions $d\geq 3$. This is because for the directed polymer, the analogue
of our Theorem 2.10 does not hold. Quite the contrary, if $d\geq 3$ and
$\beta$ is small, the end-point distribution of the directed polymer satisfies
the central limit theorem [22, 44, 64].

######  Remark 3.3.

The above discussion gives a justification of Conjecture 1 in the case when
$\mu<\infty$ (which implies that $\mathcal{Z}^{{\omega}}_{\beta}(t,\ast)$ has
a finite expectation). As a general rule, an environment with heavier tail is
expected to only increase geometric localization (cf. [92]).

######  Remark 3.4.

When $d=1$, known results concerning the distribution of
${\mathcal{Z}}^{\omega}_{\beta}(t,x)$ in the case of Gaussian white noise [5]
strongly suggest that, in fact, only one big ball (that is, $k=1$) is
sufficient to capture most of the mass of $P^{{\omega}}_{\beta,t}$. For a
proof of a result of this kind in the discrete polymer setup (with a special
boundary condition), we refer to [41]. In dimension $d\geq 2$, whether one or
finitely many balls are needed is a challenging open problem, even at the
heuristic level, see [10, Section 9, (1)].

###  3.3. The case of more general noise

In general, a Lévy white noise $\xi$ (on $\mathbb{R}\times\mathbb{R}^{d}$) is
a random distribution in
$\bigcap_{s>({1+d})/{2}}H^{-s}_{\mathrm{loc}}(\mathbb{R}\times\mathbb{R}^{d})$
that can be decomposed as follows:

$\xi=\xi_{1}-\xi_{2}+\alpha_{1}\xi_{3}+\alpha_{2}\mathcal{L}$ (3.8)

where $\xi_{1}$ and $\xi_{2}$ are two independent Lévy noises with respective
jump intensities ${\lambda}_{1}$ and ${\lambda}_{2}$, $\xi_{3}$ is a standard
Gaussian space-time white noise that is independent of $\xi_{1}$ and
$\xi_{2}$, and $\alpha_{1}\geq 0$ and $\alpha_{2}\in\mathbb{R}$ are constants.
In our exposition so far, we have chosen to focus on the case $\xi_{2}\equiv
0$ and $\alpha_{1}=\alpha_{2}=0$. The constant $\alpha_{2}$ is irrelevant for
any discussion of (1.1) as it only adds a multiplicative exponential in $t$ to
the solution.

Adding a Gaussian part in dimension $1$ does not pose a problem for solving
(1.1); see [35]. Given that ${\gamma}(p)$ in (1.32) exhibits the same behavior
in $\beta$ as ${\gamma}_{\beta}(p)$ in (2.15), we believe that both Theorem
2.11 and 2.12 remain valid when Gaussian noise is added. If $d\geq 2$, the SHE
with multiplicative Gaussian white noise is known to be degenerate in the
following sense. If $\xi^{({\varepsilon})}$ denotes the noise obtained by
space convolution of $\xi$ with a smooth kernel
$\theta_{{\varepsilon}}:={\varepsilon}^{-d}\theta({\varepsilon}^{-1}\cdot)$,
then the solution of

$\partial_{t}u^{({\varepsilon})}=\frac{1}{2}\Delta
u^{({\varepsilon})}+\beta\xi^{({\varepsilon})}u^{({\varepsilon})},\quad
u^{{\varepsilon}}(0,\cdot)=\delta_{0}$

satisfies $\lim_{{\varepsilon}\to 0}u^{({\varepsilon})}(t,\cdot)=0$ (not only
pointwise but also in $L^{1}(\mathbb{R}^{d})$ for any fixed $t>0$). In the
past years, important progress has been made to obtain convergence of the
solutions to non-trivial (non-Gaussian) limits for $d=2$ when $\beta$ is sent
to zero (in a specific critical window) jointly with ${\varepsilon}$, see [24,
59] and the recent breakthrough [25].

One can also consider a noise $\xi=\xi_{1}-\xi_{2}$ that includes negative
jumps. In this case, one can approximate $\xi$ by
$\xi^{a}=\xi^{a}_{1}-\xi^{a}_{2}$. Then the criterion (1.14) applied to both
jump intensities ${\lambda}_{1}$ and ${\lambda}_{2}$ still implies the
convergence of the integrals in (1.13), and many of the proofs presented in
this paper can be applied verbatim. Note that when negative jumps are present
in the noise, the solution is not necessarily positive. In particular, in this
case, $\mathcal{Z}^{{\omega}}_{\beta}(t,x)$ does not correspond to the
partition function of a polymer model.

Applying the proofs presented below, Theorems 2.1, 2.5 and 2.7 remain valid
when signed noise is considered, and the same holds true for the upper bounds
of $\mathbb{E}[|{\mathcal{Z}}^{{\omega}}_{\beta}(t,0)|^{p}]$ for $p>1$. In
fact, this translates into half of the inequalities (the upper bounds) proved
in Theorems 2.11 and 2.12. On the other hand, our proof of Proposition 2.8 and
the upper bounds on
$\mathbb{E}\big{[}|{\mathcal{Z}}^{{\omega}}_{\beta}(t,0)|^{p}\big{]}$ for
$p\in(0,1)$ rely in a crucial manner on the positivity of
${\mathcal{Z}}^{{\omega}}_{\beta}(t,0)$, and thus break down when allowing for
negative jumps. Note that the upper bounds on fractional moments of the
partition function are required for the other half of the inequalities (the
lower bounds) in Theorems 2.11 and 2.12, and in particular, for Theorem 2.10.
We believe nonetheless that intermittency occurs in the following sense.

######  Conjecture 2.

If $\xi=\xi_{1}-\xi_{2}$ is a signed Lévy white noise as above, with jump
intensities that satisfy (2.1) and
$\int^{\infty}_{0}z^{p_{0}}\,({\lambda}_{1}+{\lambda}_{2})(\mathrm{d}z)$ for
some $p_{0}<\min(2,1+\tfrac{2}{d})$, then

$\gamma_{\beta}(p):=\lim_{t\to\infty}\frac{1}{t}\log\mathbb{E}[|Z^{{\omega}}_{\beta}(t,0)|^{p}]$

is well defined for $p\in[0,p_{0}]$ and $p\mapsto\gamma_{\beta}(p)/p$ is an
increasing function of $p$ on the interval $[1,p_{0}]$.

##  4\. Preliminary tools

###  4.1. Size-biased measure

If $\mu<\infty$, note that $\bar{\mathcal{Z}}_{\beta}^{{\omega},a}(t,\ast)$ is
non-negative by (1.19) and satisfies
$\mathbb{E}[\bar{\mathcal{Z}}_{\beta}^{{\omega},a}(t,\ast)]=1$. We can
therefore define a new measure $\widetilde{\mathbb{P}}^{a}_{\beta,t}$ for the
environment by

$\widetilde{\mathbb{P}}^{a}_{\beta,t}({\omega}\in
A):=\mathbb{E}\Big{[}\bar{\mathcal{Z}}_{\beta}^{{\omega},a}(t,\ast)\mathbf{1}_{A}\Big{]}\,,$
(4.1)

which is referred to as the _size-biased measure_. Note that when
$(\bar{\mathcal{Z}}_{\beta}^{{\omega},a}(t,\ast))_{a\in(0,1)}$ is uniformly
integrable, then using martingale convergence, we can also consider
$\widetilde{\mathbb{P}}^{0}_{\beta,t}$ whose density with respect to
$\mathbb{P}$ is given by $\bar{\mathcal{Z}}^{{\omega}}_{\beta}(t,\ast)$.

Lemma 3.7 in [13] gives a useful representation of the size-biased measure
$\widetilde{\mathbb{P}}^{a}_{\beta,t}$: the distribution of ${\omega}$ under
$\widetilde{\mathbb{P}}^{a}_{\beta,t}$ is obtained by adding to ${\omega}$ an
independent Poisson process on the trajectory of a Brownian motion. More
precisely, we let ${\omega}^{\prime}$ be a Poisson point process on
$\mathbb{R}\times(0,\infty)$ with intensity
$\mathrm{d}t\otimes{\beta}z\,\lambda(\mathrm{d}z)$ and we denote its law by
$\mathbb{P}^{\prime}_{\beta}$. Furthermore, let $(B_{t})_{t\in[0,\infty)}$ be
a standard $d$-dimensional Brownian motion starting from $0$ that is
independent of both ${\omega}$ and ${\omega}^{\prime}$ and denote its
distribution by $\mathbf{Q}$. For $a\in[0,1]$, we then define

$\displaystyle\widetilde{\omega}({\omega}^{\prime},B)$
$\displaystyle:=\Big{\\{}(\tau,B_{\tau},\zeta):(\tau,\zeta)\in{\omega}^{\prime}\Big{\\}},$
$\displaystyle\widetilde{\omega}_{a,t}({\omega}^{\prime},B)$
$\displaystyle:=\Big{\\{}(\tau,B_{\tau},\zeta):(\tau,\zeta)\in{\omega}^{\prime},\tau\in[0,t]\,,\zeta\geq
a\Big{\\}}\,,$

which corresponds to putting the Poisson process ${\omega}^{\prime}$ on the
trajectory of $B$, with some restrictions on $\tau$ and $\zeta$ in the second
case. When $a=0$, the dependence in $a$ is sometimes omitted.

######  Lemma 4.1.

If $\mu<\infty$, then for any $\beta,t>0$ and $a\in(0,1]$ and for any
measurable bounded function $g$, we have

$\widetilde{\mathbb{E}}^{a}_{\beta,t}[g({\omega})]=\mathbb{E}\otimes\mathbb{E}^{\prime}_{\beta}\otimes\mathbf{Q}[\widehat{g}_{a,t}({\omega},{\omega}^{\prime},B)],$
(4.2)

where
$\widehat{g}_{a,t}({\omega},{\omega}^{\prime},B):=g({\omega}\cup\widetilde{\omega}_{a,t}({\omega}^{\prime},B))$.
If $(\bar{\mathcal{Z}}_{\beta}^{{\omega},a}(t,\ast))_{a\in(0,1)}$ is uniformly
integrable, then (4.2) is also satisfied for $a=0$.

If $a>0$, this was shown in [13, Lemma 3.7]. For $a=0$, the result can be
deduced from the case $a>0$ by observing that
$\mathbb{E}[\bar{\mathcal{Z}}^{{\omega}}_{\beta}(t,\ast)\mid\mathcal{G}_{a}]=\bar{\mathcal{Z}}^{{\omega},a}_{\beta}(t,\ast)$
for any $a\in(0,1]$, which is why (4.2) with $a=0$ is true for any bounded
$\mathcal{G}_{a}$-measurable $g$. The monotone class theorem yields (4.2) for
general $g$. Note that by Theorem 2.1,
$(\bar{\mathcal{Z}}_{\beta}^{{\omega},a}(t,\ast))_{a\in(0,1)}$ is uniformly
integrable as soon as (2.1) holds.

######  Remark 4.2.

Let us make a quick comment about notation. In (4.2) as well as in the rest of
the paper, we denote the expectation with respect to a measure $\mathbb{P}$,
possibly with decorations, by $\mathbb{E}$ with the same decorations. For
example, we write $\widetilde{\mathbb{E}}^{a}_{\beta,t}$ and
$\mathbb{E}^{\prime}_{\beta}$ for expectation with respect to
$\widetilde{\mathbb{P}}^{a}_{\beta,t}$ and $\mathbb{P}^{\prime}_{\beta}$,
respectively.

###  4.2. Decoupling inequalities

In the proofs below, we shall occasionally make use of _moment decoupling
inequalities_ for multiple Poisson integrals. The results that we review in
this section are based on [67], where discrete-time decoupling techniques from
[48, 73] are extended to continuous time. We also refer the reader to these
references for further literature (and history) on the subject. Let $N\geq 1$
be an integer, let
$\mathbb{X}:=(0,\infty)\times\mathbb{R}^{d}\times(0,\infty)$, with generic
point $w:=(t,x,z)$, and for concreteness let $M=M^{\omega}$ be the compensated
Poisson random measure
$M^{\omega}(\mathrm{d}w)=(\delta_{\omega}-\nu)(\mathrm{d}w)$ (recall that
$\nu:=\mathrm{d}t\otimes\mathrm{d}x\otimes{\lambda}(\mathrm{d}z)$).
Furthermore, suppose that $f:\mathbb{X}^{N}\to\mathbb{R}$ is measurable and
_tetrahedral_ , that is,

$f((t_{1},x_{1},z_{1}),\dots,(t_{N},x_{N},z_{N}))=0\quad\text{unless}\
t_{1}<\dots<t_{N}\,.$

In that case, subject to integrability conditions on $f$, the multiple
integral

$\int_{\mathbb{X}^{N}}f(\boldsymbol{w})\,M(\mathrm{d}w_{1})\cdots
M(\mathrm{d}w_{N}):=\int_{\mathbb{X}}\Biggl{(}\cdots\Biggl{(}\int_{\mathbb{X}}f(\boldsymbol{w})\,M(\mathrm{d}w_{1})\Biggr{)}\cdots\Biggr{)}\,M(\mathrm{d}w_{N})\,,$

with $\boldsymbol{w}=(w_{1},\dots,w_{N})$, can be defined as an iterated Itô
integral (recall Remark 1.2). The general definition of Itô integrals used in
this paper is detailed in Section 4.3. We let $({\omega}_{i})_{i=1}^{N}$
denote i.i.d. copies of our environment ${\omega}$ and we use the notations
$\mathbb{P}^{\otimes N}$ and $\mathbb{E}^{\otimes N}$ for the associated
probability and expectation, respectively. The following theorem is proved in
Appendix A.

######  Theorem 4.3.

There exists a universal constant $C>0$ such that for every $1<p\leq 2$, for
any integer $N\geq 1$ and any measurable and tetrahedral function
$f:\mathbb{X}^{N}\to\mathbb{R}$ satisfying

$\int_{\mathbb{X}^{N}}\lvert
f(\boldsymbol{w})\rvert^{p}\,\nu(\mathrm{d}w_{1})\cdots\nu(\mathrm{d}w_{N})<\infty\,,$
(4.3)

the stochastic integrals below are well defined and have finite moments of
order $p$ satisfying

$\begin{split}&\bigg{(}\frac{p-1}{C}\bigg{)}^{N}\mathbb{E}^{\otimes
N}\Biggl{[}\Biggl{\lvert}\int_{\mathbb{X}^{N}}f(\boldsymbol{w})\,M^{{\omega}_{1}}(\mathrm{d}w_{1})\cdots\,M^{{\omega}_{N}}(\mathrm{d}w_{N})\Biggr{\rvert}^{p}\Biggr{]}\\\
&\qquad\qquad\qquad\qquad\leq\mathbb{E}\Biggl{[}\Biggl{\lvert}\int_{\mathbb{X}^{N}}f(\boldsymbol{w})\,M^{\omega}(\mathrm{d}w_{1})\cdots\,M^{\omega}(\mathrm{d}w_{N})\Biggr{\rvert}^{p}\Biggr{]}\\\
&\qquad\qquad\qquad\qquad\qquad\qquad\leq\bigg{(}\frac{C}{p-1}\bigg{)}^{N}\mathbb{E}^{\otimes
N}\Biggl{[}\Biggl{\lvert}\int_{\mathbb{X}^{N}}f(\boldsymbol{w})\,M^{{\omega}_{1}}(\mathrm{d}w_{1})\cdots\,M^{{\omega}_{N}}(\mathrm{d}w_{N})\Biggr{\rvert}^{p}\Biggr{]}.\end{split}$
(4.4)

###  4.3. Stochastic integration in the absence of moments

Because ${\mathcal{Z}}^{\omega}_{\beta}(t,x)$ does not possess any finite
moments in general, the proof of Theorem 2.5 has to make use of a stochastic
integration theory that does not assume existence of any moments a priori. For
the reader’s convenience, let us give a brief review of this $L^{0}$-theory,
which was developed by [20] in its most general form (see also [37, Appendix
A] for a summary). Let $M=M^{\omega}$ denote either the Poisson measure
$\delta_{\omega}$ or its compensated version $\delta_{\omega}-\nu$.

A _predictable step process_ $H$ is of the form
$H=\sum_{i=1}^{r}a_{i}\mathbf{1}_{A_{i}}$, where $r\in\mathbb{N}$,
$a_{i}\in\mathbb{R}$, and for each $i$ we have
$A_{i}\in{\mathcal{P}}\otimes{\mathcal{B}}(\mathbb{R}^{d}\times(0,\infty))$
(where $\mathcal{P}$ is the usual predictable $\sigma$-field) and
$A_{i}\subseteq{\Omega}\times(0,T)\times[-N,N]^{d}\times(a,\infty)$ for some
$T,N,a>0$. Then, the integral of the predictable step process
$H=\sum_{i=1}^{r}a_{i}\mathbf{1}_{A_{i}}$ is canonically defined as

$\int_{\mathbb{X}}H(\boldsymbol{w})\,M(\mathrm{d}\boldsymbol{w}):=\sum_{i=1}^{r}a_{i}M(A_{i})\,.$

Denoting by ${\mathcal{S}}$ the collection of predictable step processes, we
can extend the integral to a larger subset of predictable processes by using
the metric induced by

$\lVert K\rVert_{M,p}:=\sup_{H\in{\mathcal{S}},\lvert H\rvert\leq\lvert
K\rvert}\Biggl{\lVert}\int_{\mathbb{X}}H(\boldsymbol{w})\,M(\mathrm{d}\boldsymbol{w})\Biggr{\rVert}_{L^{p}},$
(4.5)

defined for
${\mathcal{P}}\otimes{\mathcal{B}}(\mathbb{R}^{d}\times(0,\infty))$-measurable
processes $K$, where we have used the notation $\lVert
X\rVert_{L^{p}}:=\mathbb{E}[\lvert X\rvert^{p}]^{1/p}$ and $\lVert
X\rVert_{L^{0}}:=\mathbb{E}[1\wedge\lvert X\rvert]$. Such a process $K$ is
called _$L^{p}$ -integrable with respect to $M$_ if there exists a sequence
$(H_{n})_{n\in\mathbb{N}}\subseteq{\mathcal{S}}$ such that
$\lim_{n\to\infty}\lVert K-H_{n}\rVert_{M,p}=0$. The stochastic integral of
$K$ with respect to $M$ is then defined as the $L^{p}$-limit of
$\int_{\mathbb{X}}H_{n}(\boldsymbol{w})\,M(\mathrm{d}\boldsymbol{w})$, which
exists and does not depend on the choice of $(H_{n})_{n\in\mathbb{N}}$. If
$p=0$, we simply say that $K$ is _integrable with respect to $M$_. According
to [20, (2.8)], $K$ is $L^{p}$-integrable with respect to $M$ if and only if

$\begin{cases}\lVert K\rVert_{M,p}<\infty&\text{if }p>0,\\\
\displaystyle\lim_{u\to 0}\lVert uK\rVert_{M,0}=0&\text{if }p=0.\end{cases}$
(4.6)

Furthermore, by [37, Lemma A.2],

  * •

if $1\leq p<\infty$ and $M^{\omega}=\delta_{\omega}-\nu$, there are
$c=c_{p}>0$ and $C=C_{p}>0$ such that

$c\lVert
K\rVert_{M,p}\leq\mathbb{E}\Biggl{[}\biggl{(}\int_{\mathbb{X}}K(\boldsymbol{w})^{2}\,\delta_{\omega}(\mathrm{d}\boldsymbol{w})\biggr{)}^{\frac{p}{2}}\Biggr{]}^{\frac{1}{p}}\leq
C\lVert K\rVert_{M,p};$ (4.7)

  * •

if $0<p\leq 1$ and $M^{\omega}=\delta_{\omega}$, then

$\lVert K\rVert_{M,p}^{p}\leq\int_{\mathbb{X}}\mathbb{E}[\lvert
K(\boldsymbol{w})\rvert^{p}]\,\nu(\mathrm{d}\boldsymbol{w}).$

In particular, the $L^{p}$-theory, $p\geq 1$, for $\delta_{\omega}-\nu$
encompasses the $L^{p}$\- and $L^{2}$-integrals considered in [89] and [94].
Even in the case when $\delta_{\omega}-\nu$ has a finite first moment (i.e.,
is a martingale measure), the $L^{0}$-theory is more general since, for
example, it does not require the integrands have finite moments.

Let us end this section with a simple integrability criterion that we need in
the proof of Theorem 2.5. Its proof can be found in the appendix.

######  Lemma 4.4.

Suppose that for $\mathbb{P}_{\geq}$-a.e. realization of ${\omega}_{\geq}$,
the process $({\omega}_{<},t,x)\mapsto K({\omega}_{<}\cup{\omega}_{\geq},t,x)$
is $L^{p}(\mathbb{P}_{<})$-integrable with respect to $\xi_{{\omega}_{<}}$ for
some $p>0$. Then $K$ is $L^{0}(\mathbb{P})$-integrable with respect to
$\xi_{{\omega}_{<}}$.

###  4.4. A technical lemma

The following technical result will be used repeatedly in the paper. Its proof
comes from a straightforward calculation and can be found in [13, Lemma A.3].

######  Lemma 4.5.

For any $t>0$, $k\geq 0$ and $\zeta_{1},\dots,\zeta_{k+1}>0$,

$\int_{\mathfrak{X}_{k}(t)}\prod_{i=1}^{k+1}(\Delta
t_{i})^{\zeta_{i}-1}\,\mathrm{d}t_{i}=t^{\sum_{i=1}^{k+1}\zeta_{i}-1}\frac{\prod_{i=1}^{k+1}\Gamma(\zeta_{i})}{\Gamma(\sum_{i=1}^{k+1}\zeta_{i})}\,,$

where $\Delta t_{i}:=t_{i}-t_{i-1}$ as defined in (1.10), with the convention
$t_{0}=0$ and $t_{k+1}=t$.

##  5\. Qualitative properties of Lyapunov exponents

In this section, we prove the statements made in Section 2.3 concerning
existence and basic properties of moment Lyapunov exponents (Propositions 2.8
and 2.9), assuming that Theorem 2.1 as well as the bounds in Propositions 6.1
and 6.3 below are true. These are proved independently in Sections 6 and 7,
respectively.

###  5.1. Existence of Lyapunov exponents

The existence of ${\gamma}_{\beta}(p)$ readily follows from sub-/
supermultiplicative properties of the moments of the partition function.

######  Lemma 5.1.

Under assumption (1.14), for every $a\in(0,1]$, $s,t>0$ and
$p\in[0,1+\frac{2}{d})$, we have

$\displaystyle\mathbb{E}[{\mathcal{Z}}^{{\omega},a}_{\beta}(t+s,\ast)^{p}]\leq\mathbb{E}[{\mathcal{Z}}^{{\omega},a}_{\beta}(s,\ast)^{p}]\mathbb{E}[{\mathcal{Z}}^{{\omega},a}_{\beta}(t,\ast)^{p}]\qquad\text{
if }p\geq 1,$ (5.1)
$\displaystyle\mathbb{E}[{\mathcal{Z}}^{{\omega},a}_{\beta}(t+s,\ast)^{p}]\geq\mathbb{E}[{\mathcal{Z}}^{{\omega},a}_{\beta}(s,\ast)^{p}]\mathbb{E}[{\mathcal{Z}}^{{\omega},a}_{\beta}(t,\ast)^{p}]\qquad\text{
if }p\leq 1.$ (5.2)

###### Proof.

The statement being trivial if $\mu_{1,\infty}(p)=\infty$, we can assume
$\mu_{1,\infty}(p)<\infty$ for the remainder of the proof. Let
$P_{{\beta},t}^{{\omega},a}$ denote the (random) probability measure on
$\mathbb{R}^{d}$ whose density with respect to the Lebesgue measure is given
by
${\mathcal{Z}}_{\beta}^{{\omega},a}(t,x)/{\mathcal{Z}}^{{\omega},a}_{\beta}(t,\ast)$
and $X$ denote an $\mathbb{R}^{d}$-valued random variable with distribution
$P_{{\beta},t}^{{\omega},a}$. Recalling the definition (1.24) of
${\mathcal{Z}}_{\beta}^{{\omega},a}(s,x;t,\ast)$ and using the Markov property
for Brownian motion, we have

$\displaystyle{\mathcal{Z}}_{\beta}^{{\omega},a}(s+t,\ast)$
$\displaystyle=\int_{\mathbb{R}^{d}}{\mathcal{Z}}_{\beta}^{{\omega},a}(s,x){\mathcal{Z}}_{\beta}^{{\omega},a}(s,x;s+t,\ast)\,\mathrm{d}x$
$\displaystyle={\mathcal{Z}}_{{\beta}}^{{\omega},a}(s,\ast)\int_{\mathbb{R}^{d}}{\mathcal{Z}}_{\beta}^{{\omega},a}(s,x;s+t,\ast)\,P_{{\beta},s}^{{\omega},a}(\mathrm{d}x)$
$\displaystyle={\mathcal{Z}}_{{\beta}}^{{\omega},a}(s,\ast)\,E_{{\beta},s}^{{\omega},a}\Big{[}{\mathcal{Z}}_{\beta}^{{\omega},a}(s,X;s+t,\ast)\Big{]}\,.$

For $p\geq 1$, we get by Jensen’s inequality that

${\mathcal{Z}}_{\beta}^{{\omega},a}(s+t,\ast)^{p}\leq{\mathcal{Z}}_{{\beta}}^{{\omega},a}(s,\ast)^{p}\,E_{{\beta},s}^{{\omega},a}\Big{[}{\mathcal{Z}}_{\beta}^{{\omega},a}(s,X;s+t,\ast)^{p}\Big{]}\,,$

and the inequality goes in the other direction if $p\in(0,1)$. Now, recalling
(1.5) and (1.18), we have

$\begin{split}\mathbb{E}\Big{[}{\mathcal{Z}}_{{\beta}}^{{\omega},a}(s,\ast)^{p}\,E_{{\beta},s}^{{\omega},a}\Big{[}{\mathcal{Z}}_{\beta}^{{\omega},a}(s,X;s+t,\ast)^{p}\Big{]}\mathrel{\Big{|}}\mathcal{F}_{s}\Big{]}&={\mathcal{Z}}_{{\beta}}^{{\omega},a}(s,\ast)^{p}\,E_{{\beta},s}^{{\omega},a}\Big{[}\mathbb{E}\Big{[}{\mathcal{Z}}_{\beta}^{{\omega},a}(s,X;s+t,\ast)^{p}\mathrel{\Big{|}}\mathcal{F}_{s}\Big{]}\Big{]}\\\
&={\mathcal{Z}}_{{\beta}}^{{\omega},a}(s,\ast)^{p}\,\mathbb{E}\Big{[}{\mathcal{Z}}_{\beta}^{{\omega},a}(t,\ast)^{p}\Big{]}\,.\end{split}$

Altogether, we have proven that for $p\geq 1$,

$\mathbb{E}\Big{[}{\mathcal{Z}}_{\beta}^{{\omega},a}(s+t,\ast)^{p}\mathrel{\Big{|}}\mathcal{F}_{s}\Big{]}\leq{\mathcal{Z}}_{{\beta}}^{{\omega},a}(s,\ast)^{p}\,\mathbb{E}\Big{[}{\mathcal{Z}}_{\beta}^{{\omega},a}(t,\ast)^{p}\Big{]}\,,$

which yields (5.1).Repeating the same computation with reversed inequalities
when $p\in(0,1)$, we obtain (5.2). ∎

###### Proof of Proposition 2.8.

As a consequence of (2.4) in Theorem 2.1, we have

$\lim_{a\to
0}\mathbb{E}\Big{[}{\mathcal{Z}}^{{\omega},a}_{\beta}(t,\ast)^{p}\Big{]}=\mathbb{E}\Big{[}{\mathcal{Z}}^{{\omega}}_{\beta}(t,\ast)^{p}\Big{]}.$

Therefore, the previous lemma remains valid for
${\mathcal{Z}}^{{\omega}}_{\beta}(t,\ast)$ instead of
${\mathcal{Z}}^{{\omega},a}_{\beta}(t,\ast)$. An immediate consequence is that
the limit in (2.11) exists by the continuous version of Fekete’s subadditive
(or superadditive) lemma. It remains to check that $\gamma_{\beta}(p)$ is
finite when (2.1) is satisfied and $\mu_{1,\infty}(p)<\infty$. When
$p\in[1,1+\tfrac{2}{d})$, ${\gamma}_{\beta}(p)<\infty$ follows from

${\gamma}_{\beta}(p)=\lim_{n\to\infty}\frac{1}{n}\log\mathbb{E}\Big{[}{\mathcal{Z}}^{\omega}_{\beta}(n,\ast)^{p}\Big{]}\leq\lim_{n\to\infty}\frac{1}{n}\log\mathbb{E}\Big{[}{\mathcal{Z}}^{\omega}_{\beta}(1,\ast)^{p}\Big{]}^{n}=\log\mathbb{E}\Big{[}{\mathcal{Z}}^{\omega}_{\beta}(1,\ast)^{p}\Big{]},$
(5.3)

and the finiteness of
$\mathbb{E}[{\mathcal{Z}}^{\omega}_{\beta}(1,\ast)^{p}]$, see Theorem 2.1.

For $p\in(0,1)$, we prove $\gamma_{\beta}(p)>-\infty$ and
$\gamma_{\beta}(p)<\infty$ separately. For the first part, let
$\gamma^{<}_{\beta}(p)$ be the moment Lyapunov exponents that one obtains
after replacing ${\omega}$ by ${\omega}_{<}$. Thus, by an obvious comparison
and convexity (see Proposition 2.9, applied to
$\lambda_{<}(\mathrm{d}z)=\mathbf{1}_{(0,1)}(z)\,\lambda(\mathrm{d}z)$), it
follows that

$\gamma_{\beta}(p)\geq\gamma^{<}_{\beta}(p)\geq\frac{{\gamma}_{\beta}^{<}(1)-(1-\theta){\gamma}_{\beta}^{<}(1+\frac{1}{d})}{\theta}>-\infty,$

where $\theta\in(0,1)$ is such that $1=\theta p+(1-\theta)(1+\frac{1}{d})$.
Note that ${\gamma}_{\beta}^{<}(1)=0$ and that
${\gamma}_{\beta}^{<}(1+\frac{1}{d})<\infty$ thanks to (5.3).

In order to show that ${\gamma}_{\beta}(p)<\infty$ when $p\in(0,\infty)$ and
$\mu_{1,\infty}(p)<\infty$, we use Jensen’s inequality for $\mathbb{E}_{<}$ so
that for every $a\in[0,1)$,

$\mathbb{E}\Big{[}{\mathcal{Z}}^{{\omega},a}_{\beta}(t,\ast)^{p}\Big{]}\leq\mathbb{E}\Big{[}{\mathcal{Z}}^{{\omega}_{\geq}}_{\beta}(t,\ast)^{p}\Big{]}.$

To bound the right-hand side, we use the following simple inequality, which
will be used extensively in the remainder of the paper: given $\kappa\in(0,1)$
and any countable collection of non-negative numbers $(a_{i})_{i\in I}$, we
have

$\bigg{(}\sum_{i\in I}a_{i}\bigg{)}^{\kappa}\leq\sum_{i\in I}a^{\kappa}_{i}.$
(5.4)

We will refer to (5.4), which can be proved by induction (using the fact that
$(a+b)^{\kappa}\leq a^{\kappa}+b^{\kappa}$), as the subadditivity property;
note that it obviously extends to stochastic integrals with respect to
discrete measures.

Moreover, note that we have

$\rho(t,x)^{p}=t^{\nu_{p}-1}\vartheta(p)\rho(\tfrac{t}{p},x),$ (5.5)

where $\nu_{p}=1-\frac{d}{2}(p-1)$ was defined in (2.17) and
$\vartheta(p):=(2\pi)^{\nu_{p}-1}p^{-\frac{d}{2}}$. Thus, using (5.4) for the
first step, (5.5) and the fact that $\rho$ is a density for the second step,
and Lemma 4.5 for the last step, we derive

$\displaystyle\mathbb{E}\Big{[}{\mathcal{Z}}^{{\omega}_{\geq}}_{\beta}(t,\ast)^{p}\Big{]}$
$\displaystyle\leq\sum_{k=0}^{\infty}(\beta^{p}\mu_{1,\infty}(p))^{k}\int_{{\mathfrak{X}}_{k}(t)\times(\mathbb{R}^{d})^{k}}\prod_{i=1}^{k}\rho(\Delta
t_{i},\Delta x_{i})^{p}\,\mathrm{d}t_{i}\,\mathrm{d}x_{i}$
$\displaystyle=\sum_{k=0}^{\infty}(\beta^{p}\vartheta(p)\mu_{1,\infty}(p))^{k}\int_{{\mathfrak{X}}_{k}(t)}\prod_{i=1}^{k}(\Delta
t_{i})^{\nu_{p}-1}\,\mathrm{d}t_{i}=\sum_{k=0}^{\infty}\frac{(\beta^{p}\vartheta(p)\mu_{1,\infty}(p)\Gamma(\nu_{p})t^{\nu_{p}})^{k}}{\Gamma(\nu_{p}k+1)}\,.$

Using Stirling’s formula for the gamma function (or the estimate (7.13)
below), one can check that the right-hand side grows exponentially in $t$: as
a consequence, we have $\gamma_{\beta}(p)<\infty$. This ends the proof of part
(i) of the proposition.

For part (ii), we are going to prove the upper and lower bounds in (2.12)
separately. Using Jensen’s inequality for the first inequality and the
translation invariance (1.18) for the last identity, we have, for $p>1$,

$\mathbb{E}\Big{[}{\mathcal{Z}}^{{\omega}}_{\beta}(t,\ast)^{p}\Big{]}\leq\int_{\mathbb{R}^{d}}\rho(t,x)\mathbb{E}\Big{[}\Big{(}\rho(t,x)^{-1}{\mathcal{Z}}^{{\omega}}_{\beta}(t,x)\Big{)}^{p}\Big{]}\,\mathrm{d}x=\mathbb{E}\Big{[}\Big{(}\rho(t,0)^{-1}{\mathcal{Z}}^{{\omega}}_{\beta}(t,0)\Big{)}^{p}\Big{]}.$
(5.6)

Since $\rho(t,0)^{-1}$ is of order $t^{d/2}$, after taking logarithm, dividing
by $t$ and taking the limit as $t\to\infty$, we obtain

$\liminf_{t\to\infty}\frac{1}{t}\log\mathbb{E}\Big{[}{\mathcal{Z}}^{{\omega}}_{\beta}(t,0)^{p}\Big{]}\geq\gamma_{\beta}(p).$
(5.7)

The same proof yields for $p\in(0,1)$ that

$\limsup_{t\to\infty}\frac{1}{t}\log\mathbb{E}\Big{[}{\mathcal{Z}}^{{\omega}}_{\beta}(t,0)^{p}\Big{]}\leq\gamma_{\beta}(p).$
(5.8)

Let us now prove the complementary bound, first in the case $p>1$. We
introduce the probability measure $\bar{P}_{{\beta},t}^{{\omega}}$ with
Lebesgue density
${\mathcal{Z}}^{{\omega}}_{\beta}(t,x)\rho(1,x)/\int_{\mathbb{R}^{d}}{\mathcal{Z}}^{{\omega}}_{\beta}(t,x)\rho(1,x)\,\mathrm{d}x$.
Then we have

$\displaystyle{\mathcal{Z}}^{{\omega}}_{\beta}(t+1,0)$
$\displaystyle=\int_{\mathbb{R}^{d}}{\mathcal{Z}}^{{\omega}}_{\beta}(t,x)\rho(1,x){\mathcal{Z}}^{{\omega}}_{\beta}(t,x;t+1,0)\rho(1,x)^{-1}\,\mathrm{d}x$
$\displaystyle=\Bigg{(}\int_{\mathbb{R}^{d}}{\mathcal{Z}}^{{\omega}}_{\beta}(t,x)\rho(1,x)\,\mathrm{d}x\Bigg{)}\bar{E}_{{\beta},t}^{{\omega}}\Big{[}{\mathcal{Z}}^{{\omega}}_{\beta}(t,X;t+1,0)\rho(1,X)^{-1}\Big{]}.$

Proceeding as in the proof of Lemma 5.1 and using the translation invariance
(1.18), we obtain for $p>1$

$\mathbb{E}\Big{[}{\mathcal{Z}}^{{\omega}}_{\beta}(t+1,0)^{p}\Big{]}\leq\mathbb{E}\bigg{[}\bigg{(}\int_{\mathbb{R}^{d}}{\mathcal{Z}}^{{\omega}}_{\beta}(t,x)\rho(1,x)\,\mathrm{d}x\bigg{)}^{p}\bigg{]}\mathbb{E}\Big{[}{\mathcal{Z}}^{{\omega}}_{\beta}(1,0)^{p}\rho(1,0)^{-p}\Big{]}\,.$
(5.9)

In particular, we have

$\mathbb{E}\Big{[}{\mathcal{Z}}^{{\omega}}_{\beta}(t+1,0)^{p}\Big{]}\leq(2\pi)^{-\frac{pd}{2}}\mathbb{E}\Big{[}{\mathcal{Z}}^{{\omega}}_{\beta}(t,\ast)^{p}\Big{]}\mathbb{E}\Big{[}{\mathcal{Z}}^{{\omega}}_{\beta}(1,0)^{p}\rho(1,0)^{-p}\Big{]},$

which implies, for $p>1$,

$\limsup_{t\to\infty}\frac{1}{t}\mathbb{E}\Big{[}{\mathcal{Z}}^{{\omega}}_{\beta}(t+1,0)^{p}\Big{]}\leq\gamma_{\beta}(p)\,.$

When $p\in(0,1)$, repeating the proof of (5.9), but using concavity instead of
convexity, we obtain

$\mathbb{E}\Big{[}{\mathcal{Z}}^{{\omega}}_{\beta}(t+1,0)^{p}\Big{]}\geq\mathbb{E}\bigg{[}\bigg{(}\int_{\mathbb{R}^{d}}{\mathcal{Z}}^{{\omega}}_{\beta}(t,x)\rho(1,x)\,\mathrm{d}x\bigg{)}^{p}\bigg{]}\mathbb{E}\Big{[}{\mathcal{Z}}^{{\omega}}_{\beta}(1,0)^{p}\rho(1,0)^{-p}\Big{]}.$
(5.10)

Now setting $\mathcal{C}_{z}:=z+[0,1)^{d}$, we have

${\mathcal{Z}}^{{\omega}}_{\beta}(t,\ast)=\sum_{z\in\mathbb{Z}^{d}}\int_{\mathcal{C}_{z}}{\mathcal{Z}}^{{\omega}}_{\beta}(t,x)\,\mathrm{d}x\leq\sum_{z\in\mathbb{Z}^{d}}\bigg{(}\max_{x\in\mathcal{C}_{z}}\rho(t,x)\bigg{)}Y_{z}(t)$
(5.11)

with
$Y_{z}(t):=\int_{\mathcal{C}_{z}}{\mathcal{Z}}^{{\omega}}_{\beta}(t,x)\rho(t,x)^{-1}\,\mathrm{d}x.$

Note that the variables $Y_{z}$ are identically distributed by (1.18), which
together with (5.4) implies

$\mathbb{E}\Big{[}{\mathcal{Z}}^{{\omega}}_{\beta}(t,\ast)^{p}\Big{]}\leq\bigg{(}\sum_{z\in\mathbb{Z}^{d}}\max_{x\in\mathcal{C}_{z}}\rho(t,x)^{p}\bigg{)}\mathbb{E}[Y_{0}(t)^{p}]\leq
Ct^{\frac{d}{2}{(1-p)}}\mathbb{E}[Y_{0}(t)^{p}]$ (5.12)

for $t>1$. Moreover, also for $t>1$,

$\int_{\mathbb{R}^{d}}{\mathcal{Z}}^{{\omega}}_{\beta}(t,x)\rho(1,x)\,\mathrm{d}x\geq\bigg{(}\min_{x\in\mathcal{C}_{0}}\rho(1,x)\rho(t,x)\bigg{)}Y_{0}(t)\geq(2\pi)^{-d}t^{-\frac{d}{2}}Y_{0}(t)\,.$
(5.13)

Therefore, combining (5.10) with (5.12) and (5.13), we obtain that

$\mathbb{E}\Big{[}{\mathcal{Z}}^{{\omega}}_{\beta}(t+1,0)^{p}\Big{]}\geq
C^{\prime}{t^{-\frac{d}{2}}}\mathbb{E}\Big{[}{\mathcal{Z}}^{{\omega}}_{\beta}(t,\ast)^{p}\Big{]}\,.$

This allows us to conclude that, for $p\in(0,1)$,

$\liminf_{t\to\infty}\frac{1}{t}\log\mathbb{E}\Big{[}{\mathcal{Z}}^{{\omega}}_{\beta}(t+1,0)^{p}\Big{]}\geq\gamma_{\beta}(p).\qed$

###  5.2. Monotonicity and convexity properties

To show the monotonicity in $\beta$ of the Lyapunov exponents, we prove a more
general result linking monotonicity and convexity. Its proof is inspired by an
analogous result proved in the discrete setup [44, Lemma 3.3]. Recall the
definition of the renormalized partition function (1.26).

######  Lemma 5.2.

Assume that $\mu_{1,\infty}(q)<\infty$ for some $q\in[1,1+\frac{2}{d})$ and
let $\varphi:[0,\infty)\to\mathbb{R}$ be a convex function that satisfies

$\sup_{u\geq 1}\ |\varphi(u)|u^{-q}<\infty\,.$

Then
$\beta\mapsto\mathbb{E}[\varphi(\bar{\mathcal{Z}}^{{\omega},a}_{\beta}(t,0))]$
is a non-decreasing function, for any $a>0$.

###### Proof of Proposition 2.9.

As a consequence of Lemma 5.2, if $\mu_{1,\infty}(p)<\infty$ for some
$p\in[1,1+\frac{2}{d})$, then
$\beta\mapsto\mathbb{E}[\bar{\mathcal{Z}}^{{\omega},a}_{\beta}(t,0)^{p}]$ is
non-decreasing if $p>1$ and non-increasing if $p\in(0,1)$ (apply Lemma 5.2 to
the convex function $x\mapsto-x^{p}$). Letting $a\to 0$, we obtain the same
results for $\mathbb{E}[\bar{\mathcal{Z}}^{{\omega}}_{\beta}(t,0)^{p}]$
because of (2.4). Since we have

$\bar{\gamma}_{\beta}(p)=\lim_{t\to\infty}\frac{1}{t}\log\mathbb{E}[\bar{\mathcal{Z}}^{{\omega}}_{\beta}(t,0)^{p}]\,,$

this completes the proof of (ii) in Proposition 2.9. Both (i) and (iii) are
standard: The convexity of $p\mapsto{\gamma}_{\beta}(p)$ follows from that of
$p\mapsto\log\mathbb{E}[X^{p}]$, valid for an arbitrary non-negative random
variable $X$ by Hölder’s inequality; (iii) is a direct consequence of
convexity since for $1<p<p^{\prime}$ we have
$\bar{\gamma}_{\beta}(p)\leq\frac{p-1}{p^{\prime}-1}\bar{\gamma}_{\beta}(p^{\prime})$,
recalling that $\bar{\gamma}_{\beta}(1)=0$ by definition. ∎

###### Proof of Lemma 5.2.

Since $\varphi$ is convex, there exists $a$ and $b$ such that
$\varphi(u)+au+b\geq 0$ for every $u\geq 0$. Hence, replacing $\varphi(u)$ by
$\varphi(u)+au+b$, we can assume that $\varphi$ is non-negative, since
$\mathbb{E}[\bar{\mathcal{Z}}^{{\omega},a}_{\beta}(t,0)]=\rho(t,0)$ does not
depend on $\beta$. We may also assume without loss of generality that
$\varphi$ is differentiable on $[0,\infty)$ and that $\varphi^{\prime}$ is
bounded. Indeed, if this is not the case, we can find a sequence
$\varphi_{n}\uparrow\varphi$ that has these properties and then use monotone
convergence.

Recalling the notation (1.11) and setting $\bar{\kappa}_{a}=\kappa_{a}+\mu$,
we have, similarly to (1.19),

$\bar{\mathcal{Z}}^{{\omega},a}_{\beta}(t,0)=e^{-\beta\bar{\kappa}_{a}t}\sum_{k=0}^{\infty}\beta^{k}\int_{{\mathfrak{X}}_{k}(t)\times(\mathbb{R}^{d})^{k}}\rho_{t,0}(\boldsymbol{t},\boldsymbol{x})\prod_{i=1}^{k}\xi^{a,+}_{{\omega}}(\mathrm{d}t_{i},\mathrm{d}x_{i}).$

Taking the derivative with respect to $\beta$, we obtain, after an index shift
for the first term,

$\begin{multlined}\partial_{\beta}\bar{\mathcal{Z}}^{{\omega},a}_{\beta}(t,0)=e^{-\beta\bar{\kappa}_{a}t}\sum_{k=0}^{\infty}\beta^{k}\bigg{(}(k+1)\int_{{\mathfrak{X}}_{k+1}(t)\times(\mathbb{R}^{d})^{k+1}}\rho_{t,0}(\boldsymbol{t},\boldsymbol{x})\prod_{i=1}^{k+1}\xi^{a,+}_{{\omega}}(\mathrm{d}t_{i},\mathrm{d}x_{i})\\\
-\bar{\kappa}_{a}t\int_{{\mathfrak{X}}_{k}(t)\times(\mathbb{R}^{d})^{k}}\rho_{t,0}(\boldsymbol{t},\boldsymbol{x})\prod_{i=1}^{k}\xi^{a,+}_{{\omega}}(\mathrm{d}t_{i},\mathrm{d}x_{i})\bigg{)}\,.\end{multlined}\partial_{\beta}\bar{\mathcal{Z}}^{{\omega},a}_{\beta}(t,0)=e^{-\beta\bar{\kappa}_{a}t}\sum_{k=0}^{\infty}\beta^{k}\bigg{(}(k+1)\int_{{\mathfrak{X}}_{k+1}(t)\times(\mathbb{R}^{d})^{k+1}}\rho_{t,0}(\boldsymbol{t},\boldsymbol{x})\prod_{i=1}^{k+1}\xi^{a,+}_{{\omega}}(\mathrm{d}t_{i},\mathrm{d}x_{i})\\\
-\bar{\kappa}_{a}t\int_{{\mathfrak{X}}_{k}(t)\times(\mathbb{R}^{d})^{k}}\rho_{t,0}(\boldsymbol{t},\boldsymbol{x})\prod_{i=1}^{k}\xi^{a,+}_{{\omega}}(\mathrm{d}t_{i},\mathrm{d}x_{i})\bigg{)}\,.$
(5.14)

The first term in the sum can be viewed as the sum of $k+1$ integrals where
the variables of integration are
$(t_{1},\dots,t_{i},s,t_{i+1},\dots,t_{k})\in{\mathfrak{X}}_{k+1}(t)$ and
$(x_{1},\dots,x_{i},y,x_{i+1},\dots,x_{k})$ for $i=0,\dots,k$. Recombining the
terms, we arrive at the identity (recall the convention $t_{0}:=0$ and
$t_{k+1}:=t$)

$(k+1)\int_{{\mathfrak{X}}_{k+1}(t)\times(\mathbb{R}^{d})^{k+1}}\rho_{t,0}(\boldsymbol{t},\boldsymbol{x})\prod_{i=1}^{k}\xi^{a,+}_{{\omega}}(\mathrm{d}t_{i},\mathrm{d}x_{i})\\\
=\int_{{\mathfrak{X}}_{k}(t)\times(\mathbb{R}^{d})^{k}}\Bigg{(}\int_{(0,t)\times\mathbb{R}^{d}}\rho_{t,0}(\boldsymbol{t},\boldsymbol{x})\rho(s,y\,|\,\boldsymbol{t},\boldsymbol{x})\,\xi^{a,+}_{{\omega}}(\mathrm{d}s,\mathrm{d}y)\Bigg{)}\prod_{i=1}^{k}\xi^{a,+}_{{\omega}}(\mathrm{d}t_{i},\mathrm{d}x_{i}),$

where

$\rho(s,y\,|\,\boldsymbol{t},\boldsymbol{x}):=\frac{\rho(s-t_{i-1},y-x_{i-1})\rho(t_{i}-s,x_{i}-y)}{\rho(t_{i}-t_{i-1},x_{i}-x_{i-1})}\quad\text{
if }s\in(t_{i-1},t_{i})$

and $\rho(t_{i},y\,|\,\boldsymbol{t},\boldsymbol{x}):=0$ for all
$i=1,\dots,k$. As
$\int_{\mathbb{R}^{d}}\rho(s,y\,|\,\boldsymbol{t},\boldsymbol{x})\,\mathrm{d}y=1$,
the second term in (5.14) corresponds to a centering of the noise
$\xi^{a,+}_{{\omega}}(\mathrm{d}s,\mathrm{d}y)$, and we have

$\begin{split}&\partial_{\beta}\bar{\mathcal{Z}}^{{\omega},a}_{\beta}(t,0)\\\
&=e^{-\beta\bar{\kappa}_{a}t}\sum_{k=0}^{\infty}\beta^{k}\int_{{\mathfrak{X}}_{k}(t)\times(\mathbb{R}^{d})^{k}}\Bigg{(}\int_{(0,t)\times\mathbb{R}^{d}}\rho_{t,0}(\boldsymbol{t},\boldsymbol{x})\rho(s,y\,|\,\boldsymbol{t},\boldsymbol{x})\,\bar{\xi}^{a}_{{\omega}}(\mathrm{d}s,\mathrm{d}y)\Bigg{)}\prod_{i=1}^{k}\xi^{a,+}_{{\omega}}(\mathrm{d}t_{i},\mathrm{d}x_{i}).\end{split}$
(5.15)

Note that we have

$|\partial_{\beta}\bar{\mathcal{Z}}^{{\omega},a}_{\beta}(t,0)|\\\ \leq
e^{-\beta\bar{\kappa}_{a}t}\sum_{k=0}^{\infty}\beta^{k}\int_{{\mathfrak{X}}_{k}(t)\times(\mathbb{R}^{d})^{k}}\Bigg{(}\int_{(0,t)\times\mathbb{R}^{d}}\rho_{t,0}(\boldsymbol{t},\boldsymbol{x})\rho(s,y\,|\,\boldsymbol{t},\boldsymbol{x})\,|\bar{\xi}^{a}_{{\omega}}|(\mathrm{d}s,\mathrm{d}y)\Bigg{)}\prod_{i=1}^{k}\xi^{a,+}_{{\omega}}(\mathrm{d}t_{i},\mathrm{d}x_{i})\,,$

where
$\lvert\bar{\xi}^{a}_{{\omega}}\rvert=\bar{\xi}^{a}_{{\omega}}+2\bar{\kappa}_{a}\mathcal{L}$.
Since we have reduced to the case where $\varphi^{\prime}$ is bounded, the
expression above implies (cf. [13, Prop. 2.5]) that for any $\beta_{0}>0$,

$\mathbb{E}\left[\sup_{\beta\in[0,\beta_{0}]}\Big{|}\partial_{\beta}\bar{\mathcal{Z}}^{{\omega},a}_{\beta}(t,0)\varphi^{\prime}(\bar{\mathcal{Z}}^{{\omega},a}_{\beta}(t,0))\Big{|}\right]<\infty.$
(5.16)

This allows to interchange derivative and expectation and by (5.15), we obtain
that

$\begin{split}\partial_{\beta}\mathbb{E}\Big{[}\varphi\big{(}\bar{\mathcal{Z}}^{{\omega},a}_{\beta}(t,0)\big{)}\Big{]}&=\mathbb{E}\Big{[}\partial_{\beta}\bar{\mathcal{Z}}^{{\omega},a}_{\beta}(t,0)\,\varphi^{\prime}\big{(}\bar{\mathcal{Z}}^{{\omega},a}_{\beta}(t,0)\big{)}\Big{]}\\\
&=e^{-\beta\bar{\kappa}_{a}t}\sum_{k=0}^{\infty}\beta^{k}\mathbb{E}\Bigg{[}\int_{{\mathfrak{X}}_{k}(t)\times(\mathbb{R}^{d})^{k}}U^{\omega}_{k}(\boldsymbol{t},\boldsymbol{x})\rho_{t,0}(\boldsymbol{t},\boldsymbol{x})\prod_{i=1}^{k}\xi^{a,+}_{{\omega}}(\mathrm{d}t_{i},\mathrm{d}x_{i})\Bigg{]}\,,\end{split}$
(5.17)

where we have set

$U_{k}^{\omega}(\boldsymbol{t},\boldsymbol{x}):=\Bigg{(}\int_{[(0,t)\setminus\\{t_{i}\\}^{k}_{i=1}]\times\mathbb{R}^{d}}\rho(s,y\,|\,\boldsymbol{t},\boldsymbol{x})\,\bar{\xi}^{a}_{{\omega}}(\mathrm{d}s,\mathrm{d}y)\Bigg{)}\varphi^{\prime}\big{(}\bar{\mathcal{Z}}^{{\omega},a}_{\beta}(t,0)\big{)}.$

Given
$(\boldsymbol{t},\boldsymbol{x},\boldsymbol{z})\in{\mathfrak{X}}_{k}(t)\times(\mathbb{R}^{d})^{k}\times(0,\infty)^{k}$,
we let ${\omega}\cup(\boldsymbol{t},\boldsymbol{x},\boldsymbol{z})$ be the
point process obtained by adding the $k$ points $(t_{i},x_{i},z_{i})_{1\leq
i\leq k}$ to ${\omega}$. By Mecke’s multivariate equation (see [77, Thm.
4.4]), we have

$\begin{split}&\mathbb{E}\Bigg{[}\int_{{\mathfrak{X}}_{k}(t)\times(\mathbb{R}^{d})^{k}}U_{k}^{\omega}(\boldsymbol{t},\boldsymbol{x})\rho_{t,0}(\boldsymbol{t},\boldsymbol{x})\prod_{i=1}^{k}\xi^{a,+}_{{\omega}}(\mathrm{d}t_{i},\mathrm{d}x_{i})\Bigg{]}\\\
&\qquad=\mathbb{E}\Bigg{[}\int_{{\mathfrak{X}}_{k}(t)\times(\mathbb{R}^{d})^{k}\times(0,\infty)^{k}}U_{k}^{\omega}(\boldsymbol{t},\boldsymbol{x})\rho_{t,0}(\boldsymbol{t},\boldsymbol{x})\prod_{i=1}^{k}z_{i}\mathbf{1}_{[a,\infty)}(z_{i})\,\delta_{{\omega}}(\mathrm{d}t_{i},\mathrm{d}x_{i},\mathrm{d}z_{i})\Bigg{]}\\\
&\qquad=\int_{{\mathfrak{X}}_{k}(t)\times(\mathbb{R}^{d})^{k}\times(0,\infty)^{k}}\mathbb{E}\Big{[}U_{k}^{{\omega}\cup(\boldsymbol{t},\boldsymbol{x},\boldsymbol{z})}(\boldsymbol{t},\boldsymbol{x})\Big{]}\rho_{t,0}(\boldsymbol{t},\boldsymbol{x})\prod_{i=1}^{k}z_{i}\mathbf{1}_{[a,\infty)}(z_{i})\,\mathrm{d}t_{i}\,\mathrm{d}x_{i}\,{\lambda}(\mathrm{d}z_{i}).\end{split}$
(5.18)

If $k$, $\boldsymbol{t}$, $\boldsymbol{x}$ and $\boldsymbol{z}$ are fixed, the
functionals

${\omega}\mapsto\varphi^{\prime}\big{(}\bar{\mathcal{Z}}^{{\omega}\cup(\boldsymbol{t},\boldsymbol{x},\boldsymbol{z}),a}_{\beta}(t,0)\big{)}\quad\text{
and
}\quad{\omega}\mapsto\int_{[(0,t)\setminus\\{t_{i}\\}^{k}_{i=1}]\times\mathbb{R}^{d}}\rho(s,y\,|\,\boldsymbol{t},\boldsymbol{x})\,\bar{\xi}^{a}_{{\omega}}(\mathrm{d}s,\mathrm{d}y)$

are non-decreasing for the inclusion order. Thus we can apply the FKG
inequality for Poisson point processes (see [65, Lemma 2.1]) and obtain that

$\mathbb{E}\Big{[}U^{{\omega}\cup(\boldsymbol{t},\boldsymbol{x},\boldsymbol{z})}_{k}(\boldsymbol{t},\boldsymbol{x})\Big{]}\geq\mathbb{E}\Bigg{[}\int_{[(0,t)\setminus\\{t_{i}\\}^{k}_{i=1}]\times\mathbb{R}^{d}}\rho(s,y\,|\,\boldsymbol{t},\boldsymbol{x})\,\bar{\xi}^{a}_{{\omega}}(\mathrm{d}s,\mathrm{d}y)\Bigg{]}\mathbb{E}\Big{[}\varphi^{\prime}(\bar{\mathcal{Z}}^{{\omega}\cup(\boldsymbol{t},\boldsymbol{x},\boldsymbol{z}),a}_{\beta}(t,0))\Big{]}=0.$

Combining this with (5.17) and (5.18), we conclude that
$\partial_{\beta}\mathbb{E}\big{[}\varphi\big{(}\bar{\mathcal{Z}}^{{\omega},a}_{\beta}(t,0)\big{)}\big{]}\geq
0$. ∎

##  6\. Moments of order $p>1$

The goal of this section is to formulate and prove Propositions 6.1 and 6.3
below, which form the core of all moment upper bounds for $p>1$ in this paper.

###  6.1. The statements

Recall that $\nu_{p}:=1-\frac{d}{2}(p-1)$ and let
$\bar{\mathcal{Z}}^{{\omega},0}_{\beta}(t,x):=\bar{\mathcal{Z}}^{{\omega}}_{\beta}(t,x)$.

####  6.1.1. The case of dimension $d=1$

We start with $d=1$, where the statement is easier to state (and easier to
prove).

######  Proposition 6.1.

Assume that $d=1$ and that (2.1) holds.

  1. (i)

If $\mu_{1,\infty}(2)<\infty$, then for any $a\in[0,1)$,

$\begin{split}\mathbb{E}\Big{[}\rho(t,x)^{-2}\bar{\mathcal{Z}}^{{\omega},a}_{\beta}(t,x)^{2}\Big{]}&=\sum_{k=0}^{\infty}\Bigg{(}\frac{\beta^{2}\mu_{a,\infty}(2)\sqrt{t}}{2}\Bigg{)}^{k}\frac{\sqrt{\pi}}{{\Gamma}((k+1)/2)}\\\
&=1+\beta^{2}\mu_{a,\infty}(2)\sqrt{\pi
t}\exp\Big{(}\tfrac{1}{4}\beta^{4}\mu_{a,\infty}(2)^{2}t\Big{)}\Phi\Big{(}\beta^{2}\mu_{a,\infty}(2)\sqrt{\tfrac{t}{2}}\Big{)},\end{split}$
(6.1)

where $\Phi$ is the standard normal distribution function.

  2. (ii)

There exists a constant $C\in(0,\infty)$ such that if
$\mu_{1,\infty}(p)<\infty$ for some $p\in(1,2)$, then for any value of
$\eta\in(0,1]$, $a\in[0,1)$ and $t,\beta>0$,

$\mathbb{E}\Big{[}\rho(t,x)^{-p}\bar{\mathcal{Z}}^{{\omega},a}_{\beta}(t,x)^{p}\Big{]}^{\frac{1}{p}}\leq\sqrt{2}\Gamma(\nu_{p})^{\frac{1}{p}}\sum_{k_{1},k_{2}=0}^{\infty}\bigg{(}\frac{C\beta}{p-1}\bigg{)}^{k_{1}+k_{2}}\Big{(}\mu_{0,\eta}(2)t^{\frac{1}{2}}\Big{)}^{\frac{k_{1}}{2}}\frac{(\mu_{\eta,\infty}(p)t^{\nu_{p}})^{\frac{k_{2}}{p}}}{\Gamma(\nu_{p}(k_{2}+1))^{\frac{1}{p}}}.$
(6.2)

  3. (iii)

There exists a constant $C\in(0,\infty)$ such that if
$\mu_{1,\infty}(p)<\infty$ for some $p\in(2,3)$, then for any value of
$a\in[0,1)$ and $t,\beta>0$,

$\begin{split}\mathbb{E}\Big{[}\rho(t&,x)^{-p}\bar{\mathcal{Z}}^{{\omega},a}_{\beta}(t,x)^{p}\Big{]}^{\frac{1}{p}}\\\
&\leq\pi^{\frac{1}{4}}\Gamma(\nu_{p})^{\frac{1}{p}}\sum_{k_{2}=0}^{\infty}\frac{\big{(}C\beta\mu_{0,\infty}(2)^{\frac{1}{2}}\Gamma(\nu_{p})^{\frac{1}{p}}t^{\frac{1}{4}}\big{)}^{k_{2}}}{\Gamma(\frac{k_{2}+1}{2})^{\frac{1}{2}}}\Bigg{(}\sum_{\ell=0}^{\infty}\frac{\big{(}C^{\prime}\beta\mu_{0,\infty}(p)^{\frac{1}{p}}t^{\frac{\nu_{p}}{p}}\Gamma(\nu_{p})^{\frac{1}{p}}\big{)}^{\ell}}{{\Gamma}(\nu_{p}(\ell+1))^{\frac{1}{p}}}\Bigg{)}^{k_{2}+1}\,.\end{split}$
(6.3)

######  Remark 6.2.

The formula (6.1) is rather straightforward to prove. Starting from (1.28) and
using that $\bar{\xi}_{{\omega}}^{a}$ is centered, the second moment is given
by

$\begin{split}\mathbb{E}\Big{[}\rho(t,x)^{-2}\bar{\mathcal{Z}}^{{\omega},a}_{\beta}(t,x)^{2}\Big{]}=1+\sum_{k=1}^{\infty}({\beta}^{2}\mu_{a,\infty}(2))^{k}\int_{\mathfrak{X}_{k}(t)\times(\mathbb{R}^{d})^{k}}\rho(t,x)^{-2}\prod_{i=1}^{k+1}\rho(\Delta
t_{i},\Delta x_{i})^{2}\,\mathrm{d}t_{i}\,\mathrm{d}x_{i}\,,\end{split}$

with the notation (1.10) for $\Delta t_{i}$ and $\Delta x_{i}$. Integrating
over $x_{1},\ldots,x_{k}\in\mathbb{R}^{d}$, we get

$\begin{split}\mathbb{E}\Big{[}\rho(t,x)^{-2}\bar{\mathcal{Z}}^{{\omega},a}_{\beta}(t,x)^{2}\Big{]}=1+\sum_{k=1}^{\infty}({\beta}^{2}\mu_{a,\infty}(2))^{k}(2\sqrt{\pi
t})\int_{\mathfrak{X}_{k}(t)}\prod_{i=1}^{k+1}\frac{1}{2\sqrt{\pi\Delta
t_{i}}}\,\mathrm{d}t_{i}\,,\end{split}$

which gives the first equality in (6.1) by Lemma 4.5. The second equality
follows from the formula
$\sum_{k=0}^{\infty}x^{k}/\Gamma((k+1)/2)=\pi^{-1/2}+2xe^{x^{2}}\Phi(\sqrt{2}x)$.
Let us also mention that this formula coincides with the one we obtain if
$\bar{\mathcal{Z}}^{{\omega},a}_{\beta}(t,x)$ in (6.1) is replaced by the
solution to the SHE (1.1) with a space-time _Gaussian_ noise with variance
$\mu_{a,\infty}(2)$ (and $u_{0}=\delta_{0}$); cf. [30, Eq. (2.31)].

####  6.1.2. The case of dimension $d\geq 2$

For the statement with $d\geq 2$, we need to introduce a few auxiliary
quantities. With the usual convention $t_{0}:=0$ and $t_{k+1}:=t$ and $\Delta
t_{i}:=t_{i}-t_{i-1}$, we define $\Lambda(0,t,p):=1$ and

$\Lambda(k,t,p):=t^{1-\nu_{p}}\int_{\mathfrak{X}_{k}(t)}(\Delta
t_{k+1})^{\nu_{p}-1}\prod_{i=1}^{k}G_{p}(\Delta t_{i})\,\mathrm{d}t_{i}$ (6.4)

for $k\geq 1$, where

$G_{p}(s):=\begin{cases}s^{\frac{1}{3}\nu_{p}-1}\quad&\text{ if }t\leq 1,\\\
s^{\nu_{p}-1}&\text{ if }t\geq 1.\end{cases}$ (6.5)

Also, we let

$\zeta_{1}(\eta,p,t):=\Big{(}8\,\mu^{\log}_{0,\eta}\Big{(}1+\tfrac{2}{d}\Big{)}(1+\log_{+}t)\Big{)}^{\frac{p}{1+2/d}},\qquad\zeta_{2}(\eta,p):=\mu_{0,\eta}^{\log}\Big{(}1+\tfrac{2}{d}\Big{)}+\mu_{\eta,\infty}(p),$
(6.6)

where we have set $\log_{+}t:=\log(t\vee 1)$ and

$\mu_{0,\eta}^{\log}\Big{(}1+\tfrac{2}{d}\Big{)}:=\int_{(0,\eta)}z^{1+\frac{2}{d}}(3\lvert\log
z\rvert+1){\lambda}(\mathrm{d}z)$

Note that both $\zeta_{1}(\eta,p,t)$ and $\zeta_{2}(\eta,p)$ are finite if
(2.1) holds and $\mu_{1,\infty}(p)<\infty$.

######  Proposition 6.3.

Assume that $d\geq 2$ and that (2.1) holds. There exists a constant
$C\in(0,\infty)$ that only depends on $d$ such that if
$\mu_{1,\infty}(p)<\infty$ for $p\in(1,1+\frac{2}{d})$, then for any value of
$\eta\in(0,1]$, $a\in[0,1)$ and $t,\beta>0$, we have

$\mathbb{E}\Big{[}\rho(t,x)^{-p}\bar{\mathcal{Z}}^{{\omega},a}_{\beta}(t,x)^{p}\Big{]}^{\frac{1}{p}}\leq\sum_{k_{1},k_{2}=0}^{\infty}\bigg{(}\frac{C\beta}{p-1}\bigg{)}^{k_{1}+k_{2}}\zeta_{1}(\eta,p,t)^{\frac{k_{1}}{p}}\zeta_{2}(\eta,p)^{\frac{k_{2}}{p}}\Lambda(k_{2},t,p)^{\frac{1}{p}}.$
(6.7)

######  Remark 6.4.

In our proofs of Propositions 6.1 and 6.3, we keep track of the dependence in
$p$. The reason for this is that in the proof of Theorem 2.11—more precisely
in the proof of (2.16)—we need to apply (6.7) for $p$ that depends of $\beta$.
The only important point we need to make sure is that our estimates remain
uniform for $p$ in an interval around $1+\frac{2}{d}$.

####  6.1.3. Finiteness of the moments

An almost immediate consequence of the proposition is the following uniform
moment bound.

######  Corollary 6.5.

If (2.1) holds and $\mu_{1,\infty}(p)<\infty$ for some
$p\in(1,1+\frac{2}{d})$, then for any $\beta$ and $T$ there exists
$C(\beta,p,T)$ such that

$\sup_{a\in[0,1)}\sup_{(t,x)\in(0,T]\times\mathbb{R}^{d}}\mathbb{E}\Big{[}\Big{(}\rho(t,x)^{-1}\bar{\mathcal{Z}}^{{\omega},a}_{\beta}(t,x)\Big{)}^{p}\Big{]}^{\frac{1}{p}}<C(\beta,p,T).$

###### Proof.

First let us note that the bounds in Propositions 6.1 and 6.3 already are
uniform in $a$ and $x$, so we only need to check uniformity in $t$.

Let us start with the case $d=1$. The statement for $p=2$ is obvious from
(6.1). If $p\in(1,2)$, by reorganizing (6.2), we obtain that for every
$t\in[0,T]$,

$\mathbb{E}\Big{[}\rho(t,x)^{-p}\bar{\mathcal{Z}}^{{\omega},a}_{\beta}(t,x)^{p}\Big{]}^{\frac{1}{p}}\leq
C^{\prime}\Bigg{(}\sum_{k_{1}=0}^{\infty}\Big{(}C_{p}\beta\mu_{0,\eta}(2)^{\frac{1}{2}}T^{\frac{1}{4}}\Big{)}^{k_{1}}\Bigg{)}\Bigg{(}\sum_{k_{2}=0}^{\infty}\frac{\Big{(}C_{p}\beta\mu_{\eta,\infty}(p)^{\frac{1}{p}}T^{\frac{\nu_{p}}{p}}\Big{)}^{k_{2}}\Gamma(\nu_{p})^{\frac{1}{p}}}{\Gamma(\nu_{p}(k_{2}+1))^{\frac{1}{p}}}\Bigg{)},$

where $C_{p}=C/(p-1)$. The first sum is finite if one chooses $\eta$ such that
$C\beta\mu_{0,\eta}(2)^{1/2}T^{1/4}<\frac{1}{2}$ and the second one is always
finite since $\Gamma(\nu_{p}(k_{2}+1))$ grows super-exponentially. If
$p\in(2,3)$, we have from (6.3) that for every $t\in[0,T]$,

$\mathbb{E}\Big{[}\rho(t,x)^{-p}\bar{\mathcal{Z}}^{{\omega},a}_{\beta}(t,x)^{p}\Big{]}^{\frac{1}{p}}\leq
C_{p}\sum_{k_{2}=0}^{\infty}\frac{\big{(}C_{p}\beta
T^{\frac{1}{4}}\big{)}^{k_{2}}}{\Gamma(\frac{k_{2}+1}{2})^{\frac{1}{2}}}\Bigg{(}\sum_{\ell=0}^{\infty}\frac{\big{(}C^{\prime}_{p}\beta
T^{\frac{\nu_{p}}{p}}\big{)}^{\ell}}{\Gamma(\nu_{p}(\ell+1))}\Bigg{)}^{k_{2}+1}\,.$

Since $\Gamma(\nu_{p}(\ell+1))$ and $\Gamma(\frac{k_{2}+1}{2})$ both grow
super-exponentially, the two sums are finite.

When $d\geq 2$, since $\zeta_{1}$ is monotone in $t$, we have, in the same
manner,

$\mathbb{E}\Bigl{[}\rho(t,x)^{-p}\bar{\mathcal{Z}}^{{\omega},a}_{\beta}(t,x)^{p}\Bigr{]}^{\frac{1}{p}}\leq\Biggl{(}\sum_{k_{1}=0}^{\infty}\Bigl{(}C_{p}\beta\zeta_{1}(\eta,p,T)^{\frac{1}{p}}\Bigr{)}^{k_{1}}\Biggr{)}\Biggl{(}\sum_{k_{2}=0}^{\infty}\Bigl{(}C_{p}\beta\zeta_{2}(\eta,p)^{\frac{1}{p}}\Bigr{)}^{k_{2}}\Lambda(k_{2},t,p)^{\frac{1}{p}}\Biggr{)}$
(6.8)

for $t\leq T$. The first sum is finite provided that $\eta$ is chosen
sufficiently small. Considering the second term, we have

$\displaystyle\Lambda(k_{2},t,p)$ $\displaystyle\leq t^{1-\nu_{p}}(t\vee
1)^{\frac{2}{3}\nu_{p}k_{2}}\int_{\mathfrak{X}_{k_{2}}(t)}(\Delta
t_{k_{2}+1})^{\nu_{p}-1}\prod_{i=1}^{k_{2}}(\Delta
t_{i})^{\frac{1}{3}\nu_{p}-1}\,\mathrm{d}t_{i}$ $\displaystyle=(t\vee
1)^{\frac{2}{3}\nu_{p}k_{2}}t^{\frac{1}{3}\nu_{p}k_{2}}\frac{{\Gamma}(\frac{1}{3}\nu_{p})^{k_{2}}{\Gamma}(\nu_{p})}{{\Gamma}(\frac{1}{3}\nu_{p}(k_{2}+3))}\,,$

where we used Lemma 4.5 for the last identity. As a result, assuming that
$T\geq 1$, we obtain that for every $t\in[0,T]$,

$\Big{(}C_{p}\beta\zeta_{2}(\eta,p)^{\frac{1}{p}}\Big{)}^{k_{2}}\Lambda(k_{2},t,p)^{\frac{1}{p}}\leq\Big{(}C_{p}\beta\zeta_{2}(\eta,p)^{\frac{1}{p}}{\Gamma}({\textstyle\frac{1}{3}}\nu_{p})^{\frac{1}{p}}T^{\frac{\nu_{p}}{p}}\Big{)}^{k_{2}}\frac{{\Gamma}(\nu_{p})^{\frac{1}{p}}}{{\Gamma}(\frac{1}{3}\nu_{p}(k_{2}+3))^{\frac{1}{p}}},$

and since ${\Gamma}(\nu_{p}[(k_{2}/3)+1])$ grows super-exponentially, the sum
over $k_{2}$ in (6.8) is finite. ∎

###  6.2. Bounding moments in the chaos expansion: the first term

From now on, we focus on the case $x=0$, which yields no loss of generality by
(1.18). Also, in both Propositions 6.1 and 6.3 above, the case $a=0$ can be
deduced from the case $a>0$ using Fatou’s lemma: in the following, we can
always assume that $a>0$. Starting from the chaos decomposition (1.28), we can
use Minkowski’s inequality to get

$\mathbb{E}\Big{[}\rho(t,0)^{-p}\bar{\mathcal{Z}}^{{\omega},a}_{\beta}(t,0)^{p}\Big{]}^{\frac{1}{p}}\leq\sum_{k=0}^{\infty}{\beta}^{k}\mathbb{E}\left[|W_{a,k}(t)|^{p}\right]^{\frac{1}{p}}\,,$
(6.9)

where

$W_{a,0}(t):=1,\qquad
W_{a,k}(t):=\int_{{\mathfrak{X}}_{k}(t)\times(\mathbb{R}^{d})^{k}}\frac{\rho_{t,0}(\boldsymbol{t},\boldsymbol{x})}{\rho(t,0)}\prod_{i=1}^{k}\bar{\xi}^{a}_{{\omega}}(\mathrm{d}t_{i},\mathrm{d}x_{i})\quad\text{
for }k\geq 1.$ (6.10)

The estimates for $\mathbb{E}[|W_{a,k}(t)|^{p}]$ are intricate, so let us
spend some time on the case $k=1$ to illustrate the intuition behind our
proof. In what follows, we write

${\mathcal{X}}^{(k)}_{t}:={\mathfrak{X}}_{k}(t)\times(0,\infty)^{k},\quad
X^{(k)}_{t}:={\mathfrak{X}}_{k}(t)\times(\mathbb{R}^{d})^{k},\quad\mathbb{X}^{(k)}_{t}:={\mathfrak{X}}_{k}(t)\times(\mathbb{R}^{d})^{k}\times(0,\infty)^{k}.$
(6.11)

We drop the superscript $k$ when $k=1$. For simplicity, let us consider the
expansion of the free-end partition function. Because the integrals in (6.9)
are martingales in $t$ for the filtration (1.5), we can apply the
Burkholder–Davis–Gundy (BDG) inequality and obtain

$\mathbb{E}\Biggl{[}\Biggl{(}\int_{(0,t)\times\mathbb{R}^{d}}\rho(s,x)\,\bar{\xi}^{a}_{{\omega}}(\mathrm{d}s,\mathrm{d}x)\Biggr{)}^{p}\Biggr{]}\\\
\leq
C_{p}\,\mathbb{E}\Biggl{[}\Biggl{(}\int_{(0,t)\times\mathbb{R}^{d}\times[a,\infty)}(\rho(s,x)z)^{2}\,\delta_{{\omega}}(\mathrm{d}s,\mathrm{d}x,\mathrm{d}z)\Biggr{)}^{\frac{p}{2}}\Biggr{]}.$
(6.12)

Since we are tracking the dependence in $p$, it is worth noting that it is
possible to take $C_{p}=(4p)^{p}\leq 64$ if $p\leq 2$; see [49, Chapter VII,
Theorem 92]. In order to bound the right-hand side uniformly in $a$, we
replace $[a,\infty)$ by $(0,\infty)$. Let us further restrict ourselves to the
case $d\geq 3$ for simplicity (note that in particular $p<2$). By Jensen’s
inequality and the subadditivity property (5.4), we have, for
$\theta\in[p,2]$,

$\begin{split}\mathbb{E}\Bigg{[}\bigg{(}\int_{\mathbb{X}_{t}}\rho(s,x)^{2}z^{2}\,\delta_{{\omega}}(\mathrm{d}s,\mathrm{d}x,\mathrm{d}z)\bigg{)}^{\frac{p}{2}}\Bigg{]}&\leq\mathbb{E}\Bigg{[}\bigg{(}\int_{\mathbb{X}_{t}}\rho(s,x)^{2}z^{2}\,\delta_{{\omega}}(\mathrm{d}s,\mathrm{d}x,\mathrm{d}z)\bigg{)}^{\frac{\theta}{2}}\Bigg{]}^{\frac{p}{\theta}}\\\
&\leq\bigg{(}\int_{\mathbb{X}_{t}}\rho(s,x)^{\theta}z^{\theta}\,\mathrm{d}s\,\mathrm{d}x\,{\lambda}(\mathrm{d}z)\bigg{)}^{\frac{p}{\theta}}\\\
&=\vartheta(\theta)^{\frac{p}{\theta}}\bigg{(}\int_{{\mathcal{X}}_{t}}s^{\nu_{\theta}-1}z^{\theta}\,\mathrm{d}s\,{\lambda}(\mathrm{d}z)\bigg{)}^{\frac{p}{\theta}}\,,\end{split}$
(6.13)

recalling also (5.5) and $\nu_{\theta}=1-\frac{d}{2}(\theta-1)$ for the last
line. On the right-hand side, we see that when $\theta$ increases, the
integrability in $z$ around $0$ improves but the one in $s$ worsens, forcing
us to chose $\nu_{\theta}>0$ (i.e., $\theta<1+\frac{2}{d}$) to obtain a finite
integral. As a consequence, we need to assume $\mu_{0,\infty}(\theta)<\infty$
for some $\theta\in[p,1+\frac{2}{d})$. If this holds, then, in fact, the same
estimate can be applied iteratively in order to obtain bounds for any of the
multiple integrals in (6.9). Let us remark that this is essentially the
assumption (and the method) used in [89] to obtain existence, uniqueness and
moments for the solution to (2.6).

Clearly, $\mu_{0,\infty}(\theta)<\infty$ does not hold, for any $\theta>0$, if
$\xi_{\omega}$ is an $\alpha$-stable noise with ${\alpha}\in(1,2)$; recall
that we assume $\mu_{1,\infty}(p)<\infty$ for some $p>1$. In that case, at
least when $k=1$, it is easy to do better than (6.13). The key point is to
first separate $z\geq 1$ and $z<1$ and then, for $z<1$, further whether $z\leq
s^{d/2}$ or $z>s^{d/2}$. For $z\geq 1$, we can simply apply (6.13) with
$\theta=p$, that is,

$\mathbb{E}\Bigg{[}\bigg{(}\int_{\mathbb{X}_{t}}\rho(s,x)^{2}z^{2}\mathbf{1}_{\\{z\geq
1\\}}\,\delta_{{\omega}}(\mathrm{d}s,\mathrm{d}x,\mathrm{d}z)\bigg{)}^{\frac{p}{2}}\Bigg{]}\leq\vartheta(\theta)\mu_{1,\infty}(p)\nu_{p}^{-1}t^{\nu_{p}}\,.$

For $z\leq 1$, we first consider the contribution coming from $z\leq s^{d/2}$.
By Jensen’s inequality, we can take the exponent $\frac{p}{2}$ outside the
expectation on the right-hand side of (6.12), which leads to the bound

$\mathbb{E}\Bigg{[}\int_{\mathbb{X}_{t}}\mathbf{1}_{\\{z\leq s^{d/2}\wedge
1\\}}\rho(s,x)^{2}z^{2}\,\delta_{{\omega}}(\mathrm{d}s,\mathrm{d}x,\mathrm{d}z)\Bigg{]}^{\frac{p}{2}}=\Bigg{(}\int_{\mathbb{X}_{t}}\mathbf{1}_{\\{z\leq
s^{d/2}\wedge
1\\}}\rho(s,x)^{2}z^{2}\,\mathrm{d}s\,\mathrm{d}x\,{\lambda}(\mathrm{d}z)\Bigg{)}^{\frac{p}{2}}.$

Since $d\geq 3$, we have, integrating first with respect to $x$ (recall
(5.5)), then with respect to $s$ and finally with respect to $z$,

$\int_{\mathbb{X}_{t}}\mathbf{1}_{\\{z\leq s^{d/2}\wedge
1\\}}\rho(s,x)^{2}z^{2}\,\mathrm{d}s\,\mathrm{d}x\,{\lambda}(\mathrm{d}z)=\int_{{\mathcal{X}}_{t}}\frac{z^{2}\mathbf{1}_{\\{z\leq
s^{d/2}\wedge 1\\}}}{(4\pi
s)^{\frac{d}{2}}}\,\mathrm{d}s\,{\lambda}(\mathrm{d}z)\leq\frac{1}{(\frac{d}{2}-1)(4\pi)^{\frac{d}{2}}}\mu_{0,1}\Big{(}1+\tfrac{2}{d}\Big{)}.$

For the contribution to the integral coming from $z>s^{d/2}$, applying (6.13)
with $\theta=p$, we get that it is smaller than

$\displaystyle\vartheta(p)\int_{{\mathcal{X}}_{t}}\mathbf{1}_{\\{z>s^{d/2},z<1\\}}s^{\nu_{p}-1}z^{p}\,\mathrm{d}s\,{\lambda}(\mathrm{d}z)=\frac{\vartheta(p)}{\nu_{p}}\int_{(0,1)}z^{1+\frac{2}{d}}\,{\lambda}(\mathrm{d}z)=\frac{\vartheta(p)}{\nu_{p}}\mu_{0,1}\Big{(}1+\tfrac{2}{d}\Big{)}\,,$

where we have first integrated with respect to $s$, using that
$\nu_{p}=1-\frac{d}{2}(p-1)>0$ for $p<1+\frac{2}{d}$, and then with respect to
$z$. Altogether, we have shown the following bound.

######  Lemma 6.6.

In dimension $d\geq 3$, there exists a constant $C$ (which may depend on $d$
but not on $p$) such that for all $p\in(1,1+\frac{2}{d})$,

$\mathbb{E}\Bigg{[}\bigg{(}\int_{(0,t)\times\mathbb{R}^{d}}\rho(s,x)\,\bar{\xi}^{a}_{{\omega}}(\mathrm{d}s,\mathrm{d}x)\bigg{)}^{p}\Bigg{]}\leq\frac{C}{\nu_{p}}\Big{[}\mu_{0,1}\Big{(}1+\tfrac{2}{d}\Big{)}^{\frac{p}{2}}+\mu_{0,1}\Big{(}1+\tfrac{2}{d}\Big{)}+\mu_{1,\infty}(p)t^{\nu_{p}}\Big{]}.$

In other words, the single integral has a finite $p$th moment for some
$p\in(1,1+\frac{2}{d})$ if the intensity measure satisfies
$\mu_{0,1}(1+\frac{2}{d})+\mu_{1,\infty}(p)<\infty$. In fact, this condition
is necessary and sufficient by [87, Theorem 3.3], so Lemma 6.6 is optimal for
single integrals.

###  6.3. Decoupling and partitioning: Key tools in proving Propositions 6.1
and 6.3

####  6.3.1. Decoupling

When $k\geq 2$, there is no direct analogue of (6.12) since we cannot apply
the BDG inequality for the $k$-fold iterated integral. The first step of our
proof is to use the decoupling inequalities from Section 4.2 in order to
obtain, instead of $W_{a,k}(t)$, multiple Poisson integrals with respect to
$k$ independent copies of the original noise. An important advantage of the
decoupled integral is that we can change the order of integration without
losing the martingale property. Using Theorem 4.3 with the tetrahedral
function
$f(\boldsymbol{t},\boldsymbol{x},\boldsymbol{z}):=\rho(t,0)^{-1}\rho_{t,0}(\boldsymbol{t},\boldsymbol{x})\prod_{i=1}^{k}z_{i}\mathbf{1}_{\\{z_{i}\geq
a\\}}$ (the reader can check that (4.3) is satisfied whenever
$\mu_{1,\infty}(p)<\infty$), we have

$\mathbb{E}[\lvert
W_{a,k}(t)\rvert^{p}]\leq\bigg{(}\frac{C}{p-1}\bigg{)}^{k}\,\mathbb{E}^{\otimes
k}[\lvert V_{a,k}(t)\rvert^{p}]\,,$ (6.14)

where

$\begin{split}V_{a,k}(t):=&~{}\int_{X_{t}^{(k)}}\frac{\rho_{t,0}(\boldsymbol{t},\boldsymbol{x})}{{\rho(t,0)}}\prod_{i=1}^{k}\bar{\xi}^{a}_{{\omega}_{i}}(\mathrm{d}t_{i},\mathrm{d}x_{i})\\\
=&~{}\int_{\mathbb{X}_{t}^{(k)}}\frac{\rho_{t,0}(\boldsymbol{t},\boldsymbol{x})}{{\rho(t,0)}}\prod_{i=1}^{k}z_{i}\mathbf{1}_{\\{z_{i}\geq
a\\}}\,(\delta_{{\omega}_{i}}-\nu)(\mathrm{d}t_{i},\mathrm{d}x_{i},\mathrm{d}z_{i})\end{split}$
(6.15)

and ${\omega}_{i}$, for $i=1,\dots,k$, are i.i.d. copies of ${\omega}$. For
simplicity, and with a small abuse of notation, we write $\mathbb{P}$,
$\mathbb{P}^{\otimes k}$, $\mathbb{E}$ and $\mathbb{E}^{\otimes k}$ in the
remainder of the proof.

####  6.3.2. Partitioning and integrating over space

As in Section 6.2, we want to estimate the $p$th moment of the right-hand side
of (6.15) using a combination of the BDG inequality, Jensen’s inequality and
subadditivity and then integrate over space. Because we want to use an
intermediate exponent as in (6.13) that depends on the value of
$\boldsymbol{t}$ and $\boldsymbol{z}$, a first task is to decompose
$V_{a,k}(t)$ by considering a (non-random) partition $\mathfrak{P}_{k}$ of the
parameter space ${\mathcal{X}}^{(k)}_{t}$. For each element
$\mathcal{P}\in\mathfrak{P}_{k}$ of our partition, we will further determine a
partition $J_{1}(\mathcal{P})\cup J_{2}(\mathcal{P})$ of $\llbracket
k\rrbracket$ and first integrate with respect to $(t_{i},z_{i})$ with $i\in
J_{1}(\mathcal{P})$ and then with respect to $(t_{i},z_{i})$ with $i\in
J_{2}(\mathcal{P})$. We let $k_{1}$ and $k_{2}$ denote the respective
cardinalities of $J_{1}=J_{1}({\mathcal{P}})$ and
$J_{2}=J_{2}({\mathcal{P}})$.

When $J\subseteq\llbracket k\rrbracket$, we use the following notation:

$\Delta_{J}t_{i}:=t_{i}-t_{i(J,-1)}\qquad\text{and}\qquad\Delta_{J}x_{i}:=x_{i}-x_{i(J,-1)}$
(6.16)

where $x_{0}=x_{k+1}=0$, $t_{0}=0$ and $t_{k+1}=t$ as usual and where for
$i\in J\cup\\{k+1\\}$ we denote the predecessor of $i$ in $J\cup\\{0,k+1\\}$
by $i(J,-1)$ . As a result of our partitioning procedure, we obtain the
following estimate:

######  Lemma 6.7.

Given $p\in(1,2\wedge(1+\frac{2}{d}))$ and $\theta\in(p,2]$, we have for all
$a\in(0,1]$,

$\begin{split}\mathbb{E}\Big{[}\big{\lvert}V_{a,k}(t)\big{\rvert}^{p}\Big{]}^{\frac{1}{p}}\leq
C^{k}\sum_{\mathcal{P}\in\mathfrak{P}_{k}}t^{\frac{1-\nu_{p}}{p}}\Bigg{(}&\int_{{\mathcal{X}}_{t}^{(k_{2})}}\Biggl{(}\int_{{\mathcal{X}}_{t}^{(k_{1})}}\mathbf{1}_{\mathcal{P}}(\boldsymbol{t},\boldsymbol{z})\prod_{i=1}^{k+1}(\Delta
t_{i})^{\nu_{\theta}-1}\prod_{i\in
J_{1}}z^{\theta}_{i}\,\mathrm{d}t_{i}\,{\lambda}(\mathrm{d}z_{i})\Biggr{)}^{\frac{p}{\theta}}\\\
&\times\prod_{i\in
J_{2}\cup\\{k+1\\}}(\Delta_{J_{2}}t_{i})^{(\nu_{p}-1)+\frac{p}{\theta}(1-\nu_{\theta})}\prod_{i\in
J_{2}}z^{p}_{i}\,\mathrm{d}t_{i}\,{\lambda}(\mathrm{d}z_{i})\Bigg{)}^{\frac{1}{p}}.\end{split}$
(6.17)

###### Proof.

For $\mathcal{P}\in\mathfrak{P}_{k}$, let us define

$V_{a,k}(t,\mathcal{P}):=\int_{\mathbb{X}_{t}^{(k)}}\mathbf{1}_{\mathcal{P}}(\boldsymbol{t},\boldsymbol{z})\frac{\rho_{t,0}(\boldsymbol{t},\boldsymbol{x})}{{\rho(t,0)}}\prod_{i=1}^{k}z_{i}\mathbf{1}_{\\{z_{i}\geq
a\\}}\,(\delta_{{\omega}_{i}}-\nu)(\mathrm{d}t_{i},\mathrm{d}x_{i},\mathrm{d}z_{i})\,.$
(6.18)

Then, by (6.15) and Minkowski’s inequality, we have that

$\displaystyle\mathbb{E}\Big{[}\big{\lvert}V_{a,k}(t)\big{\rvert}^{p}\Big{]}^{\frac{1}{p}}$
$\displaystyle\leq\sum_{\mathcal{P}\in\mathfrak{P}_{k}}\mathbb{E}\Big{[}\big{\lvert}V_{a,k}(t,\mathcal{P})\big{\rvert}^{p}\Big{]}^{\frac{1}{p}}.$

Because the ${\omega}_{i}$’s are independent, we can, similarly to (A.4) in
the appendix, permute the integrals in (6.15) and integrate with respect to
the indices in $J_{1}$ first and $J_{2}$ afterwards. In conjunction with the
BDG inequality, subadditivity (recall $p\leq 2$) and Jensen’s inequality
(applied in the fashion as in (6.13)), we obtain

$\begin{split}&\mathbb{E}[\lvert V_{a,k}(t,\mathcal{P})\rvert^{p}]\\\
&\qquad\leq\begin{multlined}C^{k_{2}}\int_{\mathbb{X}_{t}^{(k_{2})}}\mathbb{E}\Bigg{[}\bigg{\lvert}\int_{\mathbb{X}_{t}^{(k_{1})}}\mathbf{1}_{\mathcal{P}}(\boldsymbol{t},\boldsymbol{z})\frac{\rho_{t,0}(\boldsymbol{t},\boldsymbol{x})}{{\rho(t,0)}}\prod_{i\in
J_{1}}z_{i}\mathbf{1}_{\\{z_{i}\geq
a\\}}\,(\delta_{{\omega}_{i}}-\nu)(\mathrm{d}t_{i},\mathrm{d}x_{i},\mathrm{d}z_{i})\bigg{\rvert}^{p}\Bigg{]}\\\
\times\prod_{i\in J_{2}}z_{i}^{p}\mathbf{1}_{\\{z_{i}\geq
a\\}}\,\mathrm{d}t_{i}\,\mathrm{d}x_{i}\,{\lambda}(\mathrm{d}z_{i})\end{multlined}C^{k_{2}}\int_{\mathbb{X}_{t}^{(k_{2})}}\mathbb{E}\Bigg{[}\bigg{\lvert}\int_{\mathbb{X}_{t}^{(k_{1})}}\mathbf{1}_{\mathcal{P}}(\boldsymbol{t},\boldsymbol{z})\frac{\rho_{t,0}(\boldsymbol{t},\boldsymbol{x})}{{\rho(t,0)}}\prod_{i\in
J_{1}}z_{i}\mathbf{1}_{\\{z_{i}\geq
a\\}}\,(\delta_{{\omega}_{i}}-\nu)(\mathrm{d}t_{i},\mathrm{d}x_{i},\mathrm{d}z_{i})\bigg{\rvert}^{p}\Bigg{]}\\\
\times\prod_{i\in J_{2}}z_{i}^{p}\mathbf{1}_{\\{z_{i}\geq
a\\}}\,\mathrm{d}t_{i}\,\mathrm{d}x_{i}\,{\lambda}(\mathrm{d}z_{i})\\\
&\qquad\leq
C^{k}\int_{\mathbb{X}_{t}^{(k_{2})}}\bigg{(}\int_{\mathbb{X}_{t}^{(k_{1})}}\mathbf{1}_{\mathcal{P}}(\boldsymbol{t},\mathbf{z})\frac{\rho_{t,0}(\boldsymbol{t},\boldsymbol{x})^{\theta}}{\rho(t,0)^{\theta}}\prod_{i\in
J_{1}}z^{\theta}_{i}\,\mathrm{d}t_{i}\,\mathrm{d}x_{i}\,{\lambda}(\mathrm{d}z_{i})\bigg{)}^{\frac{p}{\theta}}\prod_{i\in
J_{2}}z^{p}_{i}\,\mathrm{d}t_{i}\,\mathrm{d}x_{i}\,{\lambda}(\mathrm{d}z_{i}).\end{split}$
(6.19)

######  Remark 6.8.

In the above integrals and for the remainder of the proof, with a small abuse
of notation, the coordinates of elements of $\mathbb{X}_{t}^{(k_{1})}$ are
indexed by $J_{1}$ instead of $\llbracket k_{1}\rrbracket$ (and similarly for
$J_{2}$). With this convention, $\mathbb{X}^{(k)}_{t}$ is a strict subset of
$\mathbb{X}_{t}^{(k_{1})}\times\mathbb{X}_{t}^{(k_{2})}$, but this is not a
problem since the indicator function $\mathbf{1}_{\mathcal{P}}$ restricts the
integral to a subset of $\mathbb{X}^{(k)}_{t}$.

The next step is to carry out integration with respect to $x_{1},\dots,x_{k}$
explicitly. First of all, notice that for any $m\geq 1$ and any $p>1$, we have
for every $\boldsymbol{t}\in\mathfrak{X}_{m}(s)$

$\int_{(\mathbb{R}^{d})^{m}}\rho_{s,y}(\boldsymbol{t},\boldsymbol{x})^{p}\prod_{i=1}^{m}\mathrm{d}x_{i}=\vartheta(p)^{m}\rho(s,y)^{p}s^{1-\nu_{p}}\prod_{i=1}^{m+1}(\Delta
t_{i})^{\nu_{p}-1}\,.$ (6.20)

This can be proved by induction on $m$ after checking the case $m=1$ by hand,
using (5.5); we leave the details to the reader.

We then successively apply (6.20) with $p=\theta$ to the segments of $J_{1}$
(i.e., maximal sets of consecutive indices in $J_{1}$). Recalling the notation
(6.16), we get

$\int_{(\mathbb{R}^{d})^{k_{1}}}\rho_{t,0}(\boldsymbol{t},\boldsymbol{x})^{\theta}\prod_{i\in
J_{1}}\mathrm{d}x_{i}=\vartheta(\theta)^{k_{1}}\prod_{i=1}^{k+1}(\Delta
t_{i})^{\nu_{\theta}-1}\prod_{j\in
J_{2}\cup\\{k+1\\}}(\Delta_{J_{2}}t_{j})^{1-\nu_{\theta}}\rho(\Delta_{J_{2}}t_{j},\Delta_{J_{2}}x_{j})^{\theta}\,.$
(6.21)

Applying (6.20) in the case $y=0$, we also get

$\int_{(\mathbb{R}^{d})^{k_{2}}}\prod_{j\in
J_{2}\cup\\{k+1\\}}\rho(\Delta_{J_{2}}t_{j},\Delta_{J_{2}}x_{j})^{p}\prod_{i\in
J_{2}}\mathrm{d}x_{i}=\vartheta(p)^{k_{2}}\rho(t,0)^{p}t^{1-\nu_{p}}\prod_{j\in
J_{2}\cup\\{k+1\\}}(\Delta_{J_{2}}t_{j})^{\nu_{p}-1}\,.$ (6.22)

We conclude the proof of Lemma 6.7 by inserting (6.21) and (6.22) in (6.19). ∎

###  6.4. The proof of Proposition 6.1

The second moment was computed in Remark 6.2. In order to prove (6.2), we only
need to separate small and large values of $z$. We consider a partition
$\mathfrak{P}_{k}$ indexed by the subsets $J\subseteq\llbracket k\rrbracket$
and we define

$\mathcal{P}(J):=\\{(\boldsymbol{t},\boldsymbol{z})\in{\mathcal{X}}^{(k)}_{t}\
:\ z_{i}<\eta\text{ for all }i\in J\text{ and }z_{i}\geq\eta\text{ for all
}i\in\llbracket k\rrbracket\setminus J\\}$ (6.23)

as well as $J_{1}:=J$ and $J_{2}:=\llbracket k\rrbracket\setminus J$. Applying
Lemma 6.7 with $\theta=2$ and noting that $\nu_{2}=\frac{1}{2}$ and
$\nu_{p}=1-\frac{1}{2}(p-1)$ in dimension $d=1$, we obtain

$\begin{split}\mathbb{E}&\Big{[}\big{\lvert}V_{a,k}(t)\big{\rvert}^{p}\Big{]}^{\frac{1}{p}}\leq
C^{k}\sum_{J\subseteq\llbracket
k\rrbracket}t^{\frac{1-\nu_{p}}{p}}\mu_{0,\eta}(2)^{\frac{k_{1}}{2}}\mu_{\eta,\infty}(p)^{\frac{k_{2}}{p}}\\\
&\times\Bigg{(}\int_{(0,t)^{k_{2}}}\Biggl{(}\int_{(0,t)^{k_{1}}}\mathbf{1}_{\\{t_{1}<\dots<t_{k}\\}}\prod_{i=1}^{k+1}(\Delta
t_{i})^{-\frac{1}{2}}\prod_{i\in
J_{1}}\mathrm{d}t_{i}\Biggr{)}^{\frac{p}{2}}\prod_{i\in
J_{2}\cup\\{k+1\\}}(\Delta_{J_{2}}t_{i})^{\frac{2-p}{4}}\prod_{i\in
J_{2}}\mathrm{d}t_{i}\Bigg{)}^{\frac{1}{p}}.\end{split}$ (6.24)

Integrating successively over the segments of $J_{1}$ and writing
$\Delta_{J_{2}}(i)=i-i(J,-1)$ for the distance between $i\in
J_{2}\cup\\{k+1\\}$ and the previous index in $J_{2}$ (recall (6.16)), we get
from Lemma 4.5 that

$\int_{(0,t)^{k_{1}}}\mathbf{1}_{\mathfrak{X}_{k}(t)}(\boldsymbol{t})\prod_{i=1}^{k+1}(\Delta
t_{i})^{-\frac{1}{2}}\prod_{i\in
J_{1}}\mathrm{d}t_{i}=\mathbf{1}_{\mathfrak{X}_{k_{2}}(t)}((t_{i})_{i\in
J_{2}})\prod_{i\in
J_{2}\cup\\{k+1\\}}(\Delta_{J_{2}}t_{i})^{\frac{1}{2}\Delta_{J_{2}}(i)-1}\frac{{\Gamma}(\frac{1}{2})^{\Delta_{J_{2}}(i)}}{{\Gamma}(\frac{1}{2}\Delta_{J_{2}}(i))}\,.$
(6.25)

Now, if we bound $(\Delta_{J_{2}}t_{i})^{(\Delta_{J_{2}}(i)-1)/2}\leq
t^{(\Delta_{J_{2}}(i)-1)/2}$ and
${\Gamma}(\frac{1}{2}\ell)\geq\frac{1}{2}\sqrt{\pi}$ for any $\ell\geq 1$, we
get that

$\int_{(0,t)^{k_{1}}}\mathbf{1}_{\mathfrak{X}_{k}(t)}(\boldsymbol{t})\prod_{i=1}^{k+1}(\Delta
t_{i})^{-\frac{1}{2}}\prod_{i\in
J_{1}}\mathrm{d}t_{i}\leq\mathbf{1}_{\mathfrak{X}_{k_{2}}(t)}((t_{i})_{i\in
J_{2}})2^{k_{2}+1}(\pi t)^{\frac{1}{2}k_{1}}\prod_{i\in
J_{2}\cup\\{k+1\\}}(\Delta_{J_{2}}t_{i})^{-\frac{1}{2}}.$

Going back to (6.24) and taking the factor
$[(\Delta_{J_{2}}t_{i})^{-1/2}]^{p/2}$ over to the outer integral, we compute

$\int_{(0,t)^{k_{2}}}\mathbf{1}_{\mathfrak{X}_{k_{2}}(t)}((t_{i})_{i\in
J_{2}})\prod_{i\in
J_{2}\cup\\{k+1\\}}(\Delta_{J_{2}}t_{i})^{\frac{1-p}{2}}\prod_{i\in
J_{2}}\mathrm{d}t_{i}=t^{\nu_{p}k_{2}+\nu_{p}-1}\frac{{\Gamma}(\nu_{p})^{k_{2}+1}}{{\Gamma}(\nu_{p}(k_{2}+1))}\,,$
(6.26)

thanks again to Lemma 4.5. We therefore conclude that

$\begin{split}\mathbb{E}\Big{[}\big{\lvert}V_{a,k}(t)\big{\rvert}^{p}\Big{]}^{\frac{1}{p}}&\leq\sum_{J\subseteq\llbracket
k\rrbracket}\Big{(}\sqrt{\pi
t}\mu_{0,\eta}(2)\Big{)}^{\frac{k_{1}}{2}}\Big{(}2^{\frac{p}{2}}\mu_{\eta,\infty}(p)t^{\nu_{p}}{\Gamma}(\nu_{p})\Big{)}^{\frac{k_{2}}{p}}\frac{\sqrt{2}{\Gamma}(\nu_{p})^{\frac{1}{p}}}{{\Gamma}(\nu_{p}(k_{2}+1))^{\frac{1}{p}}}\\\
&\leq(2C)^{k}\max_{k_{1}+k_{2}=k}\Big{(}\sqrt{\pi
t}\mu_{0,\eta}(2)\Big{)}^{\frac{k_{1}}{2}}\Big{(}2^{\frac{p}{2}}\mu_{\eta,\infty}(p)t^{\nu_{p}}{\Gamma}(\nu_{p})\Big{)}^{\frac{k_{2}}{p}}\frac{\sqrt{2}{\Gamma}(\nu_{p})^{\frac{1}{p}}}{{\Gamma}(\nu_{p}(k_{2}+1))^{\frac{1}{p}}}.\end{split}$
(6.27)

Absorbing $\pi^{k_{1}/4}$ and $(2^{p/2}{\Gamma}(\nu_{p}))^{k_{2}/p}$, which is
uniformly bounded in $p\in[1,2]$, into the constant $C$ and replacing the
maximum by a sum, we derive (6.2) from (6.9), (6.14) and (6.27).

If $p\in(2,3)$, the subadditivity argument in (6.19) does not apply. Instead,
we shall use a variant of the BDG estimate (6.12) that only contains the
intensity measure
$\nu:=\mathrm{d}t\otimes\mathrm{d}x\otimes{\lambda}(\mathrm{d}z)$ instead of
$\delta_{\omega}$. For any $p\geq 2$, there exists a constant
$C^{\prime}_{p}\in(0,\infty)$ such that for all
${\mathcal{P}}\otimes{\mathcal{B}}(\mathbb{R}^{d}\times(0,\infty))$-measurable
process $K=K({\omega},w)=K({\omega},t,x,z)$,

$\mathbb{E}\Bigg{[}\bigg{\lvert}\int_{\mathbb{X}}K(w)\,(\delta_{\omega}-\nu)(\mathrm{d}w)\bigg{\rvert}^{p}\Bigg{]}\leq
C^{\prime}_{p}\left\\{\mathbb{E}\Bigg{[}\bigg{(}\int_{\mathbb{X}}K(w)^{2}\,\nu(\mathrm{d}w)\bigg{)}^{\frac{p}{2}}\Bigg{]}+\mathbb{E}\Bigg{[}\int_{\mathbb{X}}\lvert
K(w)\rvert^{p}\,\nu(\mathrm{d}w)\Bigg{]}\right\\},$ (6.28)

where $C^{\prime}_{p}=(2^{p-1}p(p-1)(p/(p-1))^{p})^{p/2}\vee
2(p/(p-1))^{p}(2^{p}+1+p)\leq 729$ when $p\in[2,3]$; see [85, Theorem 1 (b)]
(and its proof for the value of $C^{\prime}_{p}$). We also refer to [79] for a
survey of various versions and proofs (and names) of this inequality. Let us
define

$\widetilde{W}_{a,0}(t,x):=\rho(t,x),\qquad\widetilde{W}_{a,k}(t,x):=\int_{X^{(k)}_{t}}{\rho_{t,x}(\boldsymbol{t},\boldsymbol{x})}\prod_{i=1}^{k}\bar{\xi}^{a}_{{\omega}}(\mathrm{d}t_{i},\mathrm{d}x_{i})\quad\text{
for }k\geq 1,$

so that in particular $W_{a,k}(t)=\widetilde{W}_{a,k}(t,0)/\rho(t,0)$. With
this definition, applying (6.28), we have

$\displaystyle\mathbb{E}\Big{[}|\widetilde{W}_{a,k}(t,x)|^{p}\Big{]}^{\frac{1}{p}}$
$\displaystyle=\mathbb{E}\Bigg{[}\bigg{|}\int_{X_{t}}\rho(t-t_{k},0-x_{k})\widetilde{W}_{a,k-1}(t_{k},x_{k})\,\bar{\xi}^{a}_{\omega}(\mathrm{d}t_{k},\mathrm{d}x_{k})\bigg{|}^{p}\Bigg{]}^{\frac{1}{p}}$
$\displaystyle\leq
C^{\prime}\,\Bigg{\\{}\mathbb{E}\Bigg{[}\bigg{(}\mu_{0,\infty}(2)\int_{X_{t}}(\rho(t-t_{k},x_{k})\widetilde{W}_{a,k-1}(t_{k},x_{k}))^{2}\,\mathrm{d}t_{k}\,\mathrm{d}x_{k}\bigg{)}^{\frac{p}{2}}\Bigg{]}^{\frac{1}{p}}$
$\displaystyle\qquad\qquad\qquad+\bigg{(}\mu_{0,\infty}(p)\int_{X_{t}}\rho(t-t_{k},x_{k})^{p}\mathbb{E}[|\widetilde{W}_{a,k-1}(t_{k},x_{k})|^{p}]\,\mathrm{d}t_{k}\,\mathrm{d}x_{k}\bigg{)}^{\frac{1}{p}}\Bigg{\\}}\,.$

Thus, applying Minkowski’s integral inequality, we get

$\displaystyle\mathbb{E}\Big{[}|\widetilde{W}_{a,k}(t,x)|^{p}\Big{]}^{\frac{1}{p}}$
$\displaystyle\leq
C^{\prime}\Bigg{\\{}\,\mu_{0,\infty}(2)^{\frac{1}{2}}\bigg{(}\int_{X_{t}}\rho(t-t_{k},x_{k})^{2}\mathbb{E}[\lvert\widetilde{W}_{a,k-1}(t_{k},x_{k})\rvert^{p}]^{\frac{2}{p}}\,\mathrm{d}t_{k}\,\mathrm{d}x_{k}\bigg{)}^{\frac{1}{2}}$
$\displaystyle\qquad\qquad\qquad+\mu_{0,\infty}(p)^{\frac{1}{p}}\bigg{(}\int_{X_{t}}\rho(t-t_{k},x_{k})^{p}\mathbb{E}[|\widetilde{W}_{a,k-1}(t_{k},x_{k})|^{p}]\,\mathrm{d}t_{k}\,\mathrm{d}x_{k}\bigg{)}^{\frac{1}{p}}\Bigg{\\}}\,.$

Repeating this estimate and recalling (6.9), we get

$\mathbb{E}\Big{[}\bar{\mathcal{Z}}^{{\omega},a}_{\beta}(t,0)^{p}\Big{]}^{\frac{1}{p}}\leq\rho(t,0)+\sum_{k=1}^{\infty}(C^{\prime}\beta)^{k}\sum_{\boldsymbol{\theta}\in\\{2,p\\}^{k}}\mu_{0,\infty}(p)^{\frac{k_{1}}{p}}\mu_{0,\infty}(2)^{\frac{k_{2}}{2}}\lVert\rho_{t,0}(\boldsymbol{t},\boldsymbol{x})\rVert_{\boldsymbol{\theta},t},$
(6.29)

where $k_{1}$ is the number of $p$’s and $k_{2}$ the number of $2$’s in
$\boldsymbol{\theta}$, and where we have defined

$\lVert
f\rVert_{\boldsymbol{\theta},t}:=\bigg{(}\int_{X_{t}}\bigg{(}\cdots\bigg{(}\int_{X_{t}}\bigg{(}\int_{X_{t}}f(\boldsymbol{t},\boldsymbol{x})^{\theta_{1}}\,\mathrm{d}t_{1}\,\mathrm{d}x_{1}\bigg{)}^{\frac{\theta_{2}}{\theta_{1}}}\,\mathrm{d}t_{2}\,\mathrm{d}x_{2}\bigg{)}^{\frac{\theta_{3}}{\theta_{2}}}\cdots\bigg{)}^{\frac{\theta_{k}}{\theta_{k-1}}}\,\mathrm{d}t_{k}\,\mathrm{d}x_{k}\bigg{)}^{\frac{1}{\theta_{k}}}$

for $f:((0,\infty)\times\mathbb{R}^{d})^{k}\to[0,\infty)$ and
$\boldsymbol{\theta}=(\theta_{1},\dots,\theta_{k})\in[1,\infty)^{k}$.

Now in order to conclude, we want to replace
$\lVert\rho_{t,0}(\boldsymbol{t},\boldsymbol{x})\rVert_{\boldsymbol{\theta},t}$
by an integral which is analogous to that found in the on the right-hand side
of (6.19). To do so, we use the following identity, valid for any positive
function $f$ and any measures $\mu_{1}$ and $\mu_{2}$: