# Stochastic wave equation with heavy-tailed noise: Uniqueness of solutions
and past-light cone property

Juan J. Jiménez 111University of Ottawa, Department of Mathematics and
Statistics, 150 Louis Pasteur Private, Ottawa, Ontario, K1G 0P8, Canada.
E-mail address<EMAIL_ADDRESS>
(October 9, 2023)

###### Abstract

In [5] was proved the existence of a solution for the stochastic wave equation
driven by a Lévy noise, which may have infinite $p$-th moments. In this work,
we show the uniqueness of this solution using a compact support property of
the fundamental solution of the wave operator. Furthermore, we extend this
result to a general class of wave equations. Secondly, as a consequence of the
past-light cone property, we show that $p$-th moments of the solution are
bounded up to a certain stopping times that depend on a bounded region in
$\mathbb{R}^{d}$.

MSC 2020: Primary 60H15; Secondary 60G60, 60G51

Keywords: stochastic partial differential equations, random fields, space-time
Lévy white noise

##  1 Introduction

Let $(\Omega,\mathcal{F},(\mathcal{F}_{t})_{t\in\mathbb{R}_{+}},\mathbb{P})$
be a stochastic basis satisfying the usual conditions of completeness and
right-continuity. In this article, we study the stochastic wave equation with
spatial dimension $d=1,2$, given by

$\begin{cases}\dfrac{\partial^{2}u}{\partial t^{2}}(t,x)=\Delta
u(t,x)+\sigma(u(t,x))\dot{\Lambda}(t,x),\quad t>0,\,x\in\mathbb{R}^{d},\\\
u(0,x)=u_{0}(x),\quad\quad\dfrac{\partial u}{\partial
t}(0,x)=v_{0}(x),\quad\quad\quad\quad\quad x\in\mathbb{R}^{d},\\\ \end{cases}$
(1)

where $\sigma$ is a globally Liptchiz function, $u_{0}$ and $v_{0}$ are
assumed to be non-random measurable functions, and
$\Lambda=\\{\Lambda(B);B\in\mathcal{B}_{b}(\mathbb{R}_{+}\times\mathbb{R}^{d})\\}$
is a pure-jump Lévy space-time white noise satisfying

$\Lambda(B)=b|B|+\int_{B\times\\{|z|\leq
1\\}}z\tilde{J}(dt,dx,dz)+\int_{B\times\\{|z|>1\\}}zJ(dt,dx,dz),$ (2)

where $\mathcal{B}_{b}(\mathbb{R}_{+}\times\mathbb{R}^{d})$ is the class of
Borel sets in $\mathbb{R}_{+}\times\mathbb{R}^{d}$ with finite Lebesgue
measure, $b\in\mathbb{R}$, $|B|$ is the Lebesgue measure of $B$ in
$\mathbb{R}^{d+1}$, $J$ is a Poisson random measure on
$\mathbb{R}_{+}\times\mathbb{R}^{d}\times\mathbb{R}$ with intensity
$m(dt,dx,dz)=dtdx\nu(dz)$, and $\tilde{J}$ is the compensated Poisson random
measure of $J$ given by $\tilde{J}=J-m$. Here, $\nu$ is a Lévy measure defined
on $\mathbb{R}$ i.e. $\nu$ satisfies

$\int_{\mathbb{R}}(|z|^{2}\wedge
1)\nu(dz)<+\infty\quad\text{and}\quad\nu(\\{0\\})=0.$ (3)

We say that a random field $\phi=\\{\phi(t,x);t\geq 0,\,x\in\mathbb{R}^{d}\\}$
is predictable if it is measurable with respect to the $\sigma$-field
$\tilde{\mathcal{P}}=\mathcal{P}_{0}\times\mathcal{B}(\mathbb{R}^{d})$, where
$\mathcal{P}_{0}$ is the predictable $\sigma$-field on
$\Omega\times\mathbb{R}_{+}$, and $\mathcal{B}(\mathbb{R}^{d})$ is the Borel
$\sigma$-field in $\mathbb{R}^{d}$. We denote by $\mathcal{P}$ the collection
of predictable processes. A predictable random field $u=\\{u(t,x)\,;\,t\geq
0,\,x\in\mathbb{R}^{d}\\}$ is a mild solution of (1) if it satisfies the
stochastic integral equation

$u(t,x)=w(t,x)+\int_{0}^{t}\int_{\mathbb{R}^{d}}G_{t-s}(x-y)\sigma(u(s,y))\Lambda(ds,dy),$
(4)

where $G_{t}(x)$ is the fundamental solution of the wave operator, i.e.

$G_{t}(x)=\begin{cases}\frac{1}{2}\mathds{1}_{\\{|x|<t\\}}\qquad\qquad\qquad\text{if}\quad
d=1,\\\
\frac{1}{2\pi}\frac{1}{\sqrt{t^{2}-|x|^{2}}}\mathds{1}_{\\{|x|<t\\}}\qquad\text{if}\quad
d=2,\\\ \end{cases}$ (5)

and $w$ is the solution of the wave equation $\frac{\partial^{2}u}{\partial
t^{2}}-\Delta u=0$ on $\mathbb{R}_{+}\times\mathbb{R}^{d}$, with the same
initial conditions of (1), i.e.

$w(t,x)=(G_{t}\ast v_{0})(x)+\frac{\partial}{\partial t}(G_{t}\ast u_{0})(x).$
(6)

We assume the following conditions for the initial conditions $u_{0}$ and
$v_{0}$.

######  Assumption 1.

We assume that $u_{0}$ and $v_{0}$ are deterministic functions satisfying:

  * •

If $d=1$, $u_{0}$ is locally bounded and continuous, and $v_{0}$ is locally
bounded and measurable.

  * •

If $d=2$, $u_{0}\in C^{1}(\mathbb{R}^{2})$ and there exists
$q_{0}\in(2,\infty]$ such that $u_{0}$, $\nabla u_{0}$ $v_{0}\in
L_{\text{loc}}^{q_{0}}(\mathbb{R}^{2}).$

Observe that under Assumption 1,

$\sup_{t\in[0,T]}\sup_{|x|\leq R}|w(t,x)|<+\infty\text{ for all
$T,R\in\mathbb{R}_{+}.$}$ (7)

Regarding the theory of stochastic integration, we use the theory developed in
[7] based on the concept of Daniell mean. The stochastic integral on the
right-hand side of (4) is defined in Appendix A. This integration theory has
been successfully applied for studying SPDEs driven by heavy-tail noise in [5,
10, 11, 12].

Throughout this work, we will use the following notation:

  * •

$\tilde{\mathcal{P}}_{b}$ is the collection of all sets
$A\in\tilde{\mathcal{P}}$ such that there exists $k\in\mathbb{N}$ with
$A\subset\Omega\times[0,k]\times[-k,k]^{d}$,

  * •

$||X||_{p}:=\mathbb{E}|X|^{p}$ for $0<p<1$ and
$||X||_{p}:=(\mathbb{E}|X|^{p})^{\frac{1}{p}}$ if $p>1$,

  * •

$||X||_{0}:=\mathbb{E}[|X|\wedge 1]$, and $||X||_{\infty}:=\inf\\{C\geq
0\,;\,P\\{|X|\leq C\\}=1\\}$,

  * •

$[\\![R,S]\\!]:=\\{(\omega,t)\in\Omega\times\mathbb{R}_{+};R(\omega)\leq t\leq
S(\omega)\\}$ for two $\mathcal{F}_{t}$-stopping times $R$ and $S$,

  * •

$(\\!(R,S]\\!]:=\\{(\omega,t)\in\Omega\times\mathbb{R}_{+};R(\omega)<t\leq
S(\omega)\\}$ for two $\mathcal{F}_{t}$-stopping times $R$ and $S$,

  * •

for $p\in(0,\infty]$, $B^{p}$ is the set of all $\phi\in\mathcal{P}$ such that

$||\phi||_{p,T}:=\sup_{(t,x)\in[0,T]\times\mathbb{R}^{d}}||\phi(t,x)||_{p}<+\infty,$

for all $T\in\mathbb{R}_{+}$,

  * •

for $p\in(0,\infty]$, $B_{\text{loc}}^{p}$ is the set of all
$\phi\in\mathcal{P}$ such that

$||\phi||_{p,T,R}:=\sup_{t\in[0,T]}\sup_{|x|\leq R}||\phi(t,x)||_{p}<+\infty,$

for all $T,R\in\mathbb{R}_{+}$.

  * •

If $\tau$ is a $\mathbb{F}$-stopping time, we write $\phi\in
B_{\text{loc}}^{p}(\tau)$ if $\phi\,\mathds{1}_{[\\![0,\tau]\\!]}\in
B_{\text{loc}}^{p}$ i.e.

$||\phi\mathds{1}_{[\\![0,\tau]\\!]}(t)||_{p,T,R}:=\sup_{t\in[0,T]}\sup_{|x|\leq
R}||\phi(t,x)\mathds{1}_{[\\![0,\tau]\\!]}(t)||_{p}<+\infty,$

for all $T,R\in\mathbb{R}_{+}$,

  * •

$B_{r}(x):=\\{y\in\mathbb{R}^{d};|x-y|<r\\}$.

The existence and uniqueness of solutions of the stochastic wave equation (1)
with replaced $\Lambda$ by a Gaussian noise $W$ has been widely studied since
the seminal lectures notes [20] by Walsh; see for instance [13, 14]. On the
other hand, equation (1) driven by $\Lambda$, under the condition

$\int_{|z|>1}|z|^{2}\nu(dz)<+\infty,$ (8)

was studied in [2, 3, 4]. Under condition (8), $\Lambda$ is an square-
integrable martingale with discontinuities, which implies that we can apply
the theory developed for $L^{2}$-random measures in a similar way to the
Gaussian noise. However, when (8) is not satisfied, $\Lambda$ may have
infinite second moment such as the case of the $\alpha$-stable Lévy white
noise i.e when the Lévy measure has the form

$\nu_{\alpha}(dz)=[c_{+}\alpha
z^{-\alpha-1}\mathds{1}_{(0,\infty)}(z)+c_{-}\alpha(-z)^{-\alpha-1}\mathds{1}_{(-\infty,0)}(z))]dz.$
(9)

for some $c_{+},c_{-}\geq 0$ and $\alpha\in(0,2)$. In this case,
$\int_{\mathbb{R}}|z|^{2}\nu_{\alpha}(dz)=+\infty$ and
$\mathbb{E}|\Lambda(B)|^{2}$ may be infinite. In contrast with the vast amount
of literature focused on studying (1) when $\Lambda$ is a $L^{2}$-random
measure, to the best of our knowledge [5] is the only work that investigates
the stochastic wave equation (1) driven by a pure-jump Lévy white noise with
possibly infinite variance. Particularly, in [5], it was proved the existence
of a mild solution to (1) and the regularity of the paths of the solutions.

The goal of the present article is to prove the existence and uniqueness of
solutions for the stochastic wave equation (1) driven by pure-jump Lévy white
noise with possibly infinite variance. This is the first time in literature
that the uniqueness of the solution of a hyperbolic SPDE in the whole space
with heavy-tailed noise is proved. Furthermore, we extend some of the main
results for (1) to a more general wave equation. The key idea is to exploit
the compact support property of the fundamental solution $G_{t}$ of the wave
operator. More precisely, we will use the fact that for any fixed
$(t,x)\in\mathbb{R}_{+}\times\mathbb{R}^{d}$, the function
$(s,y)\in[0,t]\times\mathbb{R}^{d}\to G_{t-s}(x-y)$ has support in

$\\{(s,y)\in\mathbb{R}_{+}\times\mathbb{R}^{d};|x-y|\leq t-s\\}.$ (10)

The region defined in (10) is called past-light cone or domain of dependence.
This region plays an important role in physics since it leads to some
fundamental properties of the wave equation, for instance, the past-light cone
property (see Theorem 14.1 of [19]). Particularly, this property was proved
for the stochastic wave equation (1) driven by a colored Gaussian noise
instead of $\Lambda$ in page 59 of [14]. This present work is the first to
investigate the connection between the past-light cone property and the
stochastic wave equation driven by a heavy-tailed noise. Using a similar
strategy used for solving SPDEs in bounded regions and the fact that the
domain of dependence is a bounded region in space for a fixed value of time
$t$, we show that for any bounded domain $D$ in $\mathbb{R}^{d}$, there exists
a sequence $\\{\tau_{N}(D)\\}_{N\in\mathbb{N}}$ of stopping times for which
the $p$-th moments of the solution $u(t,x)$ of (1) are bounded for any
$(t,x)\in[0,T]\times\overline{D}$ with $\\{t\leq\tau_{N}(D)\\}$ for each
$N\in\mathbb{N}$, where $\overline{D}$ is the topological closure of $D$ in
$\mathbb{R}^{d}$.

We include few comments about the stochastic heat equation driven by
$\Lambda$:

$\begin{cases}\dfrac{\partial u}{\partial t}(t,x)=\dfrac{1}{2}\Delta
u(t,x)+\sigma(u(t,x))\dot{\Lambda}(t,x)\,\,\,&t>0,\,x\in\mathbb{R}^{d},\\\
u(0,x)=u_{0}(x)\,\,\,&x\in\mathbb{R}^{d},\\\ \end{cases}$ (11)

where $u_{0}$ is a deterministic bounded function on $\mathbb{R}^{d}$. A mild
solution of (11) is a predictable random field $u$ satisfying

$u(t,x)=w_{0}(t,x)+\int_{0}^{t}\int_{\mathbb{R}^{d}}\rho_{t-s}(x-t)\sigma(u(t,x))\Lambda(ds,dy),$

where $\rho_{t}(x)=(2\pi t)^{-d/2}\exp{(-\frac{|x|^{2}}{2\,t})}$ and

$w_{0}(t,x)=\int_{0}^{t}\int_{\mathbb{R}^{d}}\rho_{t-s}(x-y)u_{0}(y)dy.$

In [18], it was proved that the equation (11) has a unique solution which
satisfies

$\sup_{(t,x)\in[0,T]\times\mathbb{R}^{d}}\mathbb{E}|u(t,x)|^{p}<+\infty,$

if the Lévy measure $\nu$ satisfies $\int_{\mathbb{R}}|z|^{p}\nu(dz)<+\infty$
for some $p\in[1,2]$, and $p<1+\frac{2}{d}$.

In [10], it was showed for the first time that the heat equation (11) driven
by a Lévy noise $\Lambda$ has a mild solution with unbounded $p$-th moments.
More precisely, the main result of [10] shows that if there exist exponents
$p$ and $q$ satisfying $0<q\leq p<1+2/d$ and $p/(1+(1+\frac{2}{d}-p))<q$, such
that

$\int_{\\{|z|\leq
1\\}}|z|^{p}\nu(dz)<+\infty\,\,\,\,\text{and}\,\,\,\,\,\int_{|z|>1}|z|^{q}\nu(dz)<+\infty,$

then the stochastic heat equation,

$\begin{cases}\frac{\partial u}{\partial t}(t,x)=\frac{1}{2}\Delta
u(t,x)+\sigma(u(t,x))\dot{\Lambda}_{N}(t,x)\,\,\,&t>0,\,x\in\mathbb{R}^{d},\\\
u(0,x)=u_{0}(x)\,\,\,&x\in\mathbb{R}^{d},\\\ \end{cases}$ (12)

has a solution $u^{(N)}$ in the space $B^{p}_{\text{loc}}$ for each
$N\in\mathbb{N}$, where
$\Lambda_{N}=\\{\Lambda_{N}(B);B\in\mathcal{B}_{b}(\mathbb{R}_{+}\times\mathbb{R}^{d})\\}$
is the truncated noise given by

$\Lambda_{N}(B)=b\,|B|+\int_{B\times\\{|z|\leq
1\\}}z\tilde{J}(dt,dx,dz)+\int_{B\times\\{1<|z|\leq N\,h(x)\\}}zJ(dt,dx,dz),$
(13)

where

$h(x)=1+|x|^{\eta},$ (14)

with $\eta>d/q$. Moreover,
$u(t,x)=u^{(N)}(t,x)\mathds{1}_{{[\\![0,\tau_{N}]\\!]}}(t)$ is a solution to
(11), and $\tau_{N}$ is the stopping time given by

$\tau_{N}:=\inf\Bigg{\\{}T\in\mathbb{R}_{+}:\int_{0}^{T}\int_{\mathbb{R}^{d}}\int_{\mathbb{R}}\mathds{1}_{\\{|z|>Nh(x)\\}}J(dt,dx,dz)>0\Bigg{\\}}$
(15)

for each $N\in\mathbb{N}$.

To the best of our knowledge, the uniqueness of a solution to (11) remains an
open problem for a general Liptchiz function $\sigma$, although a positive
answer can be given when $\sigma(u)=\beta u$ with $\beta\in\mathbb{R}$ (see
[6]). To be precise, in [10], the existence of a mild solution $u$ for the
stochastic heat equation (11) was obtained by proving that there exists a non-
trivial solution $u^{(N)}$ of (12) in $B^{p}_{\text{loc}}$. This was derived
using the Picard iteration sequence $\\{u_{n}^{(N)}\\}_{n\in\mathbb{N}}$ given
by $u^{(N)}_{n}=\mathcal{J}_{N}^{(n)}(\psi_{0})$ for each $N\in\mathbb{N}$,
where the starting point is $u_{0}^{(N)}=\psi_{0}\in B^{p}$, $\mathcal{J}_{N}$
is the integral operator defined by

$\mathcal{J}_{N}(\phi)(t,x):=w(t,x)+\int_{0}^{t}\int_{\mathbb{R}^{d}}\rho_{t-s}(x-y)\sigma(\phi(s,y))\Lambda_{N}(ds,dy),$
(16)

for any $\phi\in\mathcal{P}$, and
$\mathcal{J}_{N}^{(n)}(\phi):=\mathcal{J}_{N}^{(n-1)}(\mathcal{J}_{N}(\phi))$
for each $n\in\mathbb{N}$. Then, $\\{u_{n}^{(N)}\\}_{n\in\mathbb{N}}$ is a
sequence in $B^{p}_{\text{loc}}$ and converges to a non-trivial fixed point
$u^{(N)}$ of the operator $\mathcal{J}_{N}$ in $B^{p}_{\text{loc}}$. However,
it is not possible to find a complete sub-space of $B^{p}_{\text{loc}}$ where
$\mathcal{J}_{N}$ is a self-map as mentioned on page 13 in [10]. For this
reason, the initial point $\psi_{0}$ of the Picard iteration sequence has to
be an element of $B^{p}$. A different strategy was used in [6], where it was
proved that there exists a unique solution to (11) when $\nu(-\infty,0)=0$ and
$\sigma(u)=\beta u$, with $\beta>0$.

##  2 Existence and uniqueness

In this section, we will show that the stochastic wave equation (1) has a
unique random field that lies in $B_{\text{loc}}^{p}(\tau_{N})$ for each
$N\in\mathbb{N}$. Before the main results of this section, we present some
preliminary results.

For the following result, we assume that
$\mathcal{G}:\mathbb{R}_{+}\times\mathbb{R}^{d}\mapsto\mathbb{R}$ is a
measurable non-negative function. For a fixed
$(t,x)\in\mathbb{R}_{+}\times\mathbb{R}^{d}$ and $\phi\in\mathcal{P}$, we
define the random field given by

$\mathcal{I}^{(t,x)}(\phi)(s,y):=\mathcal{G}_{t-s}(x-y)\phi(s,y)\mathds{1}_{t\geq
s}.$

Also, $||\cdot||_{\Lambda_{N},p}$ denotes the Daniell mean of $\Lambda_{N}$.

######  Lemma 2.1 (Lemma 3.3 in [10]).

Let fixed $T>0$ and $N\in\mathbb{N}$. Assume that there exist $0<q\leq p$ such
that

$\int_{0}^{T}\int_{\mathbb{R}^{d}}(\mathcal{G}_{t}^{p}(x)+\mathcal{G}_{t}(x)\mathds{1}_{p\geq
1})dxdt<+\infty,\,\,\,\int_{\\{|z|\leq
1\\}}|z|^{p}\nu(dz)<+\infty,\,\,\text{and}\,\,\,\,\,\int_{|z|>1}|z|^{q}\nu(dz)<+\infty.$

Also, if $p<1$, we assume that $b=\int_{\\{|z|\leq 1\\}}z\nu(dz)$. Then, we
have: (i) For any $(t,x)\in[0,T]\times\mathbb{R}^{d}$, there exists a constant
$C_{T}\in\mathbb{R}_{+}$ such that for all $\phi\in\mathcal{P}$, we have

$\begin{split}\\\
\mathbb{E}\Big{|}\int_{0}^{t}\int_{\mathbb{R}^{d}}\mathcal{G}_{t-s}&(x-y)\phi(s,y)\Lambda_{N}(ds,dy)\Big{|}^{p}\leq||\mathcal{I}^{(t,x)}(\phi)||_{\Lambda_{N},p}^{p\vee
1}\\\ &\leq
C_{T}\int_{0}^{t}\int_{\mathbb{R}^{d}}(\mathcal{G}_{t-s}^{p}(x-y)+\mathcal{G}_{t-s}(x-y)\mathds{1}_{p\geq
1})\mathbb{E}|\phi(s,y)|^{p}h(y)^{p-q}dyds.\\\ \end{split}$ (17)

(ii) For any $(t,x)\in[0,T]\times\mathbb{R}^{d}$, there exists a constant
$C_{T}\in\mathbb{R}_{+}$ such that for all
$\phi_{1},\,\phi_{2}\in\mathcal{P}$, it holds:

$\begin{split}&\mathbb{E}\Big{|}\int_{0}^{t}\int_{\mathbb{R}^{d}}\mathcal{G}_{t-s}(x-y)(\phi_{1}(s,y)-\phi_{2}(s,y))\Lambda_{N}(ds,dy)\Big{|}^{p}\leq||\mathcal{I}^{(t,x)}(\phi_{1})-\mathcal{I}^{(t,x)}(\phi_{2})||_{\Lambda_{N},p}^{p\vee
1}\\\ &\leq
C_{T}\int_{0}^{t}\int_{\mathbb{R}^{d}}(\mathcal{G}_{t-s}^{p}(x-y)+\mathcal{G}_{t-s}(x-y)\mathds{1}_{p\geq
1})\mathbb{E}|\phi_{1}(s,y)-\phi_{2}(s,y)|^{p}h(y)^{p-q}dyds.\end{split}$ (18)

Note that the first inequality in (17) is due to the contraction property of
the Daniell mean, while the second inequality in (17) follows by Theorem A.1
in Appendix in [12] in conjunction with the proof of Lemma 3.3 in [10]. Note
that the case $p\geq 2$ follows using the maximal inequality (15) in [3].

For showing the existence and uniqueness of solutions of the stochastic wave
equation (1), we require that the Lévy measure $\nu$ of $\Lambda$ satisfies
the following assumption.

######  Assumption 2.

(i) If $d=1,$ we assume that there exists $q\in(0,\infty)$ such that

$\int_{\\{|z|>1\\}}|z|^{q}\nu(dz)<+\infty.$ (19)

(ii) If $d=2$, there exists $0<q\leq p<2$ such that

$\int_{\\{|z|\leq
1\\}}|z|^{p}\nu(dz)+\int_{\\{|z|>1\\}}|z|^{q}\nu(dz)<+\infty.$ (20)

In addition, if $p<1$, we assume that $b=\int_{\\{|z|\leq 1\\}}z\nu(dz)$.

Notice that the fundamental solution (5) of the wave equation satisfies

$\int_{0}^{t}\int_{\mathbb{R}^{d}}G_{t-s}^{p}(x-y)dyds=\begin{cases}2^{-p}\,t^{2}\quad\quad\quad\quad\,\text{for
any $p>0$, if }\quad d=1,\\\
\frac{(2\pi)^{1-p}}{(2-p)(3-p)}\,t^{3-p}\quad\text{for any $p\in(0,2)$,
if}\quad d=2.\end{cases}$ (21)

for all $t\in\mathbb{R}_{+}$. We denote
$g_{p}(t,x)=G_{t}^{p}(x)+G_{t}(x)\mathds{1}_{p\geq 1}$.

######  Remark 2.2.

In the case of equation (1) in dimension $d=1$, we do not require condition
(20) since $\int_{0}^{T}\int_{\mathbb{R}}G_{t}^{p}(x)dxdt<+\infty$ for all
$p>0$, and

$\int_{\\{|z|\leq 1\\}}|z|^{p}\nu(dz)\leq\int_{\\{|z|\leq
1\\}}|z|^{2}\nu(dz)<+\infty,\quad\text{for all $p\geq 2$.}$

So, we can choose an arbitrary $p\geq 2$ for the main result. On the other
hand, for (1) in dimension $d=2$, we have to consider (20) due to the fact
that $\int_{0}^{T}\int_{\mathbb{R}}G_{t}^{p}(x)dxdt<+\infty$ for all
$p\in(0,2).$

######  Lemma 2.3.

For any $\beta_{1}>-1,\ldots,\beta_{n}>-1$,

$\int_{T_{n}(T)}\prod_{j=1}^{n}(t_{j+1}-t_{j})^{\beta_{j}}dt_{1}\ldots
dt_{n}=\frac{\prod_{j=1}^{n}\Gamma(\beta_{j}+1)}{\Gamma(\sum_{j=1}^{n}\beta_{j}+n+1)}t^{\sum_{j=1}^{n}\beta_{j}+n},$

where $T_{n}(t)=\\{(t_{1},\ldots,t_{n})\in(0,t)^{n}:t_{1}<\ldots<t_{n}\\}$ and
$t=t_{n+1}$.

In the present work, we will follow a strategy similar to [10] for the
existence of a mild solution of the stochastic heat equation (11). More
precisely, we first show that the stochastic wave equation driven by the
truncated Lévy noise $\Lambda_{N}$:

$\begin{cases}\dfrac{\partial^{2}u}{\partial t^{2}}(t,x)=\Delta
u(t,x)+\sigma(u(t,x))\dot{\Lambda}_{N}(t,x),\quad
t>0,\,x\in\mathbb{R}^{d},\,\\\ u(0,x)=u_{0}(x),\quad\quad\dfrac{\partial
u}{\partial t}(0,x)=v_{0}(x),\quad\quad\quad\quad\quad x\in\mathbb{R}^{d},\\\
\end{cases}$ (22)

has a unique solution $u^{(N)}$ in $B^{p}_{\text{loc}}$ for each
$N\in\mathbb{N}$. Then, we prove that we can paste together the solutions
$\\{u^{(N)}\\}_{N\in\mathbb{N}}$ to obtain a solution $u$ of equation (1).

A mild solution of (22) satisfies

$u(t,x)=w(t,x)+\int_{0}^{t}\int_{\mathbb{R}^{d}}G_{t-s}(x-y)\sigma(u(s,y))\Lambda_{N}(ds,dy).$
(23)

We define the operator $\mathcal{T}_{N}:\mathcal{P}\to\mathcal{P}$ by

$\mathcal{T}_{N}(\phi)(t,x):=w(t,x)+\int_{0}^{t}\int_{\mathbb{R}^{d}}G_{t-s}(x-y)\sigma(\phi(s,y))\Lambda_{N}(ds,dy),$
(24)

for any $\phi\in\mathcal{P}$. By Lemma 6.6 of [11], the process
$\mathcal{T}_{N}(\phi)$ admits a predictable modification. We will always work
with this modification.

By using the compact support property of $G_{t}$, we will be able to show that
$\mathcal{T}_{N}$ is a self-map in $B^{p}_{\text{loc}}$. Then, the existence
and uniqueness of a mild solution of (22) will follow from the Banach fixed
point theorem, as long as we prove that $\mathcal{T}_{N}$ satisfies the
hypotheses of this theorem. Note that this strategy cannot be applied for the
heat equation (11) since the operator $\mathcal{J}_{N}$ given by (16) is well-
defined from $B^{p}$ to $B_{\text{loc}}^{p}$, but may not be a self-map in
$B_{\text{loc}}^{p}$.

######  Theorem 2.4.

Assume that Assumptions 1 and 2 are satisfied. Then, for any fixed
$N\in\mathbb{N}$, equation (22) has a unique solution
$u^{(N)}=\\{u^{(N)}(t,x);t\geq 0,x\in\mathbb{R}^{d}\\}$ that satisfies

$\sup_{t\in[0,T]}\sup_{|x|\leq R}\mathbb{E}|u^{(N)}(t,x)|^{p}<+\infty,$

for any $T>0$ and $R>0$, where $p\geq 2$ is arbitrary if $d=1$, and $p$ is the
exponent from Assumption 2 if $d=2$.

###### Proof.

Step 1 : ($\mathcal{T}_{N}$ is a self-map in $B^{p}_{\text{loc}}$) Note that
for any $0\leq s<t\leq T$ and $|x|\leq R$, we have

$\text{supp}(G_{t-s}(x-\cdot))\subseteq\overline{B_{T}(x)}\subset\overline{B_{T+R}(0)}.$
(25)

Hence,

$h(y)^{p-q}<C_{T}(1+|R|^{\gamma})\quad\text{ for all
$y\in\overline{B_{T+R}(0)}$,}$ (26)

where $\gamma=\eta(p-q)$. Then, by Lemma 2.1, (26), and (25), we have for any
$\phi\in B_{\text{loc}}^{p}$,

$\begin{split}&\mathbb{E}\Bigg{|}\int_{0}^{t}\int_{\mathbb{R}^{d}}G_{t-s}(x-y)\sigma(\phi(s,y))\Lambda_{N}(ds,dy)\Bigg{|}^{p}\\\
&\leq
C_{t}\int_{0}^{t}\int_{B_{T}(x)}g_{p}(t-s,x-y)\,(1+\mathbb{E}|\phi(s,y)|^{p})h(y)^{p-q}dyds\\\
&\leq
C_{T}(1+|R|^{\gamma})\int_{0}^{t}\int_{B_{T}(x)}g_{p}(t-s,x-y)(1+\mathbb{E}|\phi(s,y)|^{p})dyds\\\
&\leq
C_{p,T,R}\int_{0}^{t}\int_{B_{T+R}(0)}g_{p}(t-s,x-y)(1+\mathbb{E}|\phi(s,y)|^{p})dyds\\\
&\leq C_{p,T,R}(1+\sup_{s\in[0,T]}\sup_{|y|\leq
R+T}\mathbb{E}|\phi(s,y)|^{p})\int_{0}^{T}\int_{B_{T+R}(0)}g_{p}(t-s,x-y)dyds.\\\
\end{split}$ (27)

Therefore, using (7), we obtain:

$\sup_{t\in[0,T]}\sup_{|x|\leq
R}\mathbb{E}|\mathcal{T}_{N}(\phi)(t,x)|^{p}<+\infty\qquad\text{for all
$T,R\in\mathbb{R}_{+}$.}$ (28)

Step 2: (Convergence of the Picard’s iterations) In this step, we consider the
Picard’s iteration $\\{u_{n}^{(N)}(t,x),t\geq 0,x\in\mathbb{R}^{d}\\}$ given
by: $u_{0}^{(N)}(t,x)=\Psi_{0}(t,x)$, where $\Psi_{0}$ is any arbitrary
element of $B^{p}_{\text{loc}}$, and
$u_{n}^{(N)}=\mathcal{T}_{N}(u^{(N)}_{n-1})$ i.e.

$u_{n}^{(N)}(t,x)=w(t,x)+\int_{0}^{t}\int_{\mathbb{R}^{d}}G_{t-s}(x-y)\sigma(u_{n-1}^{(N)}(s,y))\Lambda_{N}(ds,dy),\,\,\,\,\,\text{for
$n\in\mathbb{N}.$}$ (29)

By (28), it follows that $u_{n}^{(N)}\in B^{p}_{\text{loc}}$ for all
$n\in\mathbb{N}$ using induction over $n$. Next, we will show that
$\\{u_{n}^{(N)}\\}_{n\in\mathbb{N}}$ is a Cauchy sequence in
$B^{p}_{\text{loc}}$. By Lemma 2.1-(ii), we have:

$\begin{split}\mathbb{E}|u_{n}^{(N)}(t,x)-&u_{n-1}^{(N)}(t,x)|^{p}\leq\\\
&C_{T}\int_{0}^{t}\int_{\mathbb{R}^{d}}g_{p}(t-s,x-y)\mathbb{E}|u_{n-1}^{(N)}(s,y)-u_{n-2}^{(N)}(s,y)|^{p}h(y)^{p-q}dyds.\\\
\end{split}$ (30)

Iterating (30), we get:

$\begin{split}&\mathbb{E}|u_{n}^{(N)}(t,x)-u_{n-1}^{(N)}(t,x)|^{p}\leq\\\
&C_{T}^{n}\int_{T_{n}(t)}\int_{(\mathbb{R}^{d})^{n}}\prod_{i=1}^{n}g_{p}(t_{i+1}-t_{i},x_{i+1}-x_{i})\prod_{i=1}^{n}h(x_{i})^{p-q}(\mathbb{E}|u_{1}^{(N)}(t_{1},x_{1})|^{p}+\mathbb{E}|\Psi_{0}(t_{1},x_{1})|^{p})d\textbf{x}d\textbf{t},\\\
\end{split}$ (31)

where $\textbf{t}=(t_{1},\ldots,t_{n})$, $\textbf{x}=(x_{1},\ldots,x_{n})$ and
we set $t_{n+1}=t$ and $x_{n+1}=x$. Note that the integrand of the integral in
the right hand side of (31) contains the indicator of the set

$\\{\textbf{x}\in(\mathbb{R}^{d})^{n}\,:\,|x_{i+1}-x_{i}|<t_{i+1}-t_{i},\,\,\text{for}\,\,i=1,\ldots,n\\},$

and hence if $|x|\leq R,$ the integral in (31) can be considered on the
restricted domain

$\\{\textbf{x}\in(\mathbb{R}^{d})^{n}\,:\,|x_{i}|\leq
R+T\,\,\text{for}\,\,i=1,\ldots,n\\},$ (32)

since

$|x-x_{i}|\leq|x-x_{n}|+\sum_{k=i}^{n-1}|x_{k+1}-x_{k}|\leq(t-t_{n})+\sum_{k=i}^{n-1}(t_{k+1}-t_{k})=t-t_{i}\leq
t<T.$ (33)

Then, by (32), it follows that

$\begin{split}&\mathbb{E}|u_{n}^{(N)}(t,x)-u_{n-1}^{(N)}(t,x)|^{p}\leq
C_{T}^{n}\Big{(}\sup_{t\in[0,T]}\sup_{|x|\leq
R+T}\mathbb{E}|u_{1}^{(N)}(t,x)|^{p}+\mathbb{E}|\Psi_{0}(t,x)|^{p}\Big{)}\\\
&\qquad\qquad\qquad\qquad\qquad\qquad\int_{T_{n}(t)}\int_{(\mathbb{R}^{d})^{n}}\prod_{i=1}^{n}g_{p}(t_{i+1}-t_{i},x_{i+1}-x_{i})\prod_{i=1}^{n}h(x_{i})^{p-q}d\textbf{x}d\textbf{t}.\end{split}$
(34)

On the restricted domain given by (32), we have

$\prod_{i=1}^{n}h(x_{i})^{p-q}\leq\prod_{i=1}^{n}\Big{[}1+(R+T)^{\eta}\Big{]}^{p-q}\leq
C_{T}^{n}(1+|R|^{n\gamma}).$ (35)

Hence, by (34) and (35), we obtain:

$\begin{split}&\mathbb{E}|u_{n}^{(N)}(t,x)-u_{n-1}^{(N)}(t,x)|^{p}\leq
C_{T,R}^{n}\int_{T_{n}(t)}\int_{(\mathbb{R}^{d})^{n}}\prod_{i=1}^{n}g_{p}(t_{i+1}-t_{i},x_{i+1}-x_{i})d\textbf{x}d\textbf{t}:=C^{n}_{T,R}A_{n}^{(p)}(t).\\\
\end{split}$ (36)

Observe that $A_{n}^{(p)}(t)$ does not depend of $x$. If $p<1$, by (21) and
Lemma 2.3, we get:

$A_{n}^{(p)}(t)=C_{p}^{n}\int_{T_{n}(t)}\prod_{j=1}^{n}(t_{j+1}-t_{j})^{\alpha}d\textbf{t}=C_{p}^{n}\frac{t^{(\alpha+1)n}}{\Gamma((\alpha+1)n+1)},$

where $C_{p}$ is a constant that depends on $p$, and

$\alpha=\begin{cases}1\quad\text{if}\quad d=1,\\\ 2-p\quad\text{if}\quad
d=2.\end{cases}$

Hence,

$\text{if
$p<1$,}\quad\quad\sup_{t\in[0,T]}A_{n}^{(p)}(t)\leq\begin{cases}C_{p}^{n}\frac{T^{2n}}{(n!)^{2}}\qquad\,\,\,\,\,\text{when}\qquad
d=1,\\\ C_{p}^{n}\frac{T^{(3-p)n}}{(n!)^{3-p}}\qquad\text{when}\qquad
d=2.\end{cases}$ (37)

Assume $p\geq 1$. If $d=1$, it holds that $G_{t}^{p}(x)=2^{1-p}G_{t}(x)$, so
we can proceed in the same way as for $p<1$, which implies
$\sup_{t\in[0,T]}A_{n}^{(p)}(t)\leq
2^{n(1-p)}C_{p}^{n}\frac{T^{2n}}{(n!)^{2}}$. If $d=2$, by (21), we have:

$\int_{\mathbb{R}^{2}}g_{p}(t,x)dx=c_{p}t^{2-p}+t\leq(c_{p}+T^{p-1})t^{2-p},\qquad\text{with}\quad
c_{p}=\frac{(2\pi)^{1-p}}{2-p},$

and hence
$\sup_{t\in[0,T]}A_{n}^{(p)}(t)\leq(c_{p}+T^{p-1})^{n}C_{p}^{n}\frac{T^{(3-p)n}}{(n!)^{3-p}}$.
To summarize,

$\text{if $p\geq
1$,}\quad\quad\sup_{t\in[0,T]}A_{n}^{(p)}(t)\leq\begin{cases}2^{n(1-p)}C_{p}^{n}\frac{T^{2n}}{(n!)^{2}}\qquad\qquad\,\,\,\,\,\,\,\,\text{when}\qquad
d=1,\\\
(c_{p}+T^{p-1})^{n}C_{p}^{n}\frac{T^{(3-p)n}}{(n!)^{3-p}}\qquad\text{when}\qquad
d=2.\end{cases}$ (38)

Therefore, for both cases $p<1$ and $p\geq 1$, it holds

$\sum_{n\geq 1}C_{T,R,p}^{n}\sup_{t\in[0,T]}A_{n}^{(p)}(t)<+\infty.$ (39)

Consequently, due to (36) and (39), $\\{u_{n}^{(N)}\\}_{n\in\mathbb{N}}$ is a
Cauchy sequence in $B^{p}_{\text{loc}}$. Hence, there exists an element
$u^{(N)}\in B^{p}_{\text{loc}}$ such that
$u_{n}^{(N)}\xrightarrow[]{B^{p}_{\text{loc}}}u^{(N)}$ as $n\to+\infty$.

Step 3: (Existence of solution). In this step, we verify that indeed $u^{(N)}$
satisfies (23). We apply Lemma 2.1-(ii) with $\mathcal{G}_{t}=G_{t}$. Then,

$\begin{split}&\mathbb{E}\Big{|}\int_{0}^{t}\int_{\mathbb{R}^{d}}\mathcal{G}_{t-s}(x-y)(\sigma(u_{n}^{(N)}(s,y))-\sigma(u^{(N)}(s,y)))\Lambda_{N}(ds,dy)\Big{|}^{p}\\\
&\leq||\mathcal{I}^{(t,x)}(\sigma(u_{n}^{(N)}))-\mathcal{I}^{(t,x)}(\sigma(u^{(N)}))||_{\Lambda^{N},p}^{p\vee
1}\\\ &\leq
C_{T}\int_{0}^{t}\int_{\mathbb{R}^{d}}g_{p}(t-s,x-y)\mathbb{E}|u_{n}^{(N)}(s,y)-u^{(N)}(s,y)|^{p}h(y)^{p-q}dyds\\\
&\leq C_{T,R,p}\sup_{s\in[0,T]}\sup_{|y|\leq
T+R}\mathbb{E}|u_{n}^{(N)}(s,y)-u^{(N)}(s,y)|^{p}\int_{0}^{t}\int_{\mathbb{R}^{d}}g_{p}(t-s,x-y)dyds,\end{split}$
(40)

where for the last inequality above we used (25) and (26). Letting
$n\to+\infty$ on (40), we obtain that for fixed
$(t,x)\in\mathbb{R}_{+}\times\mathbb{R}^{d}$,
$\mathcal{I}^{(t,x)}(\sigma(u_{n}^{(N)}))$ converges to
$\mathcal{I}^{(t,x)}(\sigma(u^{(N)}))$ in the norm
$||\cdot||_{\Lambda^{N},p}^{p\vee 1}$, which implies that

$\lim_{n\to+\infty}\mathcal{T}_{N}(u_{n}^{(N)})(t,x)=\mathcal{T}_{N}(u^{(N)})(t,x)\quad\text{in
$L^{p}(\Omega).$}$

On the other hand, $u_{n}^{(N)}=\mathcal{T}_{N}(u_{n-1}^{(N)})$ for all
$n\in\mathbb{N}$ and $\\{u_{n}^{(N)}\\}_{n\in\mathbb{N}}$ converges to
$u^{(N)}$ in $B^{p}_{\text{loc}}$ as $n\to+\infty$, according to Step 2 above
since $u^{(N)}=\mathcal{T}_{N}(u^{(N)})$. This proves that $u^{(N)}$ satisfies
(23).

Step 4: (Uniqueness of solution) Assume that there exists another process
$v^{(N)}\in B^{p}_{\text{loc}}$ that satisfies (23) i.e.
$\mathcal{T}_{N}(v^{(N)})=v^{(N)}$. Then, by Lemma 2.1.-(ii), we have:

$\mathbb{E}|u_{n}^{(N)}(t,x)-v^{(N)}(t,x)|^{p}\leq
C_{T}\int_{0}^{t}\int_{\mathbb{R}^{d}}g_{p}(t-s,x-y)\mathbb{E}|u_{n-1}^{(N)}(t,x)-v^{(N)}(t,x)|^{p}h(y)^{p-q}dsdy.$

Iterating the inequality above as for (31) and following the same steps as for
(36), it holds

$\begin{split}&\mathbb{E}|u_{n}^{(N)}(t,x)-v^{(N)}(t,x)|^{p}\\\ &\leq
C_{p,T,R}^{n}\int_{T_{n}(t)}\int_{(\mathbb{R}^{d})^{n}}\prod_{i=1}^{n}g_{p}(t_{i+1}-t_{i},x_{i+1}-x_{i})(\mathbb{E}|u_{1}^{(N)}(t_{1},x_{1})-v^{(N)}(t_{1},x_{1})|^{p})d\textbf{x}d\textbf{t}\\\
&\leq\sup_{t\in[0,T]}\sup_{|x|\leq
R+T}\mathbb{E}|u_{1}^{(N)}(t,x)-v^{(N)}(t,x)|^{p}\,C_{p,T,R}^{n}A_{n}^{p}(t)\to
0,\end{split}$

as $n\to+\infty$. Hence, $u_{n}^{(N)}\to v^{(N)}$ in $B_{\text{loc}}^{p}$ as
$n\to+\infty$. On the other hand, $u_{n}^{(N)}\to u^{(N)}$ in
$B_{\text{loc}}^{p}$ as $n\to+\infty$, which implies $u^{(N)}=v^{(N)}$ in
$B_{\text{loc}}^{p}$. ∎

######  Theorem 2.5.

Let $\tau_{N}$ be the stopping time given by (15). Under the same assumptions
of Theorem 2.4 with $\eta>d/q$, equation (1) has a unique solution
$u=\\{u(t,x);t\geq 0,x\in\mathbb{R}^{d}\\}$ that satisfies

$\sup_{t\in[0,T]}\sup_{|x|\leq
R}\mathbb{E}|u(t,x)\mathds{1}_{{[\\![0,\tau_{N}]\\!]}}(t)|^{p}<+\infty,$

for all $R,T\in\mathbb{R}_{+}$ and $N\in\mathbb{N}$, where $p\geq 2$ is
arbitrary if $d=1$, and $p$ is the exponent from Assumption 2 if $d=2$.

###### Proof.

(Existence) The existence of a solution of (1) follows as in the proof of
Theorem 3.1 in [10] for a solution of the heat equation (11). First, note that
we have

$\Lambda(B)-\Lambda_{N}(B)=\int_{B\times\\{|z|>N\,h(x)\\}}zJ(dt,dx,dz)=0\quad\text{if
$t\leq\tau_{N}$}$ (41)

for any $B\in\mathcal{B}_{b}(\mathbb{R}_{+}\times\mathbb{R}^{d})$. Observe
that (41) holds for the extension of $\Lambda$ to a Lévy basis given by Remark
A.3, for which we have:
$\Lambda(B\cap({[\\![0,\tau_{N}]\\!]}\times\mathbb{R}^{d}))=\Lambda_{N}(B\cap({[\\![0,\tau_{N}]\\!]}\times\mathbb{R}^{d}))$
for all $B\in\tilde{\mathcal{P}}_{b}$. Therefore, by Lemma A.5, (41), and
$\tau_{N}\uparrow+\infty$ a.s. as $N\to+\infty$. It can be proved that the
random field given by

$u(t,x)=u^{(1)}(t,x)\mathds{1}_{[\\![0,\tau_{1}]\\!]}(t)+\sum_{N=2}^{\infty}u^{(N)}(t,x)\mathds{1}_{(\\!(\tau_{N-1},\tau_{N}]\\!]}(t),$

satisfies (1), where $u^{(N)}$ is the random field given by Theorem 2.4.

(Uniqueness) Assume that $v$ is another solution of (1) such that $v\in
B_{\text{loc}}^{p}(\tau_{N})$ for all $N\in\mathbb{N}$. By Lemma A.5, Lemma
2.1, and (41), we have:

$\begin{split}&\mathbb{E}|(u_{n}^{(N)}(t,x)-v(t,x))\mathds{1}_{{[\\![0,\tau_{N}]\\!]}}(t)|^{p}\\\
&\leq
C_{T}\int_{0}^{t}\int_{\mathbb{R}^{d}}g_{p}(t-s;x-y)\mathbb{E}|(u_{n-1}^{(N)}(s,y)-v(s,y))\mathds{1}_{{[\\![0,\tau_{N}]\\!]}}(s)|^{p}h(y)^{p-q}dyds,\\\
\end{split}$

Iterating the inequality above and using the same steps as for (36), we obtain
that for $t\in[0,T]$ and $|x|\leq R$,

$\begin{split}&\mathbb{E}|(u_{n}^{(N)}(t,x)-v(t,x))\mathds{1}_{{[\\![0,\tau_{N}]\\!]}}(t)|^{p}\leq\\\
&C_{T,R}^{n}\Big{(}\sup_{t\in[0,T]}\sup_{|x|\leq
R+T}\mathbb{E}|u_{1}^{(N)}(t,x)\mathds{1}_{{[\\![0,\tau_{N}]\\!]}}(t)|^{p}+\mathbb{E}|v(t,x)\mathds{1}_{{[\\![0,\tau_{N}]\\!]}}(t)|^{p}\Big{)}A_{n}^{(p)}(t).\end{split}$

where $A_{n}^{(p)}(t)$ is given by (36). Using the fact that
$\sup_{t\in[0,T]}A_{n}^{(p)}(t)\to 0$ as $n\to+\infty$, we conclude that

$\sup_{t\in[0,T]}\sup_{|x|\leq
R}\mathbb{E}|(u_{n}^{(N)}(t,x)-v(t,x))\mathds{1}_{{[\\![0,\tau_{N}]\\!]}}(t)|^{p}\to
0\qquad\text{as}\quad n\to+\infty,$

for all $R,\,T>0$. On the other hand, note that $u_{n}^{(N)}\to u$ in
$B^{p}_{\text{loc}}(\tau_{N})$ as $n\to+\infty$ for all $N\in\mathbb{N}$.
Then,
$u(t,x)\mathds{1}_{{[\\![0,\tau_{N}]\\!]}}(t)=v(t,x)\mathds{1}_{{[\\![0,\tau_{N}]\\!]}}(t)$
a.s., and letting $N\to+\infty$ we obtain that $u=v$ a.s. using the fact that
$\tau_{N}\uparrow+\infty$ a.s. for $N\to+\infty$. ∎

Now, we investigate the stochastic wave equation driven by a more general
heavy-tailed noise. Consider
$L=\\{L(B);B\in\mathcal{B}_{b}(\mathbb{R}_{+}\times\mathbb{R}^{d})\\}$ given
by

$L(B)=b|B|+aW(B)+\int_{B\times\\{|z|\leq
1\\}}z\tilde{J}(dt,dx,dz)+\int_{B\times\\{|z|>1\\}}zJ(dt,dx,dz),$ (42)

where $a>0$ and $W$ is a space-time Gaussian white noise i.e.
$\\{W(A);A\in\mathcal{B}_{b}(\mathbb{R}_{+}\times\mathbb{R}^{d})\\}$ is a zero
mean Gaussian process with covariance $E[W(A)W(B)]=|A\cap B|$. In the case
$d=2$, since $\int_{0}^{t}\int_{\mathbb{R}^{d}}G_{t-s}^{2}(x-y)dsdy=+\infty$,
there is no mild solution of (1) driven by $L$ instead of $\Lambda$.
Therefore, following the same steps of Theorem 2.4 and Theorem 2.5, we have
the following result. The proof of this result uses Lemma 3.3 of [10] which is
stated for a Lévy noise which has a Gaussian component

######  Corollary 2.6.

Assume that $d=1$, and Assumptions 1 and 2 hold. Then, the stochastic wave
equation

$\begin{cases}\dfrac{\partial^{2}u}{\partial t^{2}}(t,x)=\Delta
u(t,x)+\sigma(u(t,x))\dot{L}(t,x),\quad t>0,\,x\in\mathbb{R},\\\
u(0,x)=u_{0}(x),\quad\quad\dfrac{\partial u}{\partial
t}(0,x)=v_{0}(x),\quad\quad\quad\quad\quad x\in\mathbb{R},\\\ \end{cases}$
(43)

has a unique solution $u$ that satisfies

$\sup_{t\in[0,T]}\sup_{|x|\leq
R}\mathbb{E}|u(t,x)\mathds{1}_{{[\\![0,\tau_{N}]\\!]}}(t)|^{p}<+\infty,\quad\text{for
all $p\geq 2$,}$

for all $T,R\in\mathbb{R}_{+}$ and $N\in\mathbb{N}$.

On the other hand, we can extend Theorem 2.5 to a more general form of
stochastic wave equations with noise $\Lambda$ multiplied by $\sigma(t,x,u)$.
In addition, we add a non-linear drift $f(t,x,u)$. We consider the stochastic
wave equation for $d\leq 2$,

$\begin{cases}\dfrac{\partial^{2}u}{\partial t^{2}}(t,x)=\Delta
u(t,x)+f(t,x,u(t,x))+\sigma(t,x,u(t,x))\dot{\Lambda}(t,x),\quad t>0,\
x\in\mathbb{R}^{d},\\\ u(0,x)=u_{0}(x),\quad\quad\dfrac{\partial u}{\partial
t}(0,x)=v_{0}(x),\,\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\,x\in\mathbb{R}^{d},\\\
\end{cases}$ (44)

where $u_{0}$ and $v_{0}$ are the same initial conditions of (1). We impose
the following conditions on the processes $\sigma$ and $f$.

######  Assumption 3.

$\sigma$ and $f$ are functions defined as
$\Omega\times\mathbb{R}_{+}\times\mathbb{R}^{d+1}\to\mathbb{R}$ which are
measurable with respect to $\tilde{\mathcal{P}}\times\mathcal{B}(\mathbb{R})$.
In addition, we assume there exists positive processes
$\mathcal{C}_{f},\,\mathcal{C}_{\sigma}\in B_{loc}^{\infty}$ such that for all
$(t,x)\in\mathbb{R}_{+}\times\mathbb{R}^{d}$ and $z_{1},\,z_{2}\in\mathbb{R}$,
we have

$|\sigma(t,x,z_{1})-\sigma(t,x,z_{2})|\leq\mathcal{C}_{\sigma}(t,x)\,|z_{1}-z_{2}|\quad\text{a.s.},$
(45)

and

$|f(t,x,z_{1})-f(t,x,z_{2})|\leq\mathcal{C}_{f}(t,x)\,|z_{1}-z_{2}|\quad\text{a.s.}$
(46)

Denote $\sigma_{0}(t,x)=\sigma(t,x,0)$ and $f_{0}(t,x)=f(t,x,0)$.

A mild solution  of (44) is a predictable random field $u$ that satisfies the
stochastic integral equation:

$\begin{split}u(t,x)=w(t,x)&+\int_{0}^{t}\int_{\mathbb{R}^{d}}G_{t-s}(x-y)f(s,y,u(s,y))dyds\\\
&+\int_{0}^{t}\int_{\mathbb{R}^{d}}G_{t-s}(x-y)\sigma(s,y,u(s,y))\Lambda(ds,dy).\\\
\end{split}$ (47)

######  Theorem 2.7.

Assume that Assumptions 1, 2 and 3 are satisfied.

(i) If $d=1$, assume that there exists $p\geq 2$ such that $\sigma_{0}$ and
$f_{0}$ lies in $B^{p}_{\text{loc}}$. Then, equation (44) has a unique
solution $u=\\{u(t,x);t\geq 0,x\in\mathbb{R}\\}$ that satisfies

$\sup_{t\in[0,T]}\sup_{|x|\leq
R}\mathbb{E}|u(t,x)\mathds{1}_{{[\\![0,\tau_{N}]\\!]}}(t)|^{p}<+\infty,$

for all $T,R>0$ and $N\in\mathbb{N}$.

(ii) If $d=2$, assume that $\sigma_{0}$ and $f_{0}$ lies in
$B^{p}_{\text{loc}}$, where $p$ is the exponent of Assumption 2. In addition,
if $p<1$, we assume that $f(t,x,z)\equiv 0$. Then, equation (44) has a unique
solution $u=\\{u(t,x);t\geq 0,x\in\mathbb{R}^{2}\\}$ that satisfies

$\sup_{t\in[0,T]}\sup_{|x|\leq
R}\mathbb{E}|u(t,x)\mathds{1}_{{[\\![0,\tau_{N}]\\!]}}(t)|^{p}<+\infty,$

for all $T,R>0$ and $N\in\mathbb{N}$.

###### Proof.

We fix $N\in\mathbb{N}$ in the first part of the proof. First, we prove that
the operator $\Im_{N}:\mathcal{P}\to\mathcal{P}$ given by

$\begin{split}\Im_{N}(\phi)(t,x):=w(t,x)&+\int_{0}^{t}\int_{\mathbb{R}^{d}}G_{t-s}(x-y)f(s,y,\phi(s,y))dyds\\\
&+\int_{0}^{t}\int_{\mathbb{R}^{d}}G_{t-s}(x-y)\sigma(s,y,\phi(s,y))\Lambda_{N}(ds,dy),\end{split}$

is a self-map in $B_{\text{loc}}^{p}$. Indeed, note that $\Im_{N}$ is well
defined since $\sigma(t,x,\phi(t,x))$ and $f(t,x,\phi(t,x))$ are predictable
for all $\phi\in\mathcal{P}$ due to the fact that $\sigma$ and $f$ are
measurable with respect to $\tilde{\mathcal{P}}\times\mathcal{B}(\mathbb{R})$.
Hence, due to Lemma 6.2 in [11], $\Im_{N}(\phi)(t,x)$ has a predictable
modification. Now, by Lemma 2.1, the compact support property (25) of $G_{t}$,
relation (45), and Hölder’s inequality
$||XY||_{1}\leq||X||_{\infty}||Y||_{1}$, for any $t\in[0,T]$ and $|x|\leq R$
we have:

$\begin{split}&\mathbb{E}\Big{|}\int_{0}^{t}\int_{\mathbb{R}^{d}}G_{t-s}(x-y)\sigma(s,y,\phi(s,y))\Lambda_{N}(ds,dy)\Big{|}^{p}\\\
&\leq
C_{p,T}\int_{0}^{t}\int_{\mathbb{R}^{d}}g_{p}(t-s,x-y)\mathbb{E}|\sigma(s,y,\phi(s,y))|^{p}\,h(y)^{p-q}dyds\\\
&\leq
C_{p,T}\int_{0}^{t}\int_{\mathbb{R}^{d}}g_{p}(t-s,x-y)(||\mathcal{C}_{\sigma}(s,y)||_{\infty}^{p}\mathbb{E}|\phi(s,y)|^{p}+\mathbb{E}|\sigma_{0}(s,y)|^{p})h(y)^{p-q}dyds\\\
&\leq C_{p,T,R}\Big{(}\sup_{t\in[0,T]}\sup_{|x|\leq
T+R}||\mathcal{C}_{\sigma}(t,x)||_{\infty}^{p}\vee\mathbb{E}|\sigma_{0}(t,x)|^{p}\Big{)}\\\
&\qquad\qquad\qquad\qquad\Big{(}\sup_{t\in[0,T]}\sup_{|x|\leq
T+R}\mathbb{E}|\phi(t,x)|^{p}+1\Big{)}\int_{0}^{t}\int_{\mathbb{R}^{d}}g_{p}(t-s,x-y)dyds<+\infty.\\\
\end{split}$ (48)

Then, for all $R,T>0$,

$\sup_{t\in[0,T]}\sup_{|x|\leq
R}\mathbb{E}\Big{|}\int_{0}^{t}\int_{\mathbb{R}^{d}}G_{t-s}(x-y)\sigma(s,y,\phi(s,y))\Lambda_{N}(ds,dy)\Big{|}^{p}<+\infty.$
(49)

Next, we examine the integral that corresponds to the drift $f$. Recall that
if $p\in(0,1)$, we assume that $f=0$. Hence, we consider only the case $p\geq
1$. By Hölder’s inequality, relation (46), for any $t\in[0,T]$ and $|x|\leq
R$,

$\begin{split}&\mathbb{E}\Big{|}\int_{0}^{t}\int_{\mathbb{R}^{d}}G_{t-s}(x-y)f(s,y,\phi(s,y))dyds\Big{|}^{p}\\\
&\leq\Big{(}\int_{0}^{t}\int_{\mathbb{R}^{d}}G_{t-s}(x-y)dyds\Big{)}^{p-1}\int_{0}^{t}\int_{\mathbb{R}^{d}}G_{t-s}(x-y)\mathbb{E}|f(s,y,\phi(s,y))|^{p}dyds\\\
&\leq C_{T}\Big{(}\sup_{t\in[0,T]}\sup_{|x|\leq
T+R}||\mathcal{C}_{f}(t,x)||_{\infty}^{p}\vee\mathbb{E}|f_{0}(t,x)|^{p}\Big{)}\int_{0}^{t}\int_{\mathbb{R}^{d}}G_{t-s}(x-y)(\mathbb{E}|\phi(s,y)|^{p}+1)dyds\\\
&\leq
C_{T}\int_{0}^{t}\int_{\mathbb{R}^{d}}g_{p}(t-s,x-y)(\mathbb{E}|\phi(s,y)|^{p}+1)dyds\\\
&\leq C_{T}\Big{(}\sup_{t\in[0,T]}\sup_{|x|\leq
T+R}\mathbb{E}|\phi(t,x)|^{p}+1\Big{)}\int_{0}^{t}\int_{\mathbb{R}^{d}}g_{p}(t-s,x-y)dyds<+\infty.\\\
\end{split}$ (50)

Hence,

$\sup_{t\in[0,T]}\sup_{|x|\leq
R}\mathbb{E}\Big{|}\int_{0}^{t}\int_{\mathbb{R}^{d}}G_{t-s}(x-y)f(s,y,\phi(s,y))dyds\Big{|}^{p}<+\infty.$

Therefore, (49) and (50) yield

$\sup_{t\in[0,T]}\sup_{|x|\leq R}\mathbb{E}|\Im_{N}(\phi)(t,x)|^{p}<\infty.$
(51)

Now, consider the Picard’s iteration $\\{u_{n}^{(N)}\\}_{n\in\mathbb{N}}$
given by $u_{0}^{(N)}(t,x)\equiv\Psi_{0}(t,x)$ where $\Psi\in
B^{p}_{\text{loc}}$, and

$\begin{split}u_{n+1}^{(N)}(t,x)=w(t,x)&+\int_{0}^{t}\int_{\mathbb{R}^{d}}G_{t-s}(x-y)f(s,y,u_{n}^{(N)}(s,y))dyds\\\
&+\int_{0}^{t}\int_{\mathbb{R}^{d}}G_{t-s}(x-y)\sigma(s,y,u_{n}^{(N)}(s,y))\Lambda_{N}(ds,dy).\\\
\end{split}$ (52)

By (51), it follows that

$\sup_{t\in[0,T]}\sup_{|x|\leq R}\mathbb{E}|u_{n}^{(N)}(t,x)|^{p}<+\infty,$

for all $R,T>0$ and $n\in\mathbb{N}$. Similarly to (49) and (50), we obtain

$\mathbb{E}|u_{n}^{(N)}(t,x)-u_{n-1}^{(N)}(t,x)|^{p}\leq
C_{T}\int_{0}^{t}\int_{\mathbb{R}^{d}}g_{p}(t-s,x-y)\mathbb{E}|u_{n-1}^{(N)}(s,y)-u_{n-2}^{(N)}(s,y)|^{p}h(y)^{p-q}dsdy.$
(53)

Following the same steps as in the proof of Theorem 2.4, by iterating
inequality (53), we obtain:

$\mathbb{E}|u_{n}^{(N)}(t,x)-u_{n-1}^{(N)}(t,x)|^{p}\leq
C_{p,T,R}A_{n}^{(p)}(t),$ (54)

where $A_{n}^{(p)}(t)$ is defined on (36). Hence,
$\\{u_{n}^{(N)}\\}_{n\in\mathbb{N}}$ is a Cauchy sequence in
$B_{\text{loc}}^{p}$. Then, there exists a limit $u^{(N)}$ in
$B_{\text{loc}}^{p}$.

The existence and uniqueness of a solution $u$ of (44) follows in the same way
as in the proof of Theorem 2.5, by pasting the solutions
$\\{u^{(N)}\\}_{N\in\mathbb{N}}$

∎

##  3 Past-light cone property

In this section, we study the past-light cone property of the wave equation
and its relation with the Lévy noise. This property has been extensively
studied for hyperbolic PDEs in the literature. For instance, in Section 6 of
[14], it was proved that the solution of the stochastic wave equation driven
by a colored Gaussian noise remains invariant in a region of space if we
restrict the problem to this region. In this section, we prove a similar
result for the Lévy noise. Furthermore, since the construction of solutions of
SPDEs driven by a Lévy noise differs for unbounded and bounded domains in
space, we can find a new estimation of the $p$-th moments of the solution $u$
of (1) with a sequence of stopping times other than
$\\{\tau_{N}\\}_{N\in\mathbb{N}}$ defined in (15).

First, note that we can rewrite (4) as follows:

$u(t,x)=w(t,x)+\int_{0}^{t}\int_{B_{t-s}(x)}G_{t-s}(x-y)\sigma(u(s,y))\Lambda(ds,dy),$
(55)

which leads us to the following question: is it possible to solve the wave
equation (1) using a method similar to the one used for solving SPDEs in
bounded domains as in [1, 12]? For instance, in [12] it was proved that for
any $T>0$ fixed, the stochastic heat equation on a bounded domain
$D\subset\mathbb{R}^{d}$:

$\begin{cases}\dfrac{\partial u}{\partial t}(t,x)=\dfrac{1}{2}\Delta
u(t,x)+\sigma(u(t,x))\dot{\Lambda}(t,x)\,\,\,&(t,x)\in[0,T]\times D,\\\
u(t,x)=0,\,\,\,&\text{for all}\,(t,x)\in[0,T]\times\partial D,\\\
u(0,x)=u_{0}(x),\,\,\,&\text{for all}x\in D,\\\ \end{cases}$ (56)

has a unique solution $u=\\{u(t,x):t\in[0,T],x\in D\\}$ that satisfies

$\sup_{t\in[0,T]}\sup_{x\in
D}\mathbb{E}|u(t,x)\mathds{1}_{{[\\![0,\tau^{\star}_{N}]\\!]}}(t)|^{p}<+\infty,$

for all $N\in\mathbb{N}$, where $\tau^{\star}_{N}$ is the stopping time given
by

$\tau^{\star}_{N}=\inf\Bigg{\\{}t\in[0,T]:\int_{0}^{t}\int_{D}\int_{\\{|z|>N\\}}J(ds,dx,dz)>0\Bigg{\\}}.$

Since $J([0,t]\times D\times\\{|z|\geq N\\})<+\infty$, it can be proved that
$\tau^{\star}_{N}>0$ for any $N\in\mathbb{N}$, and
$\tau^{\star}_{N}\uparrow+\infty$ a.s. for $N\to+\infty$. However, the main
issue with using this stopping time for SPDEs on the whole space (such as (1)
and (11)) is that if $D=\mathbb{R}^{d}$ then $\tau^{\star}_{N}=0$ a.s. for all
$N\in\mathbb{N}$. This is due to the fact that the region
$[0,t]\times\mathbb{R}^{d}\times\\{|z|>N\\}$ may contain infinitely many
points of $J$. Consequently, if we want to solve (1) using a similar approach
as for SPDEs in a bounded domain, we have to consider a stopping time that
depends on $B_{t-s}(x)$; in other words a stopping time that depends on a
bounded space-time region. The goal of this section is to prove that the
solution $u$ of (1) given by Theorem 2.5 behaves on a bounded domain
$D\subset\mathbb{R}^{d}$ as the solution of the following integral equation:

$u(t,x)=w(t,x)+\int_{0}^{t}\int_{\mathbb{R}^{d}}G_{t-s}(x-y)\sigma(u(s,y))\mathds{1}_{\mathcal{K}^{D}(t,s)}(y)\Lambda(ds,dy),$
(57)

where

$\mathcal{K}^{D}(t,s):=\\{y\in\mathbb{R}^{d}:d(y,D)\leq t-s\\}\quad\text{for
all $s\leq t,$}$

and $d(y,D)$ denotes the distance from $y$ to the set $D$. Moreover, we prove
that for any bounded domain $D\subset\mathbb{R}^{d}$, there exists a stopping
time $\tau_{N}(D)$ such that

$\sup_{t\in[0,T]}\sup_{x\in\overline{D}}\mathbb{E}|u(t,x)\mathds{1}_{{[\\![0,\tau_{N}(D)]\\!]}}(t)|^{p}<+\infty,$

for all $T>0$, and $p$ is the same constant from Theorem 2.5. The sequence of
stopping times $\\{\tau_{N}(D)\\}_{N\in\mathbb{N}}$ is given by the following
lemma.

######  Lemma 3.1.

Let $D\subset\mathbb{R}^{d}$ be a bounded domain. For any $N\in\mathbb{N}$, we
define the stopping times given by:

$\tau_{N}(D):=\inf\Bigg{\\{}t\in\mathbb{R}_{+}\,:\,\int_{0}^{t}\int_{\mathbb{R}^{d}}\int_{\\{|z|>N\\}}\mathds{1}_{\mathcal{K}^{D}(t,s)}(y)J(ds,dy,dz)>0\Bigg{\\}}.$

Then, $\tau_{N}(D)>0$ a.s. for all $N\in\mathbb{N}$ and
$\tau_{N}(D)\uparrow+\infty$ a.s. when $N\uparrow+\infty$. Moreover,

$\mathbb{P}(\tau_{N}(D)>t)=e^{-\alpha_{D}(t)\nu\\{|z|>N\\}},$

where

$\alpha_{D}(t)=\int_{0}^{t}\int_{\mathbb{R}^{d}}\mathds{1}_{\mathcal{K}^{D}(t,s)}(y)dyds.$

###### Proof.

Observe that

$\int_{0}^{t}\int_{\mathbb{R}^{d}\times\\{|z|>N\\}}\mathds{1}_{\mathcal{K}^{D}(t,s)}(y)J(ds,dy,dz)=J(\mathfrak{R}_{D}(t)\times\\{|z|>N\\}),$

where

$\mathfrak{R}_{D}(t)=\\{(s,y)\in[0,t]\times\mathbb{R}^{d}\,:\,y\in\mathcal{K}^{D}(t,s)\\}.$

Additionally, notice that

$\alpha_{D}(t)\leq\begin{cases}2t(t+\beta_{D})\quad\text{if $d=1,$}\\\ \pi
t(t+\beta_{D})^{2}\quad\text{if $d=2,$}\end{cases}$

where $\beta_{D}=\sup_{x\in D}|x|$. For any fixed $t\geq 0$,
$J(\mathfrak{R}_{D}(t)\times\\{|z|>N\\})$ has a Poisson distribution with mean
$\alpha_{D}(t)\nu\\{|z|>N\\}$. Therefore,

$\mathbb{P}(\tau_{N}(D)>t)=\mathbb{P}(J(\mathfrak{R}_{D}(t)\times\\{|z|>N\\})=0)=e^{-\alpha_{D}(t)\nu\\{|z|>N\\}}.$

Note that the equality above, implies that $\mathbb{P}(\tau_{N}(D)>0)=1$ since
$\alpha_{D}(0)=0$.

Let $F_{N}^{(n)}:=\\{\tau_{N}(D)>n\\}$, for any $N\in\mathbb{N}$. Notice that
for any $n\in\mathbb{N}$ fixed,

$\mathbb{P}\Big{(}\operatorname*{\overline{lim}}_{N\to+\infty}F_{N}^{(n)}\Big{)}\geq\operatorname*{\overline{lim}}_{N\to+\infty}\mathbb{P}(F_{N}^{(n)})=1.$

Hence, $\mathbb{P}(\cap_{n\geq
1}\operatorname*{\overline{lim}}_{N\to+\infty}F_{N}^{(n)})=1$. This means that
with probability 1, for all $n\in\mathbb{N}$, there exists an integer
$N_{n}>n$ such that $\tau_{N_{n}}(D)>n$. This proves that with probability 1
$\tau_{N}(D)\to+\infty$ as $N\to+\infty$, since the sequence
$\\{\tau_{N}(D)\\}_{N\in\mathbb{N}}$ is non-decreasing. ∎

######  Theorem 3.2.

Let $D\subset\mathbb{R}^{d}$ be a bounded domain. Assume that Assumptions 1
and 2 are satisfied. Then, equation (57) has a unique solution
$u^{(D)}=\\{u^{(D)}(t,x);t\geq 0,x\in\mathbb{R}^{d}\\}$ that satisfies

$\sup_{t\in[0,T]}\sup_{|x|\leq
R}\mathbb{E}|u^{(D)}(t,x)\mathds{1}_{{[\\![0,\tau_{N}(D)]\\!]}}(t)|^{p}<+\infty,$

for all $T,R>0$ and $N\in\mathbb{N}$, where $p\geq 2$ is arbitrary if $d=1$,
and $p$ is the exponent from Assumption 2 if $d=2$. Moreover, if $u_{0}$ and
$v_{0}$ are bounded functions, then

$\sup_{(t,x)\in[0,T]\times\mathbb{R}^{d}}\mathbb{E}|u^{(D)}(t,x)\mathds{1}_{{[\\![0,\tau_{N}(D)]\\!]}}(t)|^{p}<+\infty,$

for all $T\in\mathbb{R}_{+}$ and $N\in\mathbb{N}$.

###### Proof.

Step 1: First, we show that for a fixed $N\in\mathbb{N}$ the stochastic
integral given by

$u(t,x)=w(t,x)+\int_{0}^{t}\int_{\mathbb{R}^{d}}G_{t-s}(x-y)\sigma(u(s,y))\mathds{1}_{\mathcal{K}^{D}(t,s)}(y)\overline{\Lambda}_{N}(ds,dy)$
(58)

has a unique solution in $B_{\text{loc}}^{p}$. Here, $\overline{\Lambda}_{N}$
is the noise $\Lambda_{N}$ corresponding to the case $h(x)=1$ i.e.

$\overline{\Lambda}_{N}(B)=b|B|+\int_{B\times\\{|z|\leq
1\\}}z\tilde{J}(dt,dx,dz)+\int_{B\times\\{1<|z|\leq N\\}}zJ(dt,dx,dz),$ (59)

where $B\in\mathcal{B}_{b}(\mathbb{R}_{+}\times\mathbb{R}^{d})$. Note that the
operator $\mathcal{T}_{(D,N)}:\mathcal{P}\mapsto\mathcal{P}$ defined as

$\mathcal{T}_{(D,N)}(\phi)(t,x):=w(t,x)+\int_{0}^{t}\int_{\mathbb{R}^{d}}G_{t-s}(x-y)\sigma(\phi(s,y))\mathds{1}_{\mathcal{K}^{D}(t,s)}(y)\overline{\Lambda}_{N}(ds,dy)$

is a self-map in $B^{p}_{\text{loc}}$ due to Lemma 2.1 and (25). Moreover, by
the same arguments as in the proof of Theorem 2.4, the Picard iteration
sequence $\\{u_{n}^{(D,N)}\\}_{n\in\mathbb{N}}$ given by
$u_{0}^{(D,N)}=\Psi_{0}\in B^{p}_{\text{loc}}$, and

$u_{n}^{(D,N)}(t,x)=w(t,x)+\int_{0}^{t}\int_{\mathbb{R}^{d}}G_{t-s}(x-y)\sigma(u_{n-1}^{(D,N)}(s,y))\mathds{1}_{\mathcal{K}^{D}(t,s)}(y)\overline{\Lambda}_{N}(ds,dy),\,\,\,\,\,\text{for
$n\in\mathbb{N}$},$

converges in $B_{\text{loc}}^{p}$ to a limit $u^{(D,N)}\in
B_{\text{loc}}^{p}$, which is the only solution of (58) in
$B_{\text{loc}}^{p}$.

Step 2: If $\tau_{N}(D)>t$, then $J(\mathfrak{R}_{D}(t)\times\\{|z|>N\\})=0$,
and consequently,

$\Lambda(B\cap\mathfrak{R}_{D}(t))-\overline{\Lambda}_{N}(B\cap\mathfrak{R}_{D}(t))=\int_{(B\cap\mathfrak{R}_{D}(t))\times\\{|z|>N\\}}zJ(dt,dx,dz)=0,$
(60)

for any $B\in\mathcal{B}_{b}(\mathbb{R}_{+}\times\mathbb{R}^{d})$. Observe
that relation (60) can be extended to all sets $A\in\tilde{\mathcal{P}}_{b}$:

$\Lambda(A\cap\mathfrak{R}_{(D,N)}^{\star}(t))=\overline{\Lambda}_{N}(A\cap\mathfrak{R}_{(D,N)}^{\star}(t)),$
(61)

where

$\mathfrak{R}_{(D,N)}^{\star}(t):=(\Omega\times\mathfrak{R}_{D}(t))\cap({[\\![0,\tau_{N}(D)]\\!]}\times\mathbb{R}^{d}).$

Therefore, by (60) and Lemma A.5, it follows that the random field $u^{(D)}$
given by

$u^{(D)}(t,x):=u^{(D,1)}(t,x)\mathds{1}_{{[\\![0,\tau_{1}(D)]\\!]}}(t)+\sum_{N=2}^{\infty}u^{(D,N)}(t,x)\mathds{1}_{(\\!(\tau_{N-1}(D),\tau_{N}(D)]\\!]}(t)$

is a solution of (57). The uniqueness of the solution $u^{(D)}$ follows in the
same way as in the proof of Theorem 2.5. ∎

######  Theorem 3.3.

Assume that Assumptions 1 and 2 are satisfied. If $u=\\{u(t,x)\,:\,t\geq
0,\,x\in\mathbb{R}^{d}\\}$ is the solution of (1) in Theorem 2.5. Then, for
any bounded domain $D\subset\mathbb{R}^{d}$,

$\sup_{t\in[0,T]}\sup_{x\in\overline{D}}\mathbb{E}|u(t,x)\mathds{1}_{{[\\![0,\tau_{N}(D)]\\!]}}(t)|^{p}<+\infty,$
(62)

for all $T>0$ and $N\in\mathbb{N}$, where $p$ is the same exponent of Theorem
2.5. Moreover,

$u(t,x)=u^{(D)}(t,x)\quad\text{a.s. for all $t\in[0,T]$ and
$x\in\overline{D},$}$

where $u^{(D)}(t,x)$ is the solution of (57) in Theorem 3.2.

###### Proof.

Without lost of generality, we assume that $u_{0}=v_{0}=0$. First, we will
prove that for any fixed $(t,x)\in[0,T]\times\overline{D}$ and
$N\in\mathbb{N}$,

$E|(u(t,x)-u^{(D)}(t,x))\mathds{1}_{[\\![0,\tau_{N}(D)\wedge\tau_{N}]\\!]}(t)|^{p}=0,$
(63)

by using the approximation sequences $\\{u_{n}^{(N)}\\}_{n\geq 1}$ and
$\\{u_{n}^{(D,N)}\\}_{n\geq 1}$. Recall that, in Step 2 of the proof of
Theorem 2.4, we proved that:

$E|(u_{n}^{(N)}(t,x)-u^{(N)}(t,x))\mathds{1}_{[\\![0,\tau_{N}]\\!]}(t)|^{p}\to
0\quad\text{as $n\to+\infty$,}$ (64)

while in the proof of Theorem 3.2 we showed that:

$E|(u_{n}^{(D,N)}(t,x)-u^{(D)}(t,x))\mathds{1}_{[\\![0,\tau_{N}(D)]\\!]}(t)|^{p}\to
0\quad\text{as $n\to+\infty$.}$ (65)

Obviously, (64) and (65) continue to hold if we replace $\tau_{N}$,
respectively $\tau_{N}(D)$ by $\tau_{N}\wedge\tau_{N}(D)$. Recall that
$u(t,x)=u^{(N)}(t,x)$ a.s. if $t\leq\tau_{N}$, and
$u^{(D)}(t,x)=u^{(D,N)}(t,x)$ a.s. if $t\leq\tau_{N}(D)$. So, by the
triangular inequality in $B_{\text{loc}}^{p}$, to prove (63) it is enough to
show that

$E|(u_{n}^{(N)}(t,x)-u_{n}^{(D,N)}(t,x))\mathds{1}_{[\\![0,\tau_{N}(D)\wedge\tau_{N}]\\!]}(t)|^{p}\to
0$ (66)

as $n\to+\infty$. First, we treat $u_{n}^{(D,N)}(t,x)$. By (41) and (60), note
that

$\Lambda(A\cap\mathfrak{R}_{D}(t))=\Lambda_{N}(A\cap\mathfrak{R}_{D}(t))=\overline{\Lambda}_{N}(A\cap\mathfrak{R}_{D}(t)),\quad\text{if}\quad
t\leq\tau_{N}\wedge\tau_{N}(D).$ (67)

Hence, by Lemma A.5 and (67), we have that

$\begin{split}&\mathds{1}_{[\\![0,\tau_{N}(D)\wedge\tau_{N}]\\!]}(t)u_{n}^{(D,N)}(t,x)\\\
&=\mathds{1}_{[\\![0,\tau_{N}(D)\wedge\tau_{N}]\\!]}(t)\int_{0}^{t}\int_{\mathbb{R}^{d}}G_{t-s}(x-y)\sigma(u_{n-1}^{(D,N)}(s,y))\mathds{1}_{\mathcal{K}^{D}(t,s)}(y)\overline{\Lambda}_{N}(ds,dy)\\\
&=\mathds{1}_{[\\![0,\tau_{N}(D)\wedge\tau_{N}]\\!]}(t)\int_{0}^{t}\int_{\mathbb{R}^{d}}G_{t-s}(x-y)\sigma(u_{n-1}^{(D,N)}(s,y))\mathds{1}_{\mathcal{K}^{D}(t,s)}(y)\mathds{1}_{[\\![0,\tau_{N}(D)\wedge\tau_{N}]\\!]}(s)\overline{\Lambda}_{N}(ds,dy)\\\
&=\mathds{1}_{[\\![0,\tau_{N}(D)\wedge\tau_{N}]\\!]}(t)\int_{0}^{t}\int_{\mathbb{R}^{d}}G_{t-s}(x-y)\sigma(u_{n-1}^{(D,N)}(s,y))\mathds{1}_{\mathcal{K}^{D}(t,s)}(y)\mathds{1}_{[\\![0,\tau_{N}(D)\wedge\tau_{N}]\\!]}(s)\Lambda_{N}(ds,dy)\\\
&=\mathds{1}_{[\\![0,\tau_{N}(D)\wedge\tau_{N}]\\!]}(t)\int_{0}^{t}\int_{\mathbb{R}^{d}}G_{t-s}(x-y)\sigma(u_{n-1}^{(D,N)}(s,y))\mathds{1}_{[\\![0,\tau_{N}(D)\wedge\tau_{N}]\\!]}(s)\Lambda_{N}(ds,dy),\\\
\end{split}$ (68)

where the last equality is due to the fact $G_{t-s}(x-y)$ contains the
indicator of the ball $B_{t-s}(x)$ and $B_{t-s}(x)\subset\mathcal{K}^{D}(t,s)$
for any $s\leq t$ and $x\in\overline{D}$. Next, we treat $u_{n}^{(N)}(t,x)$.
By Lemma A.5, we get:

$\begin{split}&\mathds{1}_{[\\![0,\tau_{N}(D)\wedge\tau_{N}]\\!]}(t)u_{n}^{(N)}(t,x)=\mathds{1}_{[\\![0,\tau_{N}(D)\wedge\tau_{N}]\\!]}(t)\int_{0}^{t}\int_{\mathbb{R}^{d}}G_{t-s}(x-y)\sigma(u_{n-1}^{(N)}(s,y))\Lambda_{N}(ds,dy)\\\
&=\mathds{1}_{[\\![0,\tau_{N}(D)\wedge\tau_{N}]\\!]}(t)\int_{0}^{t}\int_{\mathbb{R}^{d}}G_{t-s}(x-y)\sigma(u_{n-1}^{(N)}(s,y))\mathds{1}_{[\\![0,\tau_{N}(D)\wedge\tau_{N}]\\!]}(s)\Lambda_{N}(ds,dy).\\\
\end{split}$ (69)

Taking the difference between (68) and (69), it yields

$\begin{split}&\mathds{1}_{[\\![0,\tau_{N}(D)\wedge\tau_{N}]\\!]}(t)(u_{n}^{(N)}(t,x)-u_{n}^{(D,N)}(t,x))\\\
&=\mathds{1}_{[\\![0,\tau_{N}(D)\wedge\tau_{N}]\\!]}(t)\int_{0}^{t}\int_{\mathbb{R}^{d}}G_{t-s}(x-y)(\sigma(u_{n-1}^{(N)}(s,y))-\sigma(u_{n-1}^{(D,N)}(s,y)))\\\
&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\mathds{1}_{[\\![0,\tau_{N}(D)\wedge\tau_{N}]\\!]}(s)\Lambda_{N}(ds,dy).\\\
\end{split}$

Then, by Lemma 2.1, we get that for any $(t,x)\in[0,T]\times\overline{D}$,

$\begin{split}&E|\mathds{1}_{[\\![0,\tau_{N}(D)\wedge\tau_{N}]\\!]}(t)(u_{n}^{(N)}(t,x)-u_{n}^{(D,N)}(t,x))|^{p}\\\
&\leq
C_{T}\int_{0}^{t}\int_{\mathbb{R}^{d}}g_{p}(t-s,x-y)E|\mathds{1}_{[\\![0,\tau_{N}(D)\wedge\tau_{N}]\\!]}(s)(u_{n-1}^{(N)}(s,y)-u_{n-1}^{(D,N)}(s,y))|^{p}h(y)^{p-q}dyds.\\\
\end{split}$

Iterating the inequality above as for (31), we have

$\begin{split}&E|\mathds{1}_{[\\![0,\tau_{N}(D)\wedge\tau_{N}]\\!]}(t)(u_{n}^{(N)}(t,x)-u_{n}^{(D,N)}(t,x))|^{p}\\\
&\leq
C_{T}^{n}\int_{T_{n}(t)}\int_{(\mathbb{R}^{d})^{n}}\prod_{i=1}^{n}g_{p}(t_{i+1}-t_{i},x_{i+1}-x_{i})\prod_{i=1}^{n}h(x_{i})^{p-q}\\\
&\qquad\qquad\qquad\qquad\qquad
E|\mathds{1}_{[\\![0,\tau_{N}(D)\wedge\tau_{N}]\\!]}(t_{1})(u_{0}^{(N)}(t_{1},x_{1})-u_{0}^{(D,N)}(t_{1},x_{1}))|^{p}d\textbf{x}d\textbf{t}.\\\
\end{split}$ (70)

Here, $u_{0}^{(N)}$ and $u_{0}^{(D,N)}$ are arbitrary functions in
$B_{\text{loc}}^{p}$ which are chosen as starting points of the two Picard
iteration sequences $\\{u_{n}^{(N)}\\}_{n\geq 1}$ and
$\\{u_{n}^{(D,N)}\\}_{n\geq 1}$. Since we assumed that $x\in\overline{D}$, the
integrand of the integral on the right hand side of (70) is restricted to the
region

$\\{\textbf{x}\in(\mathbb{R}^{d})^{n}\,:\,|x_{i}|\leq\beta_{D}+T\,\,\text{for}\,\,i=1,\ldots,n.\\},$

where $\beta_{D}=\sup_{z\in\overline{D}}|z|$. Then, $h(x_{i})\leq
1+(T+\beta_{D})^{\eta}$, which implies that

$\prod_{i=1}^{n}h(x_{i})^{p-q}\leq\Big{[}1+(\beta_{D}+T)^{\eta}\Big{]}^{n(p-q)}:=C_{D,T}^{n}.$
(71)

So, by (70) and (71), it follows that

$\begin{split}&E|\mathds{1}_{[\\![0,\tau_{N}(D)\wedge\tau_{N}]\\!]}(t)(u_{n}^{(N)}(t,x)-u_{n}^{(D,N)}(t,x))|^{p}\\\
&\leq C_{T}^{n}C_{D,T}^{n}\sup_{s\in[0,T]}\sup_{|y|\leq
T+\beta_{D}}E|u_{0}^{(N)}(s,y)-u_{0}^{(D,N)}(s,y)|^{p}\\\
&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\int_{T_{n}(t)}\int_{(\mathbb{R}^{d})^{n}}\prod_{i=1}^{n}g_{p}(t_{i+1}-t_{i},x_{i+1}-x_{i})d\textbf{x}d\textbf{t}\\\
&=C_{T}^{n}C_{D,T}^{n}\sup_{s\in[0,T]}\sup_{|y|\leq
T+\beta_{D}}E|u_{0}^{(N)}(s,y)-u_{0}^{(D,N)}(s,y)|^{p}A_{n}^{(p)}(t).\\\
\end{split}$

Relation (66) will follow using the estimates (37) (for $p<1$) and (38) (for
$p\geq 1$) obtained for $A_{n}^{(p)}(t)$ above.

Finally, letting $N\to+\infty$ in (63) and using the fact that
$\tau_{N}(D)\wedge\tau_{N}\uparrow+\infty$ as $N\to+\infty$, we get

$u(t,x)=u^{(D)}(t,x)\,\text{a.s.}\qquad\text{for all
$(t,x)\in[0,T]\times\overline{D}$}.$

∎

######  Corollary 3.4.

Under the same assumptions of Theorem 3.3, the solution $u$ of (1) can be
written as

$u(t,x)=\sum_{i=0}^{\infty}\mathds{1}_{\Gamma_{i}}(x)u^{(D_{i})}(t,x),$

where $D_{0}=\\{|x|<1\\}$, $\Gamma_{0}=\\{|x|\leq 1\\}$,

$D_{i}=\\{i<|x|<i+1\\},\quad\text{and}\quad\Gamma_{i}=\\{i<|x|\leq
i+1\\}\quad\text{for all}\quad i\in\mathbb{N}.$

##  Appendix A Stochastic integration

For the reader’s convenience, we present in this section some basic elements
of the theory of stochastic integration of $L^{p}$-random measures (see [7,
8]) required to perform stochastic calculus with respect to a heavy-tailed
noise. There are different ways to define random measures. In this work, we
follow the definition of $L^{p}$-random measures presented as in [9].

######  Definition A.1 (Definition 2.1 in [9]).

Let $\\{\Omega_{k}\\}_{k\in\mathbb{N}}$ be a sequence of sets in $\mathcal{P}$
satisfying $\Omega_{K}\uparrow\tilde{\Omega}$, a map

$M:\mathcal{P}_{M}=\bigcup_{k\geq 1}\mathcal{P}\Big{|}_{\Omega_{k}}\to
L^{p}(\Omega)$

is called an $L^{p}$-random measure in $\tilde{\Omega}$ if it satisfies:

  * i)

$M(\emptyset)=0$ a.s.,

  * ii)

For every sequence $\\{A_{i}\\}_{i\in\mathbb{N}}$ of pairwise disjoint sets in
$\mathcal{P}_{M}$ with $\bigcup_{i=1}^{\infty}A_{i}\in\mathcal{P}_{M}$ we have

$M\Big{(}\bigcup_{i=1}^{\infty}A_{i}\Big{)}=\sum_{i=1}^{\infty}M(A_{i})\qquad\text{in
$L^{p}(\Omega).$}$

  * iii)

For all $A\in\mathcal{P}_{M}$ with
$A\subset\Omega\times(t,\infty)\times\mathbb{R}^{d}$ for some
$t\in\mathbb{R}$, the random variable $M(A)$ is $\mathcal{F}_{t}$-measurable.

  * iv)

For all $A\in\mathcal{P}_{M}$, $t\in\mathbb{R}$ and $F\in\mathcal{F}_{t}$, we
have

$M(A\cap(F\times(t,\infty)\times\mathbb{R}^{d}))=\mathds{1}_{F}M(A\cap(\Omega\times(t,\infty)\times\mathbb{R}^{d}))\quad
a.s.$

In [17], a Lévy basis is defined as an infinitely divisible independently
scattered random measure in $\mathbb{R}_{+}\times\mathbb{R}^{d}$. In [9, 10]
the definition of Lévy basis has been altered in order to use the theory of
stochastic integration developed in [7]. In this work, we consider the same
definition of Lévy basis as in [9, 10]. We recall this definition below.

######  Definition A.2.

A $L^{0}$-random measure $\tilde{\Lambda}:\tilde{\mathcal{P}_{b}}\to L^{0}$ is
called a Lévy basis if it satisfies:

  * i)

Let $\\{B_{i}\\}_{i\in\mathbb{N}}$ be a sequence of pairwise sets in
$\mathcal{B}_{b}(\mathbb{R}_{+}\times\mathbb{R}^{d})$, then
$\\{\tilde{\Lambda}(\Omega\times B_{i})\\}_{i\in\mathbb{N}}$ is a sequence of
independent random variables. Additionally, if
$B\in\mathcal{B}_{b}(\mathbb{R}_{+}\times\mathbb{R}^{d})$ satisfies
$B\subset(t,\infty)\times\mathbb{R}^{d}$ for some $t\in\mathbb{R}_{+}$, then
$\tilde{\Lambda}(\Omega\times B)$ is independent of $\mathcal{F}_{t}$.

  * ii)

For all $B\in\mathcal{B}_{b}(\mathbb{R}_{+}\times\mathbb{R}^{d})$,
$\tilde{\Lambda}(\Omega\times B)$ has infinitely divisible distribution.

  * iii)

For all $t\in\mathbb{R}_{+}$ and $k\in\mathbb{N}$ we have
$\tilde{\Lambda}(\Omega\times\\{t\\}\times[-k,k]^{d})=0$ a.s.

Now, we would like to extend the notion of pure-jump Lévy white noise
$\Lambda=\\{\Lambda(B);B\in\mathcal{B}_{b}(\mathbb{R}_{+}\times\mathbb{R}^{d})\\}$
defined in (2) for non-random Borel sets to a Lévy basis $\tilde{\Lambda}$,
and then consider the stochastic integration with respect to
$\tilde{\Lambda}$. The details of this extension can be found in [9, 15, 16].

######  Remark A.3.

The noise $\Lambda$ in (2) has an extension
$\tilde{\Lambda}:\tilde{\mathcal{P}}_{b}\to L^{0}$, that satisfies

$\tilde{\Lambda}(\Omega\times B)=\Lambda(B)\qquad\text{for all
$B\in\mathcal{B}_{b}(\mathbb{R}_{+}\times\mathbb{R}^{d})$}.$ (72)

This random measure $\tilde{\Lambda}$ satisfies the conditions of Definition
A.2.

Let $\mathcal{S}$ be the set of simple predictable processes of the form
$S=\sum_{i=1}^{k}\alpha_{i}\mathds{1}_{A_{i}}$ for some $a_{i}\in\mathbb{R}$
and $A_{i}\in\tilde{\mathcal{P}}_{M}$. Then, the stochastic integral of $S$
with respect to $M$ is given by:

$I^{\Lambda}(S)=\int_{0}^{\infty}\int_{\mathbb{R}^{d}}S(t,x)M(dt,dx)=\sum_{i=1}^{k}\alpha_{i}M(A_{i}).$

For any $p\geq 0$, the Daniell mean of a process $H=\\{H(t,x);t\geq
0,\,x\in\mathbb{R}^{d}\\}$ with respect to a $L^{p}$-random measure $M$ is
defined by

$||H||_{M,p}=\sup_{S\in\mathcal{S}_{\Lambda};|S|\leq|H|}||I^{M}(S)||_{p}.$

A predictable process $H$ is said to be $p$-integrable with respect to $M$ if
there exists a sequence $\\{S_{n}\\}_{n\geq 1}\subset\mathcal{S}$ such that
$||S_{n}-H||_{M,p}\to 0$ as $n\to+\infty$. In this case, the stochastic
integral of $H$ with respect to $M$ is given by:

$I^{M}(S)=\lim_{n\to+\infty}I^{M}(S_{n})\quad\text{in}\quad L^{p}(\Omega).$
(73)

We define the space

$L^{1,p}(M)=\Big{\\{}\text{the set of $p$-integrable processes with respect to
$M$}\Big{\\}}.$

In this definitions, we omit writing $p$, if $p=0$. Note that the map
$I^{M}:L^{1,p}(M)\mapsto L^{p}(\Omega)$ is a contraction. We include the
following version of the dominated convergence theorem for stochastic
integrals with respect to $M$.

######  Theorem A.4 (Theorem A.1. of [12]).

If $\\{H_{n}\\}_{n\in\mathbb{N}}$ are predictable and converge pointwise to
$H$, and $|H_{n}|\leq H_{0}$ for all $n\geq 1$ and some $H_{0}\in L^{1,p}(M)$,
then $H_{n},H\in L^{1,p}(M)$ and $||H-H_{n}||_{M,p}\to 0$ as $n\to+\infty$.

The following local property of the stochastic integral plays an important
role in the present article. We include its proof since we could not find it
in the literature.

######  Lemma A.5.

Let $\tilde{\Lambda}$ be a Lévy basis and $H(t,x)$ is a predictable process
such that there exists an increasing sequence $\\{T_{n}\\}_{n\in\mathbb{N}}$
of stopping times with $T_{n}\uparrow+\infty$ a.s. for $n\to+\infty$ and
$H(t,x)\mathds{1}_{[\\![0,T_{n}]\\!]}(t)$ is $p$-integrable with respect to
$\tilde{\Lambda}$ for all $n\in\mathbb{N}$. Then, for any stopping time $\tau$
we have:

$\mathds{1}_{[\\![0,\tau]\\!]}(t)\int_{0}^{t}\int_{\mathbb{R}^{d}}H(s,y)\tilde{\Lambda}(ds,dy)=\mathds{1}_{[\\![0,\tau]\\!]}(t)\int_{0}^{t}\int_{\mathbb{R}^{d}}H(s,y)\mathds{1}_{[\\![0,\tau]\\!]}(s)\tilde{\Lambda}(ds,dy).$
(74)

###### Proof.

Step 1: First, we prove (74) for an elementary stopping time $\tau$ and
elementary integrands of the form $H(s,y)=\mathds{1}_{A}(s,y)$ with
$A\in\tilde{\mathcal{P}}_{b}$ i.e. for any $t\in\mathbb{R}_{+}$

$\mathds{1}_{[\\![0,\tau]\\!]}(t)\int_{0}^{t}\int_{\mathbb{R}^{d}}\mathds{1}_{A}(s,y)\tilde{\Lambda}(ds,dy)=\mathds{1}_{[\\![0,\tau]\\!]}(t)\int_{0}^{t}\int_{\mathbb{R}^{d}}\mathds{1}_{A}(s,y)\mathds{1}_{[\\![0,\tau]\\!]}(s)\tilde{\Lambda}(ds,dy).$
(75)

Let $\tau$ be an elementary stopping times, and assume that $\tau$ takes
values

$\\{0=t_{0}<t_{1}<t_{2}<\ldots<t_{N_{0}+1}\\}.$

Then, the stochastic interval ${[\\![0,\tau]\\!]}$ can be decomposed as

${[\\![0,\tau]\\!]}=(\\{\tau=0\\}\times\\{t=0\\})\cup\bigcup_{n=0}^{N_{0}}\\{\tau\geq
t_{n+1}\\}\times(t_{n},t_{n+1}].$ (76)

Hence,

$\begin{split}\mathds{1}_{{[\\![0,\tau]\\!]}}&=\mathds{1}_{\\{t=0\\}\times\\{\tau=0\\}}+\sum_{n=0}^{N_{0}}\mathds{1}_{\\{\tau\geq
t_{n+1}\\}\times(t_{n},t_{n+1}]},\\\ \end{split}$ (77)

and observe that we can re-arrange the sum on (77) in the following way:

$\begin{split}\sum_{n=0}^{N_{0}}\mathds{1}_{\\{\tau\geq
t_{n+1}\\}\times(t_{n},t_{n+1}]}&=\sum_{n=0}^{N_{0}}\sum_{i=n+1}^{N_{0}}\mathds{1}_{\\{\tau=t_{i}\\}}\mathds{1}_{(t_{n},t_{n+1}]}\\\
&=\sum_{n=1}^{N_{0}+1}\mathds{1}_{\\{\tau=t_{n}\\}\times(0,t_{n}]}.\\\
\end{split}$ (78)

Then, we can re-write (77) as

$\mathds{1}_{{[\\![0,\tau]\\!]}}=\mathds{1}_{\\{t=0\\}\times\\{\tau=0\\}}+\sum_{n=1}^{N_{0}+1}\mathds{1}_{\\{\tau=t_{n}\\}\times(0,t_{n}]}.$
(79)

Then, by conditions (ii) and (iv) of Definition A.1 and Definition A.2-(iii),
we have:

$\begin{split}\int_{0}^{t}\int_{\mathbb{R}^{d}}\mathds{1}_{A}(s,y)\mathds{1}_{[\\![0,\tau]\\!]}(s)\tilde{\Lambda}(ds,dy)&=\tilde{\Lambda}(A\cap([\\![0,\tau]\\!]\times\mathbb{R}^{d})\cap(\Omega\times[0,t]\times\mathbb{R}^{d}))\\\
&=\sum_{n=0}^{N_{0}}\tilde{\Lambda}(A\cap(\\{\tau\geq
t_{n+1}\\}\times(t_{n},t_{n+1}]\times\mathbb{R}^{d})\cap(\Omega\times[0,t]\times\mathbb{R}^{d}))\\\
&=\sum_{n=0}^{N_{0}}\mathds{1}_{\\{\tau\geq
t_{n+1}\\}}\tilde{\Lambda}(A\cap(\Omega\times(t_{n},t_{n+1}]\times\mathbb{R}^{d})\cap(\Omega\times[0,t]\times\mathbb{R}^{d}))\\\
&=\sum_{n=0}^{N_{0}}\mathds{1}_{\\{\tau\geq
t_{n+1}\\}}\int_{0}^{t}\int_{\mathbb{R}^{d}}\mathds{1}_{A}(s,y)\mathds{1}_{(t_{n},t_{n+1}]}(s)\tilde{\Lambda}(ds,dy)\\\
&=\sum_{n=1}^{N_{0}+1}\mathds{1}_{\\{\tau=t_{n}\\}}\int_{0}^{t}\int_{\mathbb{R}^{d}}\mathds{1}_{A}(s,y)\mathds{1}_{(0,t_{n}]}(s)\tilde{\Lambda}(ds,dy).\\\
\end{split}$ (80)

Using the same argument of (78) and the linearity of $\tilde{\Lambda}$ on
$\tilde{\mathcal{P}}_{b}$, we can re-arrange the sum on the last equality of
(80) as it follows,

$\sum_{n=0}^{N_{0}}\mathds{1}_{\\{\tau\geq
t_{n+1}\\}}\int_{0}^{t}\int_{\mathbb{R}^{d}}\mathds{1}_{A}(s,y)\mathds{1}_{(t_{n},t_{n+1}]}(s)\tilde{\Lambda}(ds,dy)=\sum_{n=1}^{N_{0}+1}\mathds{1}_{\\{\tau=t_{n}\\}}\int_{0}^{t}\int_{\mathbb{R}^{d}}\mathds{1}_{A}(s,y)\mathds{1}_{(0,t_{n}]}(s)\tilde{\Lambda}(ds,dy).$
(81)

Additionally, for a fixed $i\in\\{1,2,\ldots,N_{0}\\}$, it holds:

$\mathds{1}_{\\{\tau=t_{i}\\}}\mathds{1}_{[\\![0,\tau]\\!]}(t)=\mathds{1}_{\\{\tau=t_{i}\\}}\mathds{1}_{(0,t_{i}]}(t).$
(82)

Hence, by (79), (80), (81), and (82), we have:

$\begin{split}\mathds{1}_{[\\![0,\tau]\\!]}(t)\int_{0}^{t}\int_{\mathbb{R}^{d}}\mathds{1}_{A}(s,y)\mathds{1}_{[\\![0,\tau]\\!]}(s)\tilde{\Lambda}(ds,dy)&=\sum_{n=1}^{N_{0}+1}\mathds{1}_{\\{\tau=t_{n}\\}}\mathds{1}_{[\\![0,\tau]\\!]}(t)\int_{0}^{t}\int_{\mathbb{R}^{d}}\mathds{1}_{A}(s,y)\mathds{1}_{(0,t_{n}]}(s)\tilde{\Lambda}(ds,dy)\\\
&=\sum_{n=1}^{N_{0}+1}\mathds{1}_{\\{\tau=t_{n}\\}}\mathds{1}_{(0,t_{n}]}(t)\int_{0}^{t}\int_{\mathbb{R}^{d}}\mathds{1}_{A}(s,y)\mathds{1}_{(0,t_{n}]}(s)\tilde{\Lambda}(ds,dy)\\\
&=\sum_{n=1}^{N_{0}+1}\mathds{1}_{\\{\tau=t_{n}\\}}\mathds{1}_{(0,t_{n}]}(t)\int_{0}^{t}\int_{\mathbb{R}^{d}}\mathds{1}_{A}(s,y)\tilde{\Lambda}(ds,dy)\\\
&=\mathds{1}_{{[\\![0,\tau]\\!]}}(t)\int_{\mathbb{R}^{d}}\mathds{1}_{A}(s,y)\tilde{\Lambda}(ds,dy).\\\
\end{split}$ (83)

Step 2: Now we prove (75) when $\tau$ is an arbitrary stopping time. Note that
there exists a sequence of elementary stopping times
$\\{\tau_{n}\\}_{n\in\mathbb{N}}$ such that $\tau_{n}\uparrow\tau$ a.s. for
$n\to+\infty$. Hence, by Theorem A.4, we have that (75) holds for $\tau$.

Step 3: We prove the case for $H(t,x)$ under the assumptions above. Since
$H(s,y)\mathds{1}_{{[\\![0,T_{n}]\\!]}}(s)\in L^{1,p}(\tilde{\Lambda})$ for
all $n\in\mathbb{N}$, there exists a sequence of simple integrands
$\\{S_{n}\\}_{n\in\mathbb{N}}$ such that
$||v^{(t)}-S_{n}^{(t)}||_{\tilde{\Lambda},p}\to 0$ as $n\to+\infty$, where
$v^{(t)}(s,y)=H(s,y)\mathds{1}_{[0,t]}(s)$ and
$S_{n}^{(t)}(s,y)=S_{n}(s,y)\mathds{1}_{[0,t]}(s)$ for each fixed
$t\in\mathbb{R}_{+}$. Hence, by linearity of $I^{\tilde{\Lambda}}$ and the
steps above, we have:

$\mathds{1}_{[\\![0,\tau]\\!]}(t)\int_{0}^{t}\int_{\mathbb{R}^{d}}S_{n}(s,y)\tilde{\Lambda}(ds,dy)=\mathds{1}_{[\\![0,\tau]\\!]}(t)\int_{0}^{t}\int_{\mathbb{R}^{d}}S_{n}(s,y)\mathds{1}_{[\\![0,\tau]\\!]}(s)\tilde{\Lambda}(ds,dy)\quad\text{for
all $n\in\mathbb{N}$.}$

Letting $n\to+\infty$, it follows (74). ∎

## Acknowledgments

The author is deeply grateful to Raluca M. Balan for her valuable comments and
some discussion.

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