lil

# The nonlinear Schrödinger Equation driven by jump processes

Anne de Bouard and Erika Hausenblas  A. de Bouard, CMAP, Ecole polytechnique,
CNRS, Université Paris-Saclay, 91128 Palaiseau, France  E. Hausenblas,
Lehrstuhl Angewandte Mathematik, Montanuniversitat Leoben, Franz-Josef-Strasse
18, 8700 Leoben, Austria

###### Abstract.

The main result of the paper is the existence of a solution of the nonlinear
Schrödinger equation with a Lévy noise with infinite activity. To be more
precise, let $A=\Delta$ be the Laplace operator with $D(A)=\\{u\in
L^{2}({\mathbb{R}}^{d}):\Delta u\in L^{2}({\mathbb{R}}^{d})\\}$. Let
$Z\hookrightarrow L^{2}({\mathbb{R}}^{d})$ be a function space and $\eta$ be a
Poisson random measure on $Z$, let $g:{\mathbb{R}}\to\mathbb{C}$ and
$h:{\mathbb{R}}\to\mathbb{C}$ be some given functions, satisfying certain
conditions specified later. Let $\alpha\geq 1$ and $\lambda\geq 0$. We are
interested in the solution of the following equation

(0.4) $\displaystyle\hskip
56.9055pt\left\\{\begin{array}[]{rcl}\lx@intercol\@ADDCLASS{ltx_eqn_lefteqn}$\displaystyle
i\,du(t,x)-\Delta u(t,x)\,dt+\lambda|u(t,x)|^{\alpha-1}u(t,x)\,dt\hskip
56.9055pt$\mbox{}\hfil\lx@intercol&&\\\
&=&\int_{Z}u(t,x)\,g(z(x))\,\tilde{\eta}(dz,dt)+\int_{Z}u(t,x)\,h(z(x))\,\gamma(dz,dt),\\\
u(0)&=&u_{0}.\end{array}\right.$

First we consider the case, where the Lévy process is a compound Poisson
process. With the help of this result we can tackle the general case, and show
that (0.4) has a solution.

This work was supported by the FWF-Project P17273-N12 and the ANR project
Stosymap (ANR 2011-B501-015-03)

Keywords and phrases: Stochastic integral of jump type, stochastic partial
differential equations, Poisson random measures, Lévy processes, Schrödinger
Equation.

AMS subject classification (2002): Primary 60H15; Secondary 60G57.

##  1\. Introduction

We consider in the present paper the problem of existence of solutions for the
nonlinear Schrödinger equation with Lévy noise. To be more precise, let
$A=\Delta$ be the Laplace operator with $D(A)=\\{u\in
L^{2}({\mathbb{R}}^{d}):\Delta u\in L^{2}({\mathbb{R}}^{d})\\}$. Let
$Z\hookrightarrow L^{2}({\mathbb{R}}^{d})$ be a function space and $\eta$ be a
Poisson random measure on $Z$, let $g:{\mathbb{R}}\to\mathbb{C}$ and
$h:{\mathbb{R}}\to\mathbb{C}$ be some given functions, satisfying certain
conditions specified later. Let $\alpha\geq 1$ and $\lambda\in{\mathbb{R}}$.
We are interested in the following equation

(1.4) $\displaystyle\hskip
56.9055pt\left\\{\begin{array}[]{rcl}\lx@intercol\@ADDCLASS{ltx_eqn_lefteqn}$\displaystyle
i\,du(t,x)-Au(t,x)\,dt+\lambda|u(t,x)|^{\alpha-1}u(t,x)\,dt\hskip
56.9055pt$\mbox{}\hfil\lx@intercol&&\\\
&=&\int_{Z}u(t,x)\,g(z(x))\,\tilde{\eta}(dz,dt)+\int_{Z}u(t,x)\,h(z(x))\,\gamma(dz,dt),\\\
u(0)&=&u_{0}.\end{array}\right.$

Our aim is to investigate the conditions on the nonlinearity, on the space $Z$
and on the complex valued functions $c$ and $g$, under which there exists a
weak or martingale solution to (1.4).

The Nonlinear Schrödinger equation (NLS) is a universal model that describes
the propagation of nonlinear waves in dispersive media. It may e.g. appear as
a so-called modulation equation, describing the complex enveloppe of a highly
oscillating field in nonlinear optics, and in particular in fiber optics (see
[2, 22]). It may also be derived from the water wave problem, thanks to
scaling and perturbation arguments, to describe the propagation of surface
waves of finite amplitude in deep fluids (see [16, 27]). The propagation of
nonlinear dispersive waves in nonhomogeneous or random media (or taking
account of temperature effects) can be modelled by the nonlinear equation with
a random force, or a random potential (see e.g. [1, 4, 18]).

When the stochastic perturbation is a Wiener process, the equation is well
treated and existence and uniqueness of the solution is known, under
reasonable assumptions on the noise correlation and on the nonlinearity. For
more information see [5, 12, 13, 14, 15]. The case where the nonlinear
Schrödinger equation is perturbed by a Lévy process is much less treated in
the literature. In [25, 26], the authors consider the NLS equation with
randomly distributed, but isolated jumps. In the context of fiber optics, the
model would describe random amplification of the signal at random (but
isolated) locations along the fiber (see [21]). In that situation the
existence and uniqueness of solutions is easily deduced from the classical
results known in the deterministic case, and the motivations in [25, 26] were
to obtain the evolution law of some physical observables of the solution.

Here, we consider the more general case where the noise is an infinite
dimensional Lévy process, with possibly non isolated jumps, and we investigate
the existence of martingale solutions. Before stating the precise result, let
us introduce some notations.

######  Notation 1.1.

For $k\in\mathbb{N}_{0}$ we denote by $H^{k}_{p}({\mathbb{R}}^{d})$ the
classical Sobolev spaces defined in [19, Chapter 3, Definition 3.1]. For
$\delta\geq 0$ and $p\in(0,\infty]$ let
$L^{p}_{\delta}({\mathbb{R}}^{d})=\\{v\in L^{p}({\mathbb{R}}^{d})$ with
$\int_{{\mathbb{R}}^{d}}(1+|x|^{2})^{\frac{\delta}{2}}\,|u(x)|^{p}\,dx<\infty\\}$.

For any index $p\in[1,\infty]$ we denote throughout the paper the conjugate
element by $p^{\prime}$. In particular, we have
$\tfrac{1}{p}+\tfrac{1}{p^{\prime}}=1$. For complex valued functions $u$ and
$v$ in $L^{2}({\mathbb{R}}^{d})$, we denote by $\langle u,v\rangle$ the (real)
inner product

$\langle u,v\rangle=\Re\int_{{\mathbb{R}}^{d}}u(x)\overline{v(x)}\,dx.$

Given a Banach space $E$ and a number $R>0$, we denote by $B_{E}(R)$ all
elements with norm smaller or equal to $R>0$, i.e. $B_{E}(R):=\\{x\in
E,|x|_{E}\leq R\\}$.

Suppose that $(Z,{\mathcal{Z}})$ is a measurable space. By $M(Z)$,
respectively $M_{+}(Z)$, we will denote the set of all $\mathbb{R}$,
respectively $[0,\infty]$-valued measures on $(Z,{\mathcal{Z}})$. By
${\mathcal{M}}(Z)$, respectively ${\mathcal{M}}_{+}(Z)$, we will denote the
$\sigma$-field on $M(Z)$, respectively $M_{+}(Z)$, generated by functions

$i_{B}:M(Z)\ni\mu\mapsto\mu(B)\in{\mathbb{R}},$

respectively by functions

$i_{B}:M_{+}(Z)\ni\mu\mapsto\mu(B)\in[0,\infty],$

for all $B\in{\mathcal{Z}}$. Similarly, by $M_{I}(Z)$ we will denote the
family of all $\overline{\mathbb{N}}$-valued measures on $(Z,{\mathcal{Z}})$
$(\overline{\mathbb{N}}=\mathbb{N}\cup\\{\infty\\})$, and by
${\mathcal{M}}_{I}(Z)$ the $\sigma$-field on $M_{I}(Z)$ generated by functions
$i_{B}:M(Z)\ni\mu\mapsto\mu(B)\in\overline{\mathbb{N}}$, $B\in{\mathcal{Z}}$.

Finally, by ${\mathcal{Z}}\otimes\mathcal{B}({\mathbb{R}_{+}})$ we denote the
product $\sigma$-field on $Z\times\mathbb{R}_{+}$ and by $\nu\otimes\lambda$
we denote the product measure of $\nu$ and the Lebesgue measure $\lambda$.

##  2\. Preliminaries and main result

Throughout the whole paper, we assume that
$\mathfrak{A}=(\Omega,{{\mathcal{F}}},{{\mathbb{F}}},{\mathbb{P}})$ is a
complete filtered probability space with right continuous filtration
$\\{{{\mathcal{F}}}_{t}\\}_{t\geq 0}$, denoted by by ${{\mathbb{F}}}$. The
following definitions are presented here for the sake of completeness because
the notion of time homogeneous random measure is introduced in many, not
always equivalent ways.

######  Definition 2.1.

(see [20], Def. I.8.1) Let $(Z,{{\mathcal{Z}}})$ be a measurable space.
A Poisson random measure $\eta$ on $(Z,{{\mathcal{Z}}})$ over
$(\Omega,{{\mathcal{F}}},{{\mathbb{F}}},{\mathbb{P}})$ is a measurable
function
$\eta:(\Omega,{{\mathcal{F}}})\to(M_{I}(Z\times{\mathbb{R}}_{+}),{{\mathcal{M}}}_{I}(Z\times{\mathbb{R}}_{+}))$,
such that

  * (i)

for each $B\in{{\mathcal{Z}}}\otimes\mathcal{B}({\mathbb{R}_{+}})$,
$\eta(B):=i_{B}\circ\eta:\Omega\to\bar{\mathbb{N}}$ is a Poisson random
variable with parameter111If $\mathbb{E}\eta(B)=\infty$, then obviously
$\eta(B)=\infty$ a.s.. $\mathbb{E}\eta(B)$;

  * (ii)

$\eta$ is independently scattered, i.e. if the sets
$B_{j}\in{{\mathcal{Z}}}\otimes\mathcal{B}({\mathbb{R}_{+}})$, $j=1,\cdots,n$,
are disjoint, then the random variables $\eta(B_{j})$, $j=1,\cdots,n$, are
independent;

  * (iii)

for each $U\in{{\mathcal{Z}}}$, the $\bar{\mathbb{N}}$-valued process
$(N(t,U))_{t\geq 0}$ defined by

$N(t,U):=\eta(U\times(0,t]),\;\;t\geq 0$

is ${{\mathbb{F}}}$-adapted and its increments are independent of the past,
i.e. if $t>s\geq 0$, then $N(t,U)-N(s,U)=\eta(U\times(s,t])$ is independent of
$\mathcal{F}_{s}$.

######  Definition 2.2.

The compensator of a random measure $\eta$ on a Banach space $Z$ is the unique
predictable measure
$\gamma:{{\mathcal{Z}}}\times\mathcal{B}({\mathbb{R}}^{0}_{+})\to{\mathbb{R}}$,
such that for any $A\in{{\mathcal{Z}}}$ the process

${\mathbb{R}}_{+}^{0}\ni t\mapsto\eta(A\times[0,t])-\gamma(A\times[0,t])$

is a martingale over $\mathfrak{A}$. We will denote by $\tilde{\eta}$ the
compensated Poisson random measure defined by $\tilde{\eta}:=\eta-\gamma$.

######  Remark 2.3.

Assume that $\eta$ is a time homogeneous Poisson random measure on
$(Z,{{\mathcal{Z}}})$ over
$(\Omega,{{\mathcal{F}}},{{\mathbb{F}}},{\mathbb{P}})$. It turns out that the
compensator $\gamma$ of $\eta$ is uniquely determined and moreover

$\gamma:{{\mathcal{Z}}}\times{{\mathcal{B}}}({\mathbb{R}}^{+})\ni(A,I)\mapsto\nu(A)\times\lambda(I),$

where the $\sigma$–finite measure
$\nu:{{\mathcal{Z}}}\to{\mathbb{R}}_{+}\cup\\{\infty\\}$ is defined by
${{\mathcal{Z}}}\ni A\mapsto\nu(A):=\mathbb{E}\eta(A\times[0,1])$. The
difference between a time homogeneous Poisson random measure $\eta$ and its
compensator $\gamma$, i.e. $\tilde{\eta}=\eta-\gamma$, is called a compensated
Poisson random measure. The measure $\nu$ is called intensity measure of
$\eta$.

Let $Z\hookrightarrow L^{2}({\mathbb{R}}^{d})$ be a function space, $\nu$ a
$\sigma$–finite measure on $Z$ such that

$\nu(\\{0\\})=0,\quad\int_{Z}(|z|^{2}\wedge
1)\nu(dz)<\infty,\quad\mathrm{and}\;\nu(Z\setminus
B_{Z}(\varepsilon))<\infty\;\mathrm{for\;all}\;\varepsilon>0.$

Let $\eta$ be a time homogenous Poisson random measure on $Z$ with intensity
measure $\nu$ over $\mathfrak{A}$.

Let $g:{\mathbb{R}}\to\mathbb{C}$ and $h:{\mathbb{R}}\to\mathbb{C}$ be two
functions specified later. We will denote by $G:L^{2}({\mathbb{R}}^{d})\to
L^{2}({\mathbb{R}}^{d})$ and $H:L^{2}({\mathbb{R}}^{d})\to
L^{2}({\mathbb{R}}^{d})$ the Nemytskii operators associated to the functions
$g$ and $h$, and defined by

$(G(z))(x):=g(z(x)),\quad\mbox{and}\quad(H(z))(x):=h(z(x)),\quad z\in
Z,\,x\in{\mathbb{R}}^{d}.$

We are now interested in the following equation

(2.4) $\displaystyle\hskip
85.35826pt\left\\{\begin{array}[]{rcl}\lx@intercol\@ADDCLASS{ltx_eqn_lefteqn}$\displaystyle
i\,du(t,x)-Au(t,x)\,dt+\lambda|u(t,x)|^{\alpha-1}u(t,x)\,dt\hskip
56.9055pt$\mbox{}\hfil\lx@intercol&&\\\
&=&\int_{Z}u(t,x)\,g(z(x))\,\tilde{\eta}(dz,dt)+\int_{Z}u(t,x)\,h(z(x))\,\gamma(dz,dt),\\\
u(0)&=&u_{0},\end{array}\right.$

Let us denote by $({{\mathcal{T}}}(t))_{t\geq 0}$ the group of isometries
generated by the operator $-iA$. As is classical in the framework of evolution
equations, we will consider a mild solution of equation (2.4), whose
definition is given below.

######  Definition 2.4.

Let $E$ be a Banach space. We call $u$ an $E$–valued solution to Equation
(2.4), if and only if $u\in\mathbb{D}(0,T;E)$ ${\mathbb{P}}$-a.s., the terms

$\int_{0}^{t}{{\mathcal{T}}}(t-s)|u(s)|^{\alpha-1}u(s)\,ds,\quad\int_{0}^{t}\int_{Z}\left|{{\mathcal{T}}}(t-s)u(s)\,G(z)\right|^{2}\nu(dz)\,ds$

and

$\int_{0}^{t}\int_{Z}{{\mathcal{T}}}(t-s)u(s)\,H(z)\nu(dz)\,ds,$

are well defined for any $t\in[0,T]$ in $E$ and $u$ solves ${\mathbb{P}}$-a.s.
the integral equation

$\displaystyle u(t)$ $\displaystyle=$
$\displaystyle{{\mathcal{T}}}(t)u_{0}+i\lambda\int_{0}^{t}{{\mathcal{T}}}(t-s)|u(s)|^{\alpha-1}u(s)\,ds$
$\displaystyle-i\int_{0}^{t}\int_{Z}{{\mathcal{T}}}(t-s)u(s)\,G(z)\,\tilde{\eta}(dz,ds)-i\int_{0}^{t}\int_{Z}{{\mathcal{T}}}(t-s){u(s)H(z)\gamma(dz,ds)}.$

However, for a Lévy noise with infinite activity, we could not show the
existence of a unique strong solution, only the existence of a martingale
solution. A concept, defined in the following.

######  Definition 2.5.

Let $E$ be a separable Banach space. Let $(Z,{{\mathcal{Z}}})$ be a measurable
space and $\nu$ a $\sigma$–finite measure on $(Z,{{\mathcal{Z}}})$. Suppose
that $G$ and $H$ are a densely defined function from $Z$ to $E$. Let $u_{0}\in
E$.
A martingale solution on $E$ to the Problem (2.4) is a system

(2.6)
$\left(\Omega,{{{\mathcal{F}}}},{\mathbb{P}},{{\mathbb{F}}},\\{\eta(t)\\}_{t\geq
0},\\{u(t)\\}_{t\geq 0}\right)$

such that

  * (i)

$(\Omega,{{{\mathcal{F}}}},{{\mathbb{F}}},{\mathbb{P}})$ is a complete
filtered probability space with filtration
${{\mathbb{F}}}=\\{{{{\mathcal{F}}}}_{t}\\}_{t\geq 0}$,

  * (ii)

${\\{\eta(t)\\}_{t\geq 0}}$ is a time homogeneous Poisson random measure on
$(Z,{{\mathcal{B}}}(Z))$ over
$(\Omega,{{{\mathcal{F}}}},{{\mathbb{F}}},{\mathbb{P}})$ with intensity
measure $\nu$,

  * (iii)

$u=\\{u(t)\\}_{t\geq 0}$ is a $E$–valued mild solution to the Problem (2.4).

In order to be able to show the existence of a solution, the space $Z$, the
Lévy measure $\nu$ and the functions $g,h:{\mathbb{R}}^{d}\to\mathbb{C}$ have
to satisfy certain conditions. In particular, they have to satisfy the
following hypothesis.

######  Hypothesis 1.

First, we assume that $Z$ a function space and $\nu$ a Lévy measure on $Z$
such that

  1. (i)

$Z$ is continuously embedded in the Sobolev space
$W^{1}_{\infty}({\mathbb{R}}^{d})$;

  2. (ii)

the Lévy measure $\nu$ satisfies the following integrability conditions

    1. (a)

$C_{0}(\nu):=\int_{Z}|z|_{L^{\infty}}^{2}\nu(dz)<\infty;\phantom{\Big{|}}$

    2. (b)

$C_{1}(\nu):=\int_{Z}|z|_{H^{1}_{\infty}}^{2}\nu(dz)<\infty;\phantom{\Big{|}}$

    3. (c)

$C_{2}(\nu):=\int_{Z}\sup_{x\in{{\mathbb{R}}^{d}}}|x|^{2}|z(x)|^{2}\nu(dz)<\infty;\phantom{\Big{|}}$

    4. (d)

$C_{3}(\nu):=\int_{Z}|z|_{L^{\infty}}^{4}\nu(dz)<\infty.\phantom{\Big{|}}$

In addition the functions $g:{\mathbb{R}}\to\mathbb{C}$ and
$h:{\mathbb{R}}\to\mathbb{C}$ are satisfying the following items:

  1. (iii)

$g$, $h$ and their first order derivatives are of linear growth, i.e. there
exist some constants $C_{g}$ and $C_{h}$ such that

$|g(\xi)|,|g^{\prime}(\xi)|\leq
C_{g}|\xi|\quad\mbox{and}\quad|h(\xi)|,|h^{\prime}(\xi)|\leq
C_{h}|\xi|,\quad\forall\xi\in{\mathbb{R}}.$

  2. (iv)

$g(0)=0$ and $h(0)=0$;

######  Remark 2.6.

Hypothesis 1 implies that the Nemitskii operators $G$ and $H$ associated to
$g$ and $h$ map $L^{\infty}({\mathbb{R}}^{d})$ into
$L^{\infty}({\mathbb{R}}^{d})$, and their Frechet derivative $\nabla G$ and
$\nabla H$ also map $L^{\infty}({\mathbb{R}}^{d})$ into
$L^{\infty}({\mathbb{R}}^{d})$.

To show the existence of the solution to the nonlinear Schrödinger equation
with Lévy noise of infinite activity, we use the technical Lemma 4.1 below,
which gives existence and uniqueness of the solution to (2.4) where the Levy
process is a compound Poisson Process, i.e. if the Lévy process has only
finite activity. Then, we use a cut off of the small jumps with a cut off
parameter $\varepsilon>0$ in order to get a noise with finite activity, and we
apply Lemma 4.1 to get the existence of a unique solution of (2.4) with the
cut-off noise of finite activity. In a second step, we show the existence of a
limit as $\varepsilon\to 0$. Here, uniform bounds on the
$L^{2}({\mathbb{R}}^{d})$ norm and $H^{1}_{2}({\mathbb{R}}^{d})$ norm play an
important role. Similarly to the deterministic setting, this uniform bounds
are obtained by controlling the mass and energy in average. Let us define the
mass by

(2.7) $\displaystyle\mathcal{E}(u):=\int_{{\mathbb{R}}^{d}}|u(x)|^{2}\,dx,$

and the energy by

(2.8) $\displaystyle\mathcal{H}(u):=\frac{1}{2}\int_{{\mathbb{R}}^{d}}|\nabla
u(x)|^{2}\,dx+{\lambda\over\alpha+1}\int_{{\mathbb{R}}^{d}}|u(x)|^{\alpha-1}{u(x)\,{\overline{u(x)}}\,}dx.$

Under certain constrains on $g$, $h$ and $u_{0}$, the mass will be conserved.
For this purpose, we introduce following hypothesis:

######  Hypothesis 2.

Let us assume

  1. (i)

$\Im(g(\xi))=\Im(h(\xi)),\quad\xi\in{\mathbb{R}}$;

  2. (ii)

$|1{-i}g(\xi)|=1,\quad\xi\in{\mathbb{R}}.$

######  Hypothesis 3.

Let us assume

$\displaystyle 2\Im(h(\xi))+|g(\xi)|^{2}=0,\quad\xi\in{\mathbb{R}}.$

In fact, Hypothesis 1 gives only conditions under which the solution exists.
Hypothesis 2 and Hypothesis 3 give the conditions for the conservation of
${{\mathcal{E}}}(u)$ ${\mathbb{P}}$–a.s. or in mean. We now state our main
result.

######  Theorem 2.7.

Let $\eta$ be a time homogenous Poisson random measure on a Banach space $Z$
with Lévy measure $\nu$ satisfying Hypothesis 1. Assume $\lambda>0$,
$1\leq\alpha<1+4/(d-2)$ if $d>2$, or $1\leq\alpha<+\infty$ if $d=1,2$. Let
$u_{0}\in H^{1}_{2}({\mathbb{R}}^{d})$ with

$\int_{{\mathbb{R}}^{d}}x^{2}|u_{0}(x)|^{2}\,dx<\infty,$

then for any $\gamma<1$ there exists a
$H^{\gamma}_{2}({\mathbb{R}}^{d})$–valued martingale mild solution to (2.4),
such that

  1. (1)

for any $T>0$, there exists a constant
$C=C(T,C_{0}(\nu),C_{3}(\nu),C_{g},C_{h})>0$ such that

$\mathbb{E}\sup_{0\leq s\leq T}|u(s)|^{2}_{L^{2}}\leq
C\,(1+|u_{0}|^{2}_{L^{2}}).$

  2. (2)

for any $T>0$, there exists a constant
$C=C(T,C_{0}(\nu),C_{1}(\nu),C_{g},C_{h})>0$ such that

$\mathbb{E}\sup_{0\leq t\leq T}{{\mathcal{H}}}(u(t))\leq
C\,\left(1+{{\mathcal{H}}}(u_{0})\right).$

  3. (3)

for any $T>0$, there exists a constant $C=C(T,C_{2}(\nu),C_{g},C_{h})>0$ such
that

$\mathbb{E}\sup_{0\leq t\leq
T}\int_{{\mathbb{R}}^{d}}x^{2}|u(t,x)|^{2}\,dx\leq
C\,\big{(}1+\int_{{\mathbb{R}}^{d}}x^{2}|u_{0}(x)|^{2}\,dx\big{)}.$

In addition,

  1. (1)

if Hypothesis 2 is satisfied, then we have for all $t\geq 0$,

$|u(t)|^{2}_{L^{2}}=|u_{0}|^{2}_{L^{2}},\quad{\mathbb{P}}-\mbox{a.s.};$

  2. (2)

if Hypothesis 3 is satisfied, then we have for all $t\geq 0$,

$\mathbb{E}|u(t)|^{2}_{L^{2}}=|u_{0}|^{2}_{L^{2}}.$

######  Example 2.8.

Let $\xi\in{\mathbb{R}}^{d}$ be fixed and
$\theta:{\mathbb{R}}^{d}\times{\mathbb{R}}\to{\mathbb{R}}$. Then
$g_{\xi}:{\mathbb{R}}\ni z\mapsto i(e^{i\theta(\xi,z)}-1)$, and
$h_{\xi}(z)=i\left(\cos(\theta(\xi,z))-1\right)$ satisfy assumption 2.

The proof of Theorem 2.7 is presented in Section 5, the technical Lemma 4.1 is
presented in Section 4. First, in Section 3 we summarize some deterministic
preliminaries which we need for the proof. In the Appendix we collect several
results, which we used within the proof.

##  3\. Deterministic Preliminaries

In this section we shortly introduce some propositions and lemmata, which are
necessary to show our main results. But before starting let us introduce some
definitions. The group $({{\mathcal{T}}}(t))_{t\geq 0}$, corresponding to the
Cauchy problem

$\displaystyle\left\\{\begin{array}[]{rcl}i\partial_{t}u(t,x)-\Delta
u(t,x)&=&0,\\\ u(0,x)&=&\phi(x),\quad
x\in{\mathbb{R}}^{d},\,t\in{\mathbb{R}},\end{array}\right.$

can be expressed explicitly in Fourier variables, i.e.

${{\mathcal{F}}}\left({{\mathcal{T}}}(t)\phi\right)(\xi)=e^{4\pi^{2}i|\xi|^{2}t}\hat{\phi}(\xi),\quad\xi\in{\mathbb{R}}^{d},\,t\in{\mathbb{R}}.$

We recall some well known deterministic results.

######  Lemma 3.1.

If $t\not=0$, $1/p+1/p^{\prime}=1$ and $p^{\prime}\in[1,2]$, then
${{\mathcal{T}}}(t):L^{p^{\prime}}({\mathbb{R}}^{d})\to
L^{p}({\mathbb{R}}^{d})$ is continuous and

$\left|{{\mathcal{T}}}(t)\phi\right|_{L^{p}}\leq
c\,|t|^{-\frac{d}{2}\left(\frac{1}{p^{\prime}}-\frac{1}{p}\right)}\left|\phi\right|_{L^{p^{\prime}}}={c\,|t|^{-d\left(\frac{1}{2}-\frac{1}{p}\right)}}\left|\phi\right|_{L^{p^{\prime}}}.$

A pair $(p,q)$ is called admissible if

(3.5) $\displaystyle\left\\{\begin{array}[]{rcl}2\leq p<{2d\over d-2},&\mbox{
if }&d\geq 3,\\\ 2\leq p<\infty,&\mbox{ if }&d=2,\\\ 2\leq p\leq\infty,&\mbox{
if
}&d=1,\end{array}\right\\}\quad\mbox{and}\quad\frac{2}{q}=\frac{d}{2}-\frac{d}{p}.$

Let $s\in{\mathbb{R}}$, $1\leq p,q\leq\infty$. For an interval
$I\subset[0,\infty)$ let $L^{q}(I;H_{p}^{s}({\mathbb{R}}^{d}))$ be the space
of all measurable functions $f:I\times{\mathbb{R}}^{d}\to{\mathbb{R}}$ such
that

$|f|_{L^{q}(I;H^{s}_{p})}:=\left(\int_{I}|f(t)|_{H_{p}^{s}}^{q}\,dt\right)^{\frac{1}{q}}<\infty.$

Let us define the convolution operator

(3.6)
$\displaystyle\mathfrak{T}u(t):=\int_{0}^{t}{{\mathcal{T}}}(t-r)u(r)\,dr,\quad
t\geq 0.$

By means of Lemma 3.1, the following Corollary can be proven.

######  Corollary 3.2.

(see [10, Theorem 2.3.3]) Let
$(p_{0},q_{0}),(p_{1},q_{1})\in[2,\infty)\times[2,\infty)$ be two admissible
pairs. Then for all $T>0$ and $s\in{\mathbb{R}}$ we have

$\left|\mathfrak{T}u\right|_{L^{q_{0}}(0,T;H^{s}_{p_{0}})}\leq
C\left|u\right|_{L^{q_{1}^{\prime}}(0,T;H^{s}_{p_{1}^{\prime}})}.$

In order to treat the nonlinearity, let $F:\mathbb{C}\to\mathbb{C}$ be given
by $F(u)=|u|^{\alpha-1}u$ and let $\mathfrak{F}$ be the convolution operator
given by

(3.7)
$\displaystyle\mathfrak{F}u(t):=\int_{0}^{t}{{\mathcal{T}}}(t-r)F(u(r))\,dr.$

######  Remark 3.3.

We would like to mention that the Nemitskii operator associated to $F$,
defined by

${{\mathcal{F}}}(u)(x):=F(u(x)),\quad u:{\mathbb{R}}^{d}\to\mathbb{C},$

is a continuous operator from $L^{\alpha+1}({\mathbb{R}}^{d})$ to
$L^{\alpha+1\over\alpha}({\mathbb{R}}^{d})$ and hence from
$H^{1}({\mathbb{R}}^{d})$ to $L^{\alpha+1\over\alpha}({\mathbb{R}}^{d})$.

######  Proposition 3.4.

Assume $1<\alpha<1+4/d$. Let $p=\alpha+1$ and $q=4(\alpha+1)/d(\alpha-1)$.
Then we have for all $T>0$ and any admissible pairs $(m,l)$,

$\left|\mathfrak{F}u\right|_{L^{l}(0,T;L^{m})}\leq{C}(T)\,\left|u\right|^{\alpha}_{L^{q}(0,T;L^{p})},$

with $C(T)\to 0$ as $T\to 0$.

The proposition can be extended to $H^{1}({\mathbb{R}}^{d})$.

######  Proposition 3.5.

Assume

$\begin{cases}1<\alpha<{d+2\over d-2},&\;\mathrm{if}\;d>2;\\\
1<\alpha<\infty,&\;\mathrm{if}\;d=1,2.\end{cases}$

Let

$p^{\sharp}={d(\alpha+1)\over d+\alpha-1}\quad\mbox{and}\quad
q^{\sharp}={4(\alpha+1)\over(d-2)(\alpha-1)}.$

Then we have for all $T>0$,

$\left|\mathfrak{F}u\right|_{L^{q^{\sharp}}(0,T;H^{1}_{p^{\sharp}})}\leq
C\,\left|u\right|^{\alpha}_{L^{q^{\sharp}}(0,T;H^{1}_{p^{\sharp}})}.$

###### Proof.

The proof is similar to the proof of Proposition 3.4. One only has to take
into account the following estimate, which follows from Hölder’s inequality :

$||u|^{\alpha-1}\nabla u|_{L^{{p}^{\sharp^{\prime}}}}\leq
c\,||u|^{\alpha-1}|_{L^{l}}|\nabla u|_{L^{p^{\sharp}}}\leq
c\,|u|_{L^{({\alpha-1})l}}^{\alpha-1}|\nabla u|_{L^{p^{\sharp}}}\leq c|\nabla
u|_{L^{p^{\sharp}}}^{\alpha}.$

Here $\frac{1}{l}=1-\frac{2}{p^{\sharp}}$ and
$\frac{1}{(\alpha-1)l}=\frac{1}{p^{\sharp}}-\frac{1}{d}$. Therefore,
$(\alpha+1)/p^{\sharp}=(d+\alpha-1)/d$. ∎

##  4\. Existence and uniqueness results for finite Lévy measure

In this section we show the existence and uniqueness of a solution to (2.4)
for a finite Lévy measure. Here, the representation of the Lévy process as a
finite sum over its jumps is essential. Using this representation, the
existence and uniqueness of the solution in a pathwise sense can be shown.

######  Technical Lemma 4.1.

Let us assume that the Lévy measure $\nu$ is finite, in particular
$\nu(Z)=\rho$, and that the Hypothesis 1 is satisfied. Let $u_{0}\in
L^{2}(\Omega;H^{1}_{2}({\mathbb{R}}^{d}))$ be fixed and
${\mathcal{F}}_{0}$-measurable.

Then, if $\lambda>0$ and

$1\leq\alpha<\begin{cases}1+4/(d-2)&\mbox{ for }d>2,\\\ +\infty&\mbox{ for
}d=1\mbox{ or }2,\end{cases}$

Equation (2.4) has a unique mild $H^{1}_{2}({\mathbb{R}}^{d})$–valued solution
$u$; in particular, $u$ is ${\mathbb{P}}$-a.s. càdlàg in
$H^{1}_{2}({\mathbb{R}}^{d})$. In addition, for any $T>0$ there exists a
constant $C=C(T,C_{0}(\nu),C_{3}(\nu),C_{g},C_{h})$ such that

$\mathbb{E}\sup_{0\leq t\leq T}|u(t)|_{L^{2}}^{2}\leq
C\,\mathbb{E}|u_{0}|_{L^{2}}^{2}$

and there exists a constant $C=C(T,C_{0}(\nu),C_{1}(\nu),C_{g},C_{h})>0$ such
that

(4.1) $\mathbb{E}\sup_{0\leq t\leq T}{{\mathcal{H}}}(u(t))\leq
C\,(1+\mathbb{E}{{\mathcal{H}}}(u_{0})).$

  1. a.)

If Hypothesis 2 is satisfied, then for any $t>0$ we have ${\mathbb{P}}$–a.s.

$|u(t)|_{L^{2}}^{2}=|u_{0}|_{L^{2}}^{2}.$

  2. b.)

If Hypothesis 3 is satisfied, then for any $t>0$ we have

$\mathbb{E}|u(t)|_{L^{2}}^{2}=\mathbb{E}|u_{0}|_{L^{2}}^{2}.$

  3. c.)

If $\mathbb{E}\int_{{\mathbb{R}}^{d}}x^{2}|u_{0}(x)|^{2}\,dx<\infty$ and
Hypothesis 1 (ii)-(c) is satisfied, then there exists a constant
$C=C(T,C_{2}(\nu),C_{g},C_{h})>0$ such that

$\mathbb{E}\sup_{0\leq t\leq
T}\int_{{\mathbb{R}}^{d}}x^{2}|u(t,x)|^{2}\,dx\leq
C\left[1+\mathbb{E}\int_{{\mathbb{R}}^{d}}x^{2}|u_{0}(x)|^{2}\,dx+\mathbb{E}{{\mathcal{H}}}(u_{0})\right].$

###### Proof.

Let $\rho=\nu(Z)$, let $\\{\tau_{n}:n\in\mathbb{N}\\}$ be a family of
independent exponential distributed random variables with parameter $\rho$,
let

(4.2) $\displaystyle T_{n}=\sum_{j=1}^{n}\tau_{j},\quad n\in\mathbb{N},$

and let $\\{N(t):t\geq 0\\}$ be the counting process defined by

$N(t):=\sum_{j=1}^{\infty}1_{[T_{j},\infty)}(t),\quad t\geq 0.$

Observe, for any $t>0$, $N(t)$ is a Poisson distributed random variable with
parameter $\rho t$. Let $\\{Y_{n}:n\in\mathbb{N}\\}$ be a family of
independent, $\nu/\rho$ distributed random variables. Then the Lévy process
$L$ given by

$L(t)=\int_{0}^{t}\int_{Z}z\,\tilde{\eta}(dz,ds),\quad t\geq 0,$

can be represented as

$L(t)=\begin{cases}-z_{\nu}t&\mbox{ for }N(t)=0,\\\
\sum_{j=1}^{N(t)}Y_{j}-z_{\nu}t&\mbox{ for }N(t)>0,\end{cases}$

where $z_{\nu}=\int_{Z}G(z)\,\nu(dz)$ (see e.g. [11, Chapter 3]).

Now, by a modification of [10, Theorem 4.4.1] there exists a solution of the
deterministic equation

(4.6) $\displaystyle\hskip
85.35826pt\left\\{\begin{array}[]{rcl}\lx@intercol\@ADDCLASS{ltx_eqn_lefteqn}$\displaystyle
i\,\partial_{t}u(t)-\Delta u(t,x)+\lambda|u(t,x)|^{\alpha-1}u(t,x)\,\hskip
56.9055pt$\mbox{}\hfil\lx@intercol&&\\\
&=&u(t,x)\int_{Z}\,h(z(x))\mathcal{}\,\nu(dz)-u(t,x)\,z_{\nu},\\\
u(0)&=&u_{0},\end{array}\right.$

which is in $C({\mathbb{R}}_{+};H^{1}_{2}({\mathbb{R}}^{d}))$, with
$u(T_{1})\in H^{1}_{2}({\mathbb{R}}^{d})$. Indeed, setting
$\mathfrak{g}(u)=\int_{Z}uh(z)\,\nu(dz)-uz_{\nu}$, it is not difficult to
check that for any $u,v\in L^{\infty}(0,T;H^{1}_{2}({\mathbb{R}}^{d}))$ we
have

$|\mathfrak{g}(u)|_{L^{\infty}(0,T;H^{1}_{2})}\leq
C_{\nu}|u|_{L^{\infty}(0,T;H^{1}_{2})}$

and

$|\mathfrak{g}(u)-\mathfrak{g}(v)|_{L^{\infty}(0,T;L^{2})}\leq
C_{\nu}|u-v|_{L^{\infty}(0,T;L^{2})},$

where the Lipschitz constant is given by

${C_{\nu}=\int_{Z}|z|_{H^{1}_{2}}\,\nu(dz)+|z_{\nu}|_{H^{1}_{2}}.}$

Hence, setting $p=\alpha+1$ and $q=4(\alpha+1)/d(\alpha-1)$ so that $(p,q)$ is
an admissible pair, one may use as in [10, Theorem 4.4.1] a fixed point in

$\begin{array}[]{rl}E:=&\\{v\in
L^{\infty}(0,T;H^{1}_{2}({\mathbb{R}}^{d}))\cap
L^{q}(0,T;H^{1}_{p}({\mathbb{R}}^{d})),\\\
&\;|v|_{L^{\infty}(0,T;H^{1}_{2})}\leq M;\;|v|_{L^{q}(0,T;H^{1}_{p})}\leq
M\\}\end{array}$

equipped with the distance

$d(u,v)=|u-v|_{L^{\infty}(0,T;L^{2})}+|u-v|_{L^{q}(0,T;L^{p})},$

and a constant $M$ depending on the initial condition (see [10, p.
95,$\uparrow 2$]). For sufficiently small $T>0$, we obtain the existence of a
unique local solution. By uniform bounds, this local solution can be
globalized.

Let us denote the solution by $u_{1}$. Since at time $T_{1}$ a jump with size
$-iu(T_{1})Y_{1}$ happens, we put $u_{2}^{0}=u_{1}(T_{1})(1-iY_{1})$ and
consider a second process, starting at time $0$ in point $u_{2}^{0}$. By
Hypothesis 1-(i), we know, $u_{2}^{0}\in H^{1}_{2}({\mathbb{R}}^{d})$. Hence,
again by Theorem [10, Theorem 4.4.1] and the previous arguments, there exists
a unique global solution of the deterministic equation

(4.10) $\displaystyle\hskip
56.9055pt\left\\{\begin{array}[]{rcl}\lx@intercol\@ADDCLASS{ltx_eqn_lefteqn}$\displaystyle
i\,\partial_{t}u(t)-\Delta u(t,x)+\lambda|u(t,x)|^{\alpha-1}u(t,x)\,\hskip
56.9055pt$\mbox{}\hfil\lx@intercol&&\\\
&=&u(t,x)\int_{Z}\,h(z(x))\mathcal{}\,\nu(dz)-u(t,x)\,z_{\nu},\\\
u(0)&=&u_{2}^{0}=u_{1}(T_{1})(1-iY_{1}).\end{array}\right.$

Let us denote the solution on $[0,T_{2}-T_{1}]$ by $u_{2}$. Iterating this
step we get a sequence of solutions $\\{u_{n}:n\in\mathbb{N}\\}$. To be more
precise, let us assume that we are given a solution $u_{n-1}$ on the time
interval $[0,T_{n-1}-T_{n-2}]$, where the family of stopping times
$\\{T_{n}:n\in\mathbb{N}\\}$ is defined in (4.2). Let
$u_{n}^{0}=u_{n-1}(T_{n-1}-T_{n-2})(1-iY_{n-1})$. Then, we denote by $u_{n}$
the solution of the following (deterministic) problem

(4.14) $\displaystyle\hskip
56.9055pt\left\\{\begin{array}[]{rcl}\lx@intercol\@ADDCLASS{ltx_eqn_lefteqn}$\displaystyle
i\,\partial_{t}u(t)-\Delta u(t,x)+\lambda|u(t,x)|^{\alpha-1}u(t,x)\,\hskip
56.9055pt$\mbox{}\hfil\lx@intercol&&\\\
&=&u(t,x)\int_{Z}\,h(z(x))\mathcal{}\,\nu(dz)-u(t,x)\,z_{\nu},\\\
u(0)&=&u_{n}^{0},\end{array}\right.$

So, for each $n\in\mathbb{N}$ we can construct a solution $u_{n}$ on the time
interval $[0,T_{n}-T_{n-1}]$. In the next step we glue these solutions
together by putting for $t\in[T_{n-1},T_{n})$, $n\in\mathbb{N}$

$u(t):=u_{n}(t-T_{n-1}).$

Let us observe, that the jumps take place at the end points at each interval
and will be taken into account, by taking as initial starting point for the
next solution $u_{n}^{0}$, the solution $u_{n-1}$ at the end point $T_{n-1}$
plus the jump. In particular, we put
$u_{n}(0)=u_{n-1}(T_{n-1}-T_{n-2})-iu_{n-1}(T_{n-1}-T_{n-2})Y_{n-1}$. It is
straightforward to show, that $u$ solves (2.4). Since
${\mathbb{P}}\left(N(T)<\infty\right)=1$, the solution $u$ is a.s. defined on
$[0,T]$. The càdlàg property follows by the fact, that $\lim_{t\downarrow
T_{j}}u(t)=u_{j}^{0}$ and the limit $\lim_{t\uparrow T_{j}}u(t)$ exists in
$H^{1}_{2}({\mathbb{R}}^{d})$.

Summing up, we have shown the existence of a unique solution $u$ belonging
${\mathbb{P}}$–a.s. to $\mathbb{D}(0,T;H^{1}_{2}({\mathbb{R}}^{d}))\cap
L^{q}(0,T;H^{1}_{p}({\mathbb{R}}^{d}))$.

Next, we will show that under the hypothesis of the Lemma, the mass may be
estimated, i.e. for $0\leq t\leq T$,

$\mathbb{E}\,\sup_{0\leq t\leq T}{{\mathcal{E}}}(u(t))=\mathbb{E}\sup_{0\leq
t\leq T}|u(t)|_{L^{2}}^{2}\leq
C\mathbb{E}|u_{0}|_{L^{2}}^{2}=C\mathbb{E}\,{{\mathcal{E}}}(u_{0}),$

where $C=C(T,C_{0}(\nu),C_{3}(\nu),C_{g},C_{h})$. In a first step we are
aiming to prove

(4.15) $\displaystyle\mathbb{E}\sup_{0\leq s\leq
t}\left[|u(s)|_{L^{2}}^{2}-|u(0)|^{2}_{L^{2}}\right]$ $\displaystyle\leq$
$\displaystyle
C\mathbb{E}\left(\int_{0}^{t}|u(s)|^{4}_{L^{2}}\,ds\right)^{\frac{1}{2}},$

where $C=C(C_{0}(\nu),C_{3}(\nu),C_{g},C_{h})$. Assume for the time being that
(4.15) is true. Then, it follows by the Hölder inequality

$\displaystyle\mathbb{E}\sup_{0\leq s\leq
t}\left[|u(s)|_{L^{2}}^{2}-|u(0)|^{2}_{L^{2}}\right]$ $\displaystyle\leq$
$\displaystyle C\sqrt{t}\mathbb{E}\sup_{0\leq s\leq t}|u(s)|_{L^{2}}^{2}.$

If $t^{\ast}$ is small enough that $C\,\sqrt{t^{\ast}}\leq\frac{1}{2}$, then

$\mathbb{E}\sup_{0\leq s\leq t^{\ast}}\left[|u(s)|_{L^{2}}^{2}\right]\leq
2\mathbb{E}|u(0)|^{2}_{L^{2}}.$

Iterating this step we get

$\mathbb{E}\sup_{0\leq s\leq T}|u(s)|_{L^{2}}^{2}\leq
C\,\mathbb{E}|u(0)|^{2}_{L^{2}},$

where $C=C(T,C_{0}(\nu),C_{3}(\nu),C_{g},C_{h})$.

Let us show estimate (4.15). If we denote by $u^{c}$ the continuous part of
$u$ and put $f(z)=(-iG(z))$, we get by the Itô formula for a twice Frechet
differentiable function $\Phi:L^{2}({\mathbb{R}}^{d})\to{\mathbb{R}}$

$\displaystyle d\Phi(u(t))=\Phi^{\prime}(u(t))du^{c}(t)$
$\displaystyle+\int_{Z}\left[\Phi(u(t^{-})(1+f(z)))-\Phi(u(t^{-})\right]\tilde{\eta}(dz,dt)$
$\displaystyle{}+\int_{Z}\left[\Phi(u(t^{-})(1+f(z)))-\Phi(u(t^{-})-\Phi^{\prime}(u(t))\,[u(t^{-})\,f(z)]\right]\gamma(dz,dt).$

First, note, since on each interval the solution belongs to
$H^{1}_{2}({\mathbb{R}}^{d})$, all terms in the above Itô formula are well
defined. Additionally, with

$\displaystyle{du(t^{-},x)}=\left(-i\Delta
u(t^{-},x)+i\lambda|u(t^{-},x)|^{\alpha-1}u(t^{-},x)\right)\,dt$
$\displaystyle{}-i\int_{Z}u(t^{-},x)g(z(x))\,\tilde{\eta}(dz,dt)-i\int_{Z}u(t^{-},x)\,h(z(x))\,\gamma(dz,dt),\quad
x\in{\mathbb{R}}^{d},\,t\geq 0,$

one obtains

$\displaystyle
d|u(t)|_{L^{2}}^{2}=-\int_{Z}\int_{{\mathbb{R}}^{d}}2u(t,x)\overline{u(t,x)}\Im(\overline{h(z(x)})\,dx\,\nu(dz)\,dt$
$\displaystyle+\int_{Z}\int_{{\mathbb{R}}^{d}}u(t^{-},x)\overline{u(t^{-},x)}\left[\left|1-ig(z(x))\right|^{2}-1\right]\,\,dx\,\tilde{\eta}(dz,dt)$
$\displaystyle+\int_{Z}\int_{{\mathbb{R}}^{d}}u(t,x)\overline{u(t,x)}\left[\left|1-ig(z(x))\right|^{2}-1+2\Im\left(\overline{g(z(x))}\right)\right]\,\,dx\,\gamma(dz,dt)$
$\displaystyle=$ $\displaystyle
2\int_{Z}\int_{{\mathbb{R}}^{d}}u(t,x)\overline{u(t,x)}\Im({h(z(x)})\,dx\,\nu(dz)\,dt$
$\displaystyle+\int_{Z}\int_{{\mathbb{R}}^{d}}u(t^{-},x)\overline{u(t^{-},x)}\left[\left|1-ig(z(x))\right|^{2}-1\right]\,\,dx\,\tilde{\eta}(dz,dt)$
$\displaystyle+\int_{Z}\int_{{\mathbb{R}}^{d}}u(t,x)\overline{u(t,x)}\left[\left|1-ig(z(x))\right|^{2}-1-2\Im\left(g(z(x))\right)\right]\,\,dx\,\gamma(dz,dt).$

To be more precise, one has by direct calculations

$\displaystyle\hskip 56.9055ptd|u(t)|_{L^{2}}^{2}$ $\displaystyle=$
$\displaystyle
2\int_{Z}\int_{{\mathbb{R}}^{d}}u(t,x)\overline{u(t,x)}\Im({h(z(x)})\,dx\,\nu(dz)\,dt$
$\displaystyle{}+\int_{Z}\int_{{\mathbb{R}}^{d}}u(t^{-},x)\overline{u(t^{-},x)}\left[\left|g(z(x))\right|^{2}+2\Im(g(z(x)))\right]\,\,dx\,\tilde{\eta}(dz,dt)$
$\displaystyle{}+\int_{Z}\int_{{\mathbb{R}}^{d}}u(t,x)\overline{u(t,x)}\left|g(z(x))\right|^{2}\,\,dx\,\gamma(dz,dt).$

An application of the Burkholder inequality and Minkowski inequality yields

$\displaystyle\mathbb{E}\sup_{0\leq s\leq
t}\left[|u(s)|_{L^{2}}^{2}-|u(0)|^{2}_{L^{2}}\right]$ $\displaystyle\leq$
$\displaystyle\mathbb{E}\left(\int_{0}^{t}\int_{Z}\left(\int_{{\mathbb{R}}^{d}}u(t,x)\overline{u(t,x)}\left[\left|g(z(x))\right|^{2}+2\Im(g(z(x)))\right]\,\,dx\right)^{2}\nu(dz)\,ds\right)^{\frac{1}{2}}$
$\displaystyle{}+\mathbb{E}\int_{0}^{t}\left|\int_{Z}2\int_{{\mathbb{R}}^{d}}u(t,x)\overline{u(t,x)}\Im({h(z(x)})\,dx\nu(dz)\right|\,ds$
$\displaystyle{}+\mathbb{E}\int_{0}^{t}\int_{Z}\int_{{\mathbb{R}}^{d}}u(s,x)\overline{u(s,x)}\left|g(z(x))\right|^{2}\,\,dx\,\gamma(dz,dt).$

Taking into account Hypothesis 1, we know that there exists a constant
$C=C(T,C_{0},C_{3},C_{g},C_{h})>0$ such that

$\displaystyle\mathbb{E}\sup_{0\leq s\leq
t}\left[|u(s)|_{L^{2}}^{2}-|u(0)|^{2}_{L^{2}}\right]$ $\displaystyle\leq$
$\displaystyle
C\mathbb{E}\left(\int_{0}^{t}|u(s)|^{4}_{L^{2}}\,ds\right)^{\frac{1}{2}},$

that is (4.15) holds and the estimate on the mass follows as explained above.

If Hypothesis 2 is satisfied, one easily deduces from (4) that
${{\mathcal{E}}}(u(t))={{\mathcal{E}}}(u(0))$ for all $t\in[0,T]$. If only
Hypothesis 3 is satisfied, then one easily deduces from (4) that
$\mathbb{E}|u(t)|_{L^{2}}^{2}=\mathbb{E}|u(0)|_{L^{2}}^{2}$ for all
$t\in[0,T]$.

In a second step we will prove that there exists a constant $C>0$ such that

(4.19) $\displaystyle\mathbb{E}\sup_{t\in[0,T]}{{\mathcal{H}}}(u(t))$
$\displaystyle\leq$ $\displaystyle
C\,\left(1+\mathbb{E}{{\mathcal{H}}}(u(0))\right).$

In order to justify the computation of the Itô formula for the Hamiltonian
${{\mathcal{H}}}$, one needs also to regularize the Hamiltonian. In
particular, one needs to regularize both terms in the Hamiltonian. One
possibility is to define $J^{\varepsilon}:=(I-\varepsilon\Delta)^{-1}$ and to
consider

${{\mathcal{H}}}_{\varepsilon}(u)={{\mathcal{H}}}(J_{\varepsilon}^{\frac{1}{2}}u)=\frac{1}{2}\int_{{\mathbb{R}}^{d}}|J_{\varepsilon}^{\frac{1}{2}}\nabla
u|^{2}+\frac{\lambda}{\alpha+1}\int_{{\mathbb{R}}^{d}}|J^{\frac{1}{2}}_{\varepsilon}u|^{\alpha+1}\,dx.$

Note, since $u$ belongs ${\mathbb{P}}$-a.s. to
$H_{2}^{1}({\mathbb{R}}^{d})\subset L^{\alpha+1}({\mathbb{R}}^{d})$ for any
$t\in[0,T]$, we know ${\mathbb{P}}$–a.s.
${{\mathcal{H}}}_{\varepsilon}(u(t))\to{{\mathcal{H}}}(u(t))$ for
$\varepsilon\to 0$. Taking into account that ${\mathbb{P}}$–a.s. $u$ belongs
to $\mathbb{D}([0,T];H^{1}_{2}({\mathbb{R}}^{d}))$, in addition, by Theorem
7.8-(b) [17], it follows that the process $[0,T]\ni
t\mapsto{{\mathcal{H}}}_{\varepsilon}(u(t))$ converges to the process
$[0,T]\ni t\mapsto{{\mathcal{H}}}(u(t))$ in $\mathbb{D}(0,T;{\mathbb{R}})$.

Let us apply the Itô formula to ${{\mathcal{H}}}_{\varepsilon}(u(t))$. First,
note that

${{\mathcal{H}}}_{\varepsilon}^{\prime}(u)\cdot
v={{\mathcal{H}}}^{\prime}(J_{\varepsilon}^{\frac{1}{2}}u)\cdot
J_{\varepsilon}^{\frac{1}{2}}v={\langle}-J_{\varepsilon}^{\frac{1}{2}}\Delta
u,J_{\varepsilon}^{\frac{1}{2}}v{\rangle}+\lambda{\langle}|J_{\varepsilon}^{\frac{1}{2}}u|^{\alpha-1}J_{\varepsilon}^{\frac{1}{2}}u,J_{\varepsilon}^{\frac{1}{2}}v{\rangle}.$

Using

$\displaystyle{du(t,x)}=\left(-i\Delta
u(t,x)+i\lambda|u(t,x)|^{\alpha-1}u(t,x)\right)\,dt$
$\displaystyle{}-i\int_{Z}u(t^{-},x)g(z(x))\,\tilde{\eta}(dz,dt)-i\int_{Z}u(t,x)\,h(z(x))\,\gamma(dz,dt),\quad
x\in{\mathbb{R}}^{d},\,t\geq 0,$ $\displaystyle{du^{c}(t,x)}=\left(-i\Delta
u(t,x)+i\lambda|u(t,x)|^{\alpha-1}u(t,x)\right)\,dt$
$\displaystyle{}-i\int_{Z}u(t,x)\,h(z(x))\,\gamma(dz,dt),\quad
x\in{\mathbb{R}}^{d},\,t\geq 0,$

and the Itô formula

$\displaystyle
d{{\mathcal{H}}}_{\varepsilon}(u)={{\mathcal{H}}}_{\varepsilon}^{\prime}(u(t))du^{c}(t)$
$\displaystyle+\int_{Z}{{\mathcal{H}}}_{\varepsilon}^{\prime}(u(t^{-}))\left[u(t^{-})(-iG(z))\right]\tilde{\eta}(dz,dt)$
$\displaystyle{}+\int_{Z}\left[{{\mathcal{H}}}_{\varepsilon}(u(t^{-})(1-iG(z)))-{{\mathcal{H}}}_{\varepsilon}(u(t^{-}))-{{\mathcal{H}}}_{\varepsilon}^{\prime}(u(t))\,[(-i)u(t^{-})\,G(z)]\right]\eta(dz,dt),$

we get

$\displaystyle{{\mathcal{H}}}_{\varepsilon}(u^{\varepsilon}(t))={{\mathcal{H}}}_{\varepsilon}(u^{\varepsilon}(0))-\int_{0}^{t}{\langle}{{\mathcal{H}}}_{\varepsilon}^{\prime}(u(s)),i\left(\Delta
u(s)-\lambda|u(s)|^{\alpha-1}u(s)\right){\rangle}ds$
$\displaystyle{}-\int_{0}^{t}\int_{Z}{\langle}{{\mathcal{H}}}_{\varepsilon}^{\prime}(u(s)),iu(s)H(z){\rangle}\nu(dz)\,ds$
$\displaystyle{}+\int_{0}^{t}\int_{Z}\left[{{\mathcal{H}}}_{\varepsilon}(u(s-)(1-iG(z)))-{{\mathcal{H}}}_{\varepsilon}(u(s-))\right]\tilde{\eta}(dz,ds)$
$\displaystyle{}+\int_{0}^{t}\int_{Z}\left[{{\mathcal{H}}}_{\varepsilon}(u(s-)(1-iG(z)))-{{\mathcal{H}}}_{\varepsilon}(u(s-))\right.$
$\displaystyle{}\phantom{\Bigg{|}}\quad\left.-{{\mathcal{H}}}_{\varepsilon}^{\prime}(u(s-))(u(s-)(-iG(z)))\right]\gamma(dz,ds).$

In order to analyse the first term, we first use the fact that
${\langle}J_{\varepsilon}^{\frac{1}{2}}{{\mathcal{H}}}^{\prime}(u),iJ_{\varepsilon}^{\frac{1}{2}}{{\mathcal{H}}}^{\prime}(u){\rangle}=0$
to write

$\displaystyle{{\mathcal{H}}}^{\prime}_{\varepsilon}(u)\cdot[i(\Delta
u-\lambda|u|^{\alpha-1}u)]$ $\displaystyle=$
$\displaystyle-{\langle}{{\mathcal{H}}}^{\prime}(J_{\varepsilon}^{\frac{1}{2}}u),iJ_{\varepsilon}^{\frac{1}{2}}{{\mathcal{H}}}^{\prime}(u){\rangle}$
$\displaystyle=$
$\displaystyle-{\langle}{{\mathcal{H}}}^{\prime}(J_{\varepsilon}^{\frac{1}{2}}u)-J_{\varepsilon}^{\frac{1}{2}}{{\mathcal{H}}}^{\prime}(u),iJ_{\varepsilon}^{\frac{1}{2}}{{\mathcal{H}}}^{\prime}(u){\rangle}.$

Note that all the terms are well defined, since
$u\in\mathbb{D}([0,T];H^{1}_{2}({\mathbb{R}}^{d}))$ implies
${{\mathcal{H}}}^{\prime}(u)\in\mathbb{D}([0,T];H^{-1}_{2}({\mathbb{R}}^{d}))$
and $J_{\varepsilon}^{\frac{1}{2}}{{\mathcal{H}}}^{\prime}(u)$ is a.s. in
$L^{\infty}(0,T;L^{2}({\mathbb{R}}^{d}))$, while
$J_{\varepsilon}^{\frac{1}{2}}u\in L^{\infty}(0,T;H^{2}_{2}({\mathbb{R}}^{d})$
so that ${{\mathcal{H}}}^{\prime}(J_{\varepsilon}^{\frac{1}{2}}u)\in
L^{\infty}(0,T;L^{2}({\mathbb{R}}^{d}))$.

Next, we prove that
$\int_{0}^{t}{\langle}{{\mathcal{H}}}^{\prime}(J_{\varepsilon}^{\frac{1}{2}}u)-J_{\varepsilon}^{\frac{1}{2}}{{\mathcal{H}}}^{\prime}(u),iJ_{\varepsilon}^{\frac{1}{2}}{{\mathcal{H}}}^{\prime}(u){\rangle}\,ds$
tends to zero as $\varepsilon$ tends to zero, for any $t$, a.s. First, note
that

(4.23)
${{\mathcal{H}}}^{\prime}(J_{\varepsilon}^{\frac{1}{2}}u)-J_{\varepsilon}^{\frac{1}{2}}{{\mathcal{H}}}^{\prime}(u)=\lambda|J_{\varepsilon}^{\frac{1}{2}}u|^{\alpha-1}J_{\varepsilon}^{\frac{1}{2}}u-\lambda
J_{\varepsilon}^{\frac{1}{2}}(|u|^{\alpha-1}u).$

Let us remind that the solution belongs ${\mathbb{P}}$-a.s. to
$\mathbb{D}(0,T;H^{1}_{2})\cap L^{q}(0,T;H_{p}^{1}({\mathbb{R}}^{d}))$. Then,
using Hölder inequalities and Sobolev embeddings, it is easily seen that the
above term is bounded independently of $\varepsilon$ in
$H_{\alpha+1/\alpha}^{1}({\mathbb{R}}^{d})\subset L^{2}({\mathbb{R}}^{d})$;
indeed, one may e.g. bound

$\left||J_{\varepsilon}^{\frac{1}{2}}u|^{\alpha-1}\nabla
J_{\varepsilon}^{\frac{1}{2}}u\right|_{L^{\frac{\alpha+1}{\alpha}}}\leq
C|J_{\varepsilon}^{\frac{1}{2}}u|_{L^{\alpha+1}}^{\alpha-1}|\nabla
J_{\varepsilon}^{\frac{1}{2}}u|_{L^{\alpha+1}}\leq
C|u|_{H^{1}_{2}}^{\alpha-1}|J_{\varepsilon}^{\frac{1}{2}}\nabla
u|_{L^{\alpha+1}}\leq C|u|_{H^{1}_{2}}^{\alpha-1}|u|_{H^{1}_{\alpha+1}},$

since $J_{\varepsilon}^{\frac{1}{2}}$ is a bounded operator – with a bound
independent of $\varepsilon$ – in $L^{\alpha+1}({\mathbb{R}}^{d})$. All the
other terms are estimated in the same way. Hence, we know by (4.23) that

(4.24)
$|{{\mathcal{H}}}^{\prime}(J_{\varepsilon}^{\frac{1}{2}}u)-J_{\varepsilon}^{\frac{1}{2}}{{\mathcal{H}}}^{\prime}(u)|_{H^{1}_{\frac{\alpha+1}{\alpha}}}\leq
C|u|_{H^{1}_{2}}^{\alpha-1}|u|_{H^{1}_{\alpha+1}}.$

With the same arguments,

(4.25) $\left||u|^{\alpha-1}u\right|_{H^{1}_{\frac{\alpha+1}{\alpha}}}\leq
C|u|_{H^{1}_{2}}^{\alpha-1}|u|_{H^{1}_{\alpha+1}}.$

Let us now decompose

$\displaystyle{\langle}{{\mathcal{H}}}^{\prime}(J_{\varepsilon}^{\frac{1}{2}}u)-J_{\varepsilon}^{\frac{1}{2}}{{\mathcal{H}}}^{\prime}(u),iJ_{\varepsilon}^{\frac{1}{2}}{{\mathcal{H}}}^{\prime}(u){\rangle}$
$\displaystyle=$
$\displaystyle-{\langle}{{\mathcal{H}}}^{\prime}(J_{\varepsilon}^{\frac{1}{2}}u)-J_{\varepsilon}^{\frac{1}{2}}{{\mathcal{H}}}^{\prime}(u),iJ_{\varepsilon}^{\frac{1}{2}}\Delta
u{\rangle}+\lambda{\langle}{{\mathcal{H}}}^{\prime}(J_{\varepsilon}^{\frac{1}{2}}u)-J_{\varepsilon}^{\frac{1}{2}}{{\mathcal{H}}}^{\prime}(u),iJ_{\varepsilon}^{\frac{1}{2}}(|u|^{\alpha-1}u){\rangle}$

and integrate the first term by parts. We then have to consider

$\displaystyle I_{\varepsilon}$ $\displaystyle=$
$\displaystyle\int_{0}^{t}{\langle}\nabla({{\mathcal{H}}}^{\prime}(J_{\varepsilon}^{\frac{1}{2}}u)-J_{\varepsilon}^{\frac{1}{2}}{{\mathcal{H}}}^{\prime}(u)),iJ_{\varepsilon}^{\frac{1}{2}}\nabla
u{\rangle}\,ds$ $\displaystyle=$
$\displaystyle\int_{0}^{t}{\langle}J_{\varepsilon}^{\frac{1}{2}}\nabla({{\mathcal{H}}}^{\prime}(J_{\varepsilon}^{\frac{1}{2}}u)-J_{\varepsilon}^{\frac{1}{2}}{{\mathcal{H}}}^{\prime}(u)),i\nabla
u{\rangle}\,ds.$

Using (4.24),
$J_{\varepsilon}^{\frac{1}{2}}\nabla({{\mathcal{H}}}^{\prime}(J_{\varepsilon}^{\frac{1}{2}}u)-J_{\varepsilon}^{\frac{1}{2}}{{\mathcal{H}}}^{\prime}(u))$
is bounded in $L^{2}(0,T;L^{\frac{\alpha+1}{\alpha}}({\mathbb{R}}^{d}))$,
independently of $\varepsilon$ by

$C|u|_{L^{\infty}(0,T;H^{1}_{2})}^{\alpha-1}|u|_{L^{2}(0,T;H^{1}_{\alpha+1})}\leq
C|u|_{L^{\infty}(0,T;H^{1}_{2})}^{\alpha-1}|u|_{L^{q}(0,T;H^{1}_{\alpha+1})}.$

Hence,
$J_{\varepsilon}^{\frac{1}{2}}\nabla({{\mathcal{H}}}^{\prime}(J_{\varepsilon}^{\frac{1}{2}}u)-J_{\varepsilon}^{\frac{1}{2}}{{\mathcal{H}}}^{\prime}(u))$
converges weakly to $0$ in $L^{2}(0,T;L^{\alpha+1/\alpha}({\mathbb{R}}^{d}))$.
Since $\nabla u\in L^{2}(0,T;L^{\alpha+1}({\mathbb{R}}^{d}))$, it follows that
$I_{\varepsilon}$ converges to $0$ as $\varepsilon\to 0$. For the second term,
we use the same argument and (4.25) : by (4.24) and the embedding
$H^{1}_{\alpha+1/\alpha}({\mathbb{R}}^{d})\subset L^{2}({\mathbb{R}}^{d})$,
the term
$J_{\varepsilon}^{\frac{1}{2}}({{\mathcal{H}}}^{\prime}(J_{\varepsilon}^{\frac{1}{2}}u)-J_{\varepsilon}^{\frac{1}{2}}{{\mathcal{H}}}^{\prime}(u))$
is bounded uniformly in $\varepsilon$, in
$L^{2}(0,T;L^{2}({\mathbb{R}}^{d}))$, by
$C|u|_{L^{\infty}(0,T;H^{1}_{2})}^{\alpha-1}|u|_{L^{2}(0,T;H^{1}_{\alpha+1})}$,
and $|u|^{\alpha-1}u\in
L^{2}(0,T;H^{1}_{\alpha+1/\alpha}({\mathbb{R}}^{d}))\subset
L^{2}(0,T;L^{2}({\mathbb{R}}^{d}))$ by (4.25), so that again,

$\int_{0}^{t}{\langle}{{\mathcal{H}}}^{\prime}(J_{\varepsilon}^{\frac{1}{2}}u)-J_{\varepsilon}^{\frac{1}{2}}{{\mathcal{H}}}^{\prime}(u),iJ_{\varepsilon}^{\frac{1}{2}}(|u|^{\alpha-1}u){\rangle}\,ds=\int_{0}^{t}{\langle}J_{\varepsilon}^{\frac{1}{2}}({{\mathcal{H}}}^{\prime}(J_{\varepsilon}^{\frac{1}{2}}u)-J_{\varepsilon}^{\frac{1}{2}}{{\mathcal{H}}}^{\prime}(u)),i|u|^{\alpha-1}u{\rangle}\,ds$

converges to $0$ as $\varepsilon\to 0$.

Going back to (4), the first term which does not vanish for $\varepsilon\to 0$
is

$\displaystyle\int_{0}^{t}\int_{Z}{\langle}{{\mathcal{H}}}_{\varepsilon}^{\prime}(u(s)),iu(s)H(z){\rangle}\nu(dz)\,ds.$

However, taking $\varepsilon\to 0$ we see by similar arguments as before that

$\displaystyle\int_{0}^{t}\int_{Z}{\langle}{{\mathcal{H}}}_{\varepsilon}^{\prime}(u(s)),iu(s)H(z){\rangle}\nu(dz)\,ds\to\int_{0}^{t}\int_{Z}{\langle}{{\mathcal{H}}}^{\prime}(u(s)),iu(s)H(z){\rangle}\nu(dz)\,ds.$

Straightforward calculations give

$\displaystyle\int_{0}^{t}\,{{\mathcal{H}}}^{\prime}(u(t)).(-i\int_{Z}u(t,x)\,H(z)\,\nu(dz))\,dt$
$\displaystyle=$ $\displaystyle-\int_{0}^{t}\langle\Delta
u(t),-i\int_{Z}u(t)\,H(z)\,\nu(dz)\rangle\,dt$
$\displaystyle{}+\lambda\int_{0}^{t}\langle|u(t)|^{\alpha-1}u(t)-i\int_{Z}u(t,x)\,h(z(x))\,\nu(dz)\rangle\,dt.$

Applying integration by parts we get for the first summand

$\displaystyle-\int_{Z}{\langle}\Delta u(t),-iu(t)H(z){\rangle}\,\nu(dz)$
$\displaystyle=$ $\displaystyle\int_{Z}\int_{{\mathbb{R}}^{d}}\Re(\nabla
u(t,x)\nabla(\overline{-iu(t,x)h(z(x))})\,dx\,\nu(dz)$ $\displaystyle=$
$\displaystyle-\int_{Z}\int_{{\mathbb{R}}^{d}}\Im(\nabla
u(t,x)\nabla(\overline{u(t,x)h(z(x))})\,dx\,\nu(dz)$ $\displaystyle=$
$\displaystyle\int_{Z}\int_{{\mathbb{R}}^{d}}|\nabla
u(t,x)|^{2}\Im({h(z(x))})\,dx\,\nu(dz)$ $\displaystyle\hskip
56.9055pt-\int_{Z}\int_{{\mathbb{R}}^{d}}\Im(\nabla
u(t,x)\overline{u(t,x)}\nabla\overline{h(z(x))})\,dx\,\nu(dz)$

Next,

$\displaystyle\lambda\int_{Z}{\langle}|u|^{\alpha-1}u,-iuH(z){\rangle}\,\nu(dz)$
$\displaystyle=$
$\displaystyle\lambda\int_{Z}\int_{{\mathbb{R}}^{d}}\Re\left(|u(t,x)|^{\alpha-1}u(t,x)(\overline{-iu(t,x)h(z(x)))}\right)\,dx\,\nu(dz)$
$\displaystyle=$
$\displaystyle-\lambda\int_{Z}\int_{{\mathbb{R}}^{d}}\Im\left(|u(t,x)|^{\alpha-1}u(t,x)\overline{u(t,x)}\overline{h(z(x))}\right)\,dx\,\nu(dz)$
$\displaystyle=$
$\displaystyle\lambda\int_{Z}\int_{{\mathbb{R}}^{d}}|u(t,x)|^{\alpha+1}\Im\left(h(z(x))\right)\,dx\,\nu(dz).$

In the next lines we calculate the terms arising due to the jumps, that is the
terms

(4.28)
$\displaystyle\int_{Z}{{\mathcal{H}}}_{\varepsilon}^{\prime}(u(t^{-}))\left[u(t^{-})(-iG(z))\right]\tilde{\eta}(dz,dt)$

and

(4.29)
$\displaystyle\int_{Z}\left[{{\mathcal{H}}}_{\varepsilon}(u(t^{-})(1-iG(z)))-{{\mathcal{H}}}_{\varepsilon}(u(t^{-}))-{{\mathcal{H}}}_{\varepsilon}^{\prime}(u(t))\,[-iu(t^{-})\,G(z)]\right]\eta(dz,dt).$

Here, again one has to take the limit. Since, the arguments are similar as
before we omit them. Applying the Burkholder inequality we get

$\displaystyle\mathbb{E}\sup_{0\leq s\leq
t}\left|\int_{Z}{{\mathcal{H}}}^{\prime}(u(s^{-}))\left[u(s^{-})(-iG(z))\right]\tilde{\eta}(dz,ds)\right|$
$\displaystyle\leq$
$\displaystyle\mathbb{E}\left(\int_{0}^{t}\int_{Z}\left|{{\mathcal{H}}}^{\prime}(u(s^{-}))\left[u(s^{-})(-iG(z))\right]\right|^{2}\nu(dz)\,ds\right)^{\frac{1}{2}}$
$\displaystyle\leq$ $\displaystyle
2\mathbb{E}\left(\int_{0}^{t}\int_{Z}\left|{\langle}\Delta
u(s),(-iu(s)G(z)){\rangle}\right|^{2}\nu(dz)\,ds\right.$
$\displaystyle\left.{}+\int_{0}^{t}\int_{Z}\left|{\langle}|u(s^{-})|^{\alpha-1}u(s^{-}),u(s^{-})(-iG(z)){\rangle}\right|^{2}\nu(dz)\,ds\right)^{\frac{1}{2}}.$

In order to calculate the inner part of the first term we compare it to (4)
and get

$\displaystyle\mathbb{E}\left(\int_{0}^{t}\int_{Z}\left|{\langle}\Delta
u(s),-iu(s)G(z){\rangle}\right|^{2}\nu(dz)\,ds\right)^{\frac{1}{2}}$
$\displaystyle\leq$ $\displaystyle
C\mathbb{E}\left(\int_{0}^{t}\int_{Z}\left(\int_{{\mathbb{R}}^{d}}|\nabla
u(s,x)|^{2}|\Im({g(z(x))})|\,dx\right)^{2}\,\nu(dz)\,ds\right.$
$\displaystyle\hskip
56.9055pt+\left.\int_{0}^{t}\int_{Z}\left(\int_{{\mathbb{R}}^{d}}|\Im(\nabla
u(s,x)\overline{u(s,x)}\nabla\overline{g(z(x))})|\,dx\right)^{2}\,\nu(dz)\right)^{\frac{1}{2}}.$

In order to calculate the inner part of the second term we compare it to (4)
and get

$\displaystyle\mathbb{E}\left(\int_{0}^{t}\int_{Z}\left|{\langle}|u(s^{-})|^{\alpha-1}u(s^{-}),u(s^{-})(-iG(z)){\rangle}\right|^{2}\nu(dz)\,ds\right)^{\frac{1}{2}}$
$\displaystyle\leq$
$\displaystyle\mathbb{E}\left(\int_{0}^{t}\int_{Z}\left(\int_{{\mathbb{R}}^{d}}|u(s,x)|^{\alpha+1}|\Im\left({g(z(x))}\right)|\,dx\right)^{2}\,\nu(dz)\,ds\right)^{\frac{1}{2}}.$

Now we are going to calculate the term (4.29). Here, taking into account that

$\int_{Z}\left|{{\mathcal{H}}}(v(1-iG(z)))-{{\mathcal{H}}}(v)-{{\mathcal{H}}}^{\prime}(v)\,[-iv\,G(z)]\right|\,\nu(dz)<\infty$

for all $v\in H^{1}_{2}({\mathbb{R}}^{d})$, taking the expectation leads on
both sides, to

$\displaystyle\mathbb{E}\int_{0}^{t}\int_{Z}\left[{{\mathcal{H}}}(u(s^{-})(1-iG(z)))-{{\mathcal{H}}}(u(s^{-}))-{{\mathcal{H}}}^{\prime}(u(s^{-}))\,[-iu(s^{-})\,G(z)]\right]\eta(dz,ds)$
$\displaystyle\leq$
$\displaystyle\mathbb{E}\int_{0}^{t}\int_{Z}\left|{{\mathcal{H}}}(u(s^{-})(1-iG(z)))-{{\mathcal{H}}}(u(s^{-}))-{{\mathcal{H}}}^{\prime}(u(s^{-}))\,[-iu(s^{-})\,G(z)]\right|\,\nu(dz)\,ds.$

First we will calculate the terms of ${{\mathcal{H}}}$ involving
$\int_{{\mathbb{R}}^{d}}|\nabla u(x)|^{2}\,dx$. Here, we will use the identity
$|a|^{2}-|b|^{2}=\Re((a-b)\overline{(a+b)})$, and taking expectation gives

$\displaystyle\frac{1}{2}\mathbb{E}\int_{0}^{t}\int_{Z}\left[\left|\nabla\left(u(s^{-})(1-iG(z))\right)\right|^{2}\right.$
$\displaystyle{}\left.-\left|\nabla\left(u(s^{-})\right)\right|^{2}-2{\langle}\nabla
u(s^{-}),\nabla(u(s^{-})(-iG(z))){\rangle}\right]\,\nu(dz)\,ds$
$\displaystyle=$
$\displaystyle\frac{1}{2}\mathbb{E}\int_{0}^{t}\int_{Z}\int_{{\mathbb{R}}^{d}}\left[\Re\left\\{\left(\nabla(u(s^{-},x)(-ig(z(x))))\right)\overline{\left(\nabla(u(s^{-},x)(2-ig(z(x)))\right)}\right\\}\right.$
$\displaystyle{}-\left.2\Re\left\\{\nabla
u(s^{-},x)\cdot\overline{\nabla(u(s^{-},x)(-ig(z(x)))}\right\\}\,\right]\,dx\,\nu(dz)\,ds$
$\displaystyle=$
$\displaystyle\frac{1}{2}\mathbb{E}\int_{0}^{t}\int_{Z}\int_{{\mathbb{R}}^{d}}\left[\left|\nabla\left(-iu(s^{-},x)g(z(x)))\right)\right|^{2}+2\Re\left(\overline{\nabla
u(s^{-},x)}\nabla(-iu(s^{-},x)g(z(x)))\right)\right\\}$
$\displaystyle{}-\left.2\Re\left\\{\nabla
u(s^{-},x)\cdot\overline{\nabla(u(s^{-},x)(-ig(z(x))))}\right\\}\,\right]\,dx\,\nu(dz)\,ds$
$\displaystyle=$
$\displaystyle\frac{1}{2}\mathbb{E}\int_{0}^{t}\int_{Z}\int_{{\mathbb{R}}^{d}}\left|\nabla\left(-iu(s^{-},x)g(z(x))\right)\right|^{2}\,dx\,\nu(dz)\,ds.$

It remains to calculate the second part of (4.29), i.e.

$\displaystyle\frac{\lambda}{\alpha+1}\mathbb{E}\int_{0}^{t}\int_{Z}\int_{{\mathbb{R}}^{d}}\left[|u(s^{-},x)(1-ig(x))|^{\alpha+1}\phantom{\Big{|}}\right.$
$\displaystyle{}\left.-|u(s^{-},x)|^{\alpha+1}-(\alpha+1)|u(s^{-},x)|^{\alpha-1}\Re\left(u(s^{-},x)\overline{u(s^{-},x)(-ig(z(x))}\right)\right]dx\,\nu(dz)\,ds.$

The Taylor formula yields

$\displaystyle\ldots$ $\displaystyle\leq$ $\displaystyle
C\int_{0}^{t}\mathbb{E}\int_{Z}\int_{{\mathbb{R}}^{d}}|u(s^{-},x)|^{\alpha+1}|g(z(x))|^{2}dx\,\nu(dz)\,ds$

Collecting altogether, taking into account the Hypothesis 1, and rearranging
the terms we see that the bound (4.30) below is satisfied ; indeed, first,
observe, since no stochastic integral is involved in the bounds, we can change
from $s^{-}$ to $s$, and we have

$\displaystyle\mathbb{E}\sup_{0\leq s\leq
t}\left[{{\mathcal{H}}}(u(s))-{{\mathcal{H}}}(u(0))\right]$
$\displaystyle\leq$
$\displaystyle\phantom{\Bigg{|}}C\,\mathbb{E}\int_{0}^{t}\int_{Z}\int_{{\mathbb{R}}^{d}}\left|\nabla
u(s,x)\right|^{2}|\Im({h(z(x))})|\,dx\,\nu(dz)\,ds\phantom{\Bigg{|}}+C\,\mathbb{E}\int_{0}^{t}\int_{Z}\int_{{\mathbb{R}}^{d}}\left\\{{\left|\Im(\nabla
u(s,x)\overline{u(s,x)}\nabla\overline{h(z(x))})\right|}\right.\phantom{\Bigg{|}}$
$\displaystyle{}\left.+|u(s,x)|^{\alpha+1}|\Im(h(z(x)))|+\left|\nabla\left(-iu(s,x)g(z(x))\right)\right|^{2}+|u(s,x)|^{\alpha+1}|g(z(x))|^{2}\right\\}\,dx\,\nu(dz)\,ds\phantom{\Bigg{|}}$
$\displaystyle{}+\mathbb{E}\left(\int_{0}^{t}\int_{Z}\left(\int_{{\mathbb{R}}^{d}}|\nabla
u(s,x)|^{2}|\Im({g(z(x))})|\,dx\right)^{2}\,\nu(dz)\,ds\right)^{\frac{1}{2}}$
$\displaystyle+\mathbb{E}\left(\int_{0}^{t}\int_{Z}\left(\int_{{\mathbb{R}}^{d}}|\Im(\nabla
u(s,x)\overline{u(s,x)}\nabla\overline{g(z(x))})|\,dx\right)^{2}\,\nu(dz)\,ds\right)^{\frac{1}{2}}$
$\displaystyle{}+\mathbb{E}\left(\int_{Z}\left(\int_{{\mathbb{R}}^{d}}\left|u(s,x)\right|^{\alpha+1}\left|g(z(x))\right|\,dx\right)^{2}\,\nu(dz)\,ds\right)^{\frac{1}{2}}.$

Next, carefully applying the Hölder inequality and, if necessary, the Young
inequality term by term, and using Hypothesis 1-(i) one finally arrives at

(4.30) $\displaystyle\mathbb{E}\sup_{0\leq s\leq
t}\left[{{\mathcal{H}}}(u(s))-{{\mathcal{H}}}(u(0))\right]$
$\displaystyle\leq$ $\displaystyle
C\left(1+\mathbb{E}\left(\int_{0}^{t}{{\mathcal{H}}}(u(s))^{2}\,ds\right)^{\frac{1}{2}}+\mathbb{E}\int_{0}^{t}{{\mathcal{H}}}(u(s))\,ds\right),$

where the constant $C>0$ depends only on $C_{0}(\nu)$, $C_{1}(\nu)$,
$C_{2}(\nu)$, and $\alpha$. We deduce

$\displaystyle\mathbb{E}\sup_{0\leq s\leq
t}\left[{{\mathcal{H}}}(u(s))-{{\mathcal{H}}}(u(0))\right]$
$\displaystyle\leq$ $\displaystyle C+C\sqrt{t}\,\mathbb{E}\sup_{0\leq s\leq
t}{{\mathcal{H}}}(u(s))+C\mathbb{E}\int_{0}^{t}{{\mathcal{H}}}(u(s))\,ds$
$\displaystyle\leq$ $\displaystyle
C+C\left(\sqrt{t}+t\right)\,\mathbb{E}\sup_{0\leq s\leq t}{{\mathcal{H}}}(s).$

Now, let $t^{\ast}$ be so small that
$C(\sqrt{t^{\ast}}+t^{\ast})\leq\frac{1}{2}$. Then, we get

$\mathbb{E}\sup_{0\leq s\leq
t^{\ast}}\left[{{\mathcal{H}}}(u(s))-{{\mathcal{H}}}(u(0))\right]\leq 2C,$

and therefore

$\mathbb{E}\sup_{0\leq s\leq
t^{\ast}}{{\mathcal{H}}}(u(s))\leq\mathbb{E}{{\mathcal{H}}}(u(0))+2C.$

Noting that $t^{\ast}$ is only depending on $C_{0}(\nu)$, $C_{1}(\nu)$,
$C_{2}(\nu)$, and $\alpha$, one may iterate the previous step on
$[t^{\ast},2t^{\ast}]$, etc, and show that (4.1) holds.

Next, we want to investigate the entity
$\int_{{\mathbb{R}}^{d}}x^{2}|u(t,x)|^{2}\,dx$. First, we will prove the
inequality

$\mathbb{E}\int_{{\mathbb{R}}^{d}}x^{2}|u(t,x)|^{2}\,dx\leq
C(T)\left[1+\mathbb{E}({{\mathcal{H}}}(u_{0}))+\mathbb{E}\int_{{\mathbb{R}}^{d}}x^{2}|u(0,x)|^{2}\,dx\right],$

where the constant $C>0$ depends only on $C_{0}(\nu)$, $C_{1}(\nu)$,
$C_{2}(\nu)$, and $\alpha$. Secondly, we will give an estimate of
$\mathbb{E}\sup_{0\leq s\leq T}\int_{{\mathbb{R}}^{d}}x^{2}|u(s,x)|^{2}\,dx$.
In particular, we have by the Itô formula

$\displaystyle d\int_{{\mathbb{R}}^{d}}|x|^{2}|u(t,x)|^{2}\,dx$
$\displaystyle=$ $\displaystyle\,2\langle x^{2}u(t),du^{c}(t)\rangle$
$\displaystyle+\int_{Z}\left[x^{2}|u(t^{-})-iu(t^{-})G(z)|^{2}-x^{2}|u(t^{-})|^{2}\right]\,\tilde{\eta}(dz,dt)$
$\displaystyle+\int_{Z}\left[x^{2}|u(t^{-})-iu(t^{-})G(z)|^{2}-x^{2}|u(t^{-})|^{2}-2{\langle}x^{2}u(t^{-}),u(t^{-})(-iG(z)){\rangle}\right]\,\nu(dz)\,dt$
$\displaystyle=$ $\displaystyle 2\langle x^{2}u,-i\Delta
u+i\lambda|u|^{\alpha-1}u\rangle\,dt-2\int_{Z}\langle
x^{2}u,iuH(z)\,\rangle\nu(dz)\,dt$
$\displaystyle+\int_{Z}\left(\int_{{\mathbb{R}}^{d}}x^{2}|u(t^{-},x)|^{2}\left[|1-ig(z(x))|^{2}-1\right]\,dx\right)\,\tilde{\eta}(dz,dt)$
$\displaystyle+\int_{Z}\int_{{\mathbb{R}}^{d}}x^{2}\Big{\\{}|u(t^{-},x)[1-ig(z(x))]|^{2}-|u(t^{-},x)|^{2}$
$\displaystyle{}-2\Re\left[x^{2}u(t^{-},x)\overline{u(t^{-},x)(-ig(z(x)))}\right]\Big{\\}}\,\nu(dz)\,dt.$

Integrating by parts,

$\displaystyle-\langle x^{2}u,i\Delta u\rangle=\langle\nabla(x^{2}u),i\nabla
u\rangle$ $\displaystyle=$ $\displaystyle 2\langle xu,i\nabla
u\rangle=-2\Im\int_{{\mathbb{R}}^{d}}\bar{u}(x)x\cdot\nabla u(x)\,dx.$

Next,

$\displaystyle\langle x^{2}u,iuH(z)\,\rangle$ $\displaystyle=$
$\displaystyle\int_{{\mathbb{R}}^{d}}\Re\left(x^{2}u(t,x)\overline{u(t,x)ih(z(x)))}\right)\,dx\,$
$\displaystyle=$
$\displaystyle-\int_{{\mathbb{R}}^{d}}|x|^{2}|u(t,x)|^{2}\Im\left(h(z(x)))\right)\,dx.$

Since, for any complex valued functions $a$ and $b$ we have
$|a|^{2}-|b|^{2}=\Re{\langle}a-b,a+b{\rangle}$, we have

$\displaystyle\int_{{\mathbb{R}}^{d}}x^{2}\left[|u(t^{-},x)[1-ig(z(x))]|^{2}-|u(t^{-},x)|^{2}-2\Re\left(u(t^{-},x)\overline{u(t^{-},x)(-ig(z(x)))}\right)\right]\,dx$
$\displaystyle=$
$\displaystyle\int_{{\mathbb{R}}^{d}}x^{2}\left[\Re\left(u(t^{-},x)(-ig(z(x))\overline{u(t^{-},x)(2-ig(z(x)))}\right)-2|u(t^{-},x)|^{2}\Re\left(\overline{-ig(z(x))}\right)\right]\,dx$
$\displaystyle=$
$\displaystyle\int_{{\mathbb{R}}^{d}}x^{2}|u(t^{-},x)|^{2}\left(2\Im\left(g(z(x))\right)-2\Re\left(i\overline{g(z(x))}\right)+|g(z(x))|^{2}\right)\,\,dx$
$\displaystyle=$
$\displaystyle\int_{{\mathbb{R}}^{d}}x^{2}|u(t^{-},x)|^{2}|g(z(x))|^{2}\,dx.$

Collecting altogether, taking expectation we get

$\displaystyle\mathbb{E}\int_{{\mathbb{R}}^{d}}x^{2}|u(s,x)|^{2}\,dx$
$\displaystyle\leq$
$\displaystyle-4\Im\int_{0}^{t}\mathbb{E}\int_{{\mathbb{R}}^{d}}\bar{u}(s,x)x\cdot\nabla
u(s,x)\,dx\,ds$
$\displaystyle{}+2\int_{0}^{t}\int_{Z}\mathbb{E}\int_{{\mathbb{R}}^{d}}|x|^{2}|u(s,x)|^{2}\Im\left(h(z(x)))\right)\,dx\,\nu(dz)\,ds$
$\displaystyle{}+\int_{0}^{t}\int_{Z}\mathbb{E}\int_{{\mathbb{R}}^{d}}|x|^{2}|u(s,x)|^{2}|g(z(x))|^{2}\,dx\,\nu(dz)\,ds.$

Taking into account Hypothesis 1-(ii)-(b), we get by the Young inequality

$\displaystyle\mathbb{E}\int_{{\mathbb{R}}^{d}}x^{2}|u(s,x)|^{2}\,dx$
$\displaystyle\leq$ $\displaystyle
C(\nu)\,\mathbb{E}\int_{0}^{t}\int_{{\mathbb{R}}^{d}}x^{2}|{u(s,x)}|^{2}\,dx\,ds+2\mathbb{E}\int_{0}^{t}\int_{{\mathbb{R}}^{d}}|\nabla
u(s,x)|^{2}\,dx\,ds.$

Since the second term is bounded by $\mathbb{E}{{\mathcal{H}}}(u(t))$, the
assertion follows by the Grownwall inequality. Next, observe that

$\displaystyle\int_{{\mathbb{R}}^{d}}x^{2}\left[|u(t^{-},x)[1-iG(z)]|^{2}-|u(t^{-},x)|^{2}\right]\,dx$
$\displaystyle=$
$\displaystyle\int_{{\mathbb{R}}^{d}}x^{2}|u(t^{-},x)|^{2}\left(2\Im(g(z(x))+|g(z(x))|^{2}\right)\,\,dx.$

Hence, we get in addition, by Burkholder inequality

$\displaystyle\mathbb{E}\sup_{0\leq s\leq
t}\int_{{\mathbb{R}}^{d}}x^{2}|u(s,x)|^{2}\,dx$ $\displaystyle\leq$
$\displaystyle-4\Im\int_{0}^{t}\int_{{\mathbb{R}}^{d}}\mathbb{E}\bar{u}(s,x)x\cdot\nabla
u(s,x)\,dx\,ds$
$\displaystyle{}+\int_{0}^{t}\int_{Z}\int_{{\mathbb{R}}^{d}}|x|^{2}\mathbb{E}|u(s,x)|^{2}\Im\left(h(z(x)))\right)\,dx\,\nu(dz)\,ds$
$\displaystyle{}+\mathbb{E}\sup_{0\leq s\leq
t}\int_{0}^{t}\int_{Z}\int_{{\mathbb{R}}^{d}}x^{2}\left[|u(t^{-},x)[1-G(z)]|^{2}-|u(t^{-},x)|^{2}\right]\,dx\,\eta(dz,ds)$
$\displaystyle\leq$
$\displaystyle-4\Im\int_{0}^{t}\int_{{\mathbb{R}}^{d}}\mathbb{E}\bar{u}(s,x)x\cdot\nabla
u(s,x)\,dx\,ds$
$\displaystyle{}+\int_{0}^{t}\int_{Z}\int_{{\mathbb{R}}^{d}}|x|^{2}\mathbb{E}|u(s,x)|^{2}\Im\left(h(z(x)))\right)\,dx\,\nu(dz)\,ds$
$\displaystyle{}+\int_{0}^{t}\int_{Z}\int_{{\mathbb{R}}^{d}}x^{2}\mathbb{E}|u(s,x)|^{2}\left(2\Im(g(z(x))+|g(z(x))|^{2}\right)\,\,dx\,\nu(dz)\,ds.$

By similar arguments as before, we can show that

$\displaystyle\mathbb{E}\sup_{0\leq s\leq
t}\int_{{\mathbb{R}}^{d}}x^{2}|u(s,x)|^{2}\,dx$ $\displaystyle\leq$
$\displaystyle
C(T)\left(\mathbb{E}\int_{{\mathbb{R}}^{d}}x^{2}|u(0,x)|^{2}\,dx+1+\mathbb{E}|\nabla
u(t)|^{2}\,dx\right).$

∎

##  5\. Existence of the solution with infinite Lévy measure - Proof of
Theorem 2.7

The proof is done in several steps. In the first step we construct a solution
by cutting of the small jumps. In this way, we get a sequence of solutions,
denoted in the following by $\\{u_{m}:m\in\mathbb{N}\\}$. Next, in the second
step, we give uniform bounds on the mass ${{\mathcal{E}}}(u_{m})$, the
Hamiltonian ${{\mathcal{H}}}(u_{m})$ and the virial $|xu_{m}|_{L^{2}}^{2}$.
Thanks to these uniform bounds we are able to prove in the third step
tightness of the laws of $\\{u_{m}:m\in\mathbb{N}\\}$ in
$\mathbb{D}(0,T;H^{\gamma}_{2}({\mathbb{R}}^{d}))$ for any $\gamma<1$. Now,
the existence of a converging subsequence follows. In order, to get again
stochastic processes we apply the Skorohod embedding Theorem. This gives us a
probability space with a family of processes converging in the almost sure
sense. Now in the last step we can show by an application of the dominated
convergence Theorem that this limit is indeed a solution to (2.4).

### Step I

In the first step we will construct an approximating sequence. Let
$\\{\varepsilon_{m}:m\in\mathbb{N}\\}$ be a sequence such that
$\varepsilon_{m}>0$ and $\varepsilon_{m}\downarrow 0$. Let $\nu_{m}$ be the
Lévy measure defined by $\nu_{m}(U):=\nu(U\setminus B_{Z}(\varepsilon_{m}))$,
$U\in{{\mathcal{B}}}(Z)$. Let $\eta$ be a time homogenous Poisson random
measure over a filtered probability space $\mathfrak{A}$ and let $\eta_{m}$ be
the time homogenous random measure given by

(5.1) $\displaystyle\eta_{m}(U\times I):=\eta(U\setminus
B_{Z}(\varepsilon_{m})\times I),\quad
U\in{{\mathcal{B}}}(Z),\,I\in{{\mathcal{B}}}([0,T]).$

Let us observe that $\eta_{m}$ has intensity measure $\nu_{m}$. We denote by
$u_{m}$ the solution to

$\displaystyle u_{m}(t)$ $\displaystyle=$
$\displaystyle{{\mathcal{T}}}(t)u_{0}+i\lambda\int_{0}^{t}{{\mathcal{T}}}(t-s)\left|u_{m}(s)\right|^{\alpha-1}\,u_{m}(s)\,ds$
$\displaystyle{}-i\int_{0}^{t}\int_{Z}{{\mathcal{T}}}(t-s)u_{m}(s)\,G(z)\,\tilde{\eta}_{m}(dz,ds)$
$\displaystyle{}-i\int_{0}^{t}\int_{Z}{{\mathcal{T}}}(t-s)u_{m}(s)\,H(z)\,\nu_{m}(dz)\,ds.$

It follows from Lemma 4.1 that for any $m$ there exists a unique solution with
$u_{m}$ to Equation (5) belonging ${\mathbb{P}}$-a.s. to
$\mathbb{D}(0,T;H^{1}({\mathbb{R}}^{d}))$.

### Step II

We will prove the following Claim.

######  Claim 5.1.

  * •

For any $T>0$ there exists a constant
$C=C(T,C_{0}(\nu),C_{3}(\nu),C_{g},C_{h})>0$ such that

$\mathbb{E}\sup_{0\leq t\leq
T}|u_{m}(t)|_{L^{2}}=C\,\left(|u_{0}|_{L^{2}}+1\right),\quad
m\in\mathbb{N},\,{\mathbb{P}}-\mbox{a.s.}$

  * •

For any $T>0$ there exists a
$C=C(T,C_{0}(\nu),C_{1}(\nu),C_{2}(\nu),C_{g},C_{h})>0$ such that

$\mathbb{E}\sup_{0\leq s\leq
T}{{\mathcal{H}}}(u_{m}(s))=C\,\left(\mathbb{E}{{\mathcal{H}}}(u(0))+1\right),\quad
m\in\mathbb{N}.$

  * •

For any $T>0$ there exists a
$C=C(T,C_{0}(\nu),C_{1}(\nu),C_{2}(\nu),C_{g},C_{h})>0$ such that

$\mathbb{E}\sup_{0\leq s\leq
T}\int_{{\mathbb{R}}^{d}}|x|^{2}|u(s,x)|^{2}\,dx=C\,\left(\mathbb{E}\int_{{\mathbb{R}}^{d}}|x|^{2}|u(0,x)|^{2}\,dx+{{\mathcal{H}}}(u(0))+1\right),\quad
m\in\mathbb{N}.$

###### Proof.

In fact the proof of Proposition 5.1 is just an application of Proposition 4.1
and taking into account, that

$C_{j}(\nu_{m})\leq C_{j}(\nu),\quad m\in\mathbb{N}\mbox{ and
}j=0,1,2\,\mbox{and}\,3.$

∎

### Step III

In this Step we show the following Claim.

######  Claim 5.2.

  1. For any $\gamma<1$ the laws of the set $\\{u_{m}:m\in\mathbb{N}\\}$ are tight in $\mathbb{D}(0,T;H^{\gamma}_{2}({\mathbb{R}}^{d}))$

###### Proof.

In order to show the assertion, we will apply Corollary B.1. We will first
prove the compact containment condition, in particular, the condition (a) in
Corollary B.1. Let $B_{0}$ be defined by

$\left\\{v\in
H^{1}_{2}({\mathbb{R}}^{d}):\int_{{\mathbb{R}}^{d}}|x|^{2}|v(x)|^{2}\,dx<\infty\right\\}$

equipped with norm

$|v|_{B_{0}}:=|v|_{H^{1}_{2}}+\left(\int_{{\mathbb{R}}^{d}}|x|^{2}|v(x)|^{2}\,dx\right)^{\frac{1}{2}}.$

Then, $B_{0}\hookrightarrow L^{2}({\mathbb{R}}^{d})$ compactly. Let us denote
the complex interpolation space
$B_{1}=(B_{0},H_{2}^{1}({\mathbb{R}}^{d}))_{\delta}$. Then, the embedding of
$B_{0}$ into $H^{1}_{2}({\mathbb{R}}^{d})$ is bounded, and the embedding of
$B_{0}$ into $L^{2}({\mathbb{R}}^{d})$ is compact. By Theorem 3.8.1 [6, p. 56]
it follows that for any $\delta<1$, $B_{0}\hookrightarrow
H^{\delta}_{2}({\mathbb{R}}^{d})$ compactly. To show condition (a) of
Corollary B.1, i.e. the compact containment condition, one can use Claim 5.1.
To be more precise, it follows from item one, two and three of Claim 5.1 that
there exists a constant $C>0$ such that

$\mathbb{E}|u_{m}(t)|_{B_{0}}\leq C,\quad m\in\mathbb{N}.$

Condition (a) follows by the Chebyschev inequality.

It remains to show, that the family $\\{u_{m}:m\in\mathbb{N}\\}$ satisfies the
second condition of Corollary B.1, i.e. (b). First, observe that we have

$\displaystyle
u_{m}(t+h)-u_{m}(t)=\left[{{\mathcal{T}}}(h)-I\right]{{\mathcal{T}}}(t)u_{0}+i\lambda\int_{t}^{t+h}{{\mathcal{T}}}(t+h-s)F(u_{m}(s))\,ds$
$\displaystyle+i\lambda\left[{{\mathcal{T}}}(h)-I\right]\int_{0}^{t}{{\mathcal{T}}}(t-s)F(u_{m}(s))\,ds$
$\displaystyle{}-i\int_{t}^{t+h}\int_{Z}{{\mathcal{T}}}(t+h-s)u_{m}(s)\,G(z)\tilde{\eta}_{m}(dz,ds)$
$\displaystyle{}-i\left[{{\mathcal{T}}}(h)-I\right]\int_{0}^{t}\int_{Z}{{\mathcal{T}}}(t-s)u_{m}(s)\,G(z)\tilde{\eta}_{m}(dz,ds)$
$\displaystyle{}-i\int_{t}^{t+h}\int_{Z}{{\mathcal{T}}}(t+h-s)u_{m}(s)\,H(z)\nu_{m}(dz)\,ds$
$\displaystyle{}-i\left[{{\mathcal{T}}}(h)-I\right]\int_{0}^{t}\int_{Z}{{\mathcal{T}}}(t-s)u_{m}(s)\,H(z)\nu_{m}(dz)\,ds$
$\displaystyle=$
$\displaystyle\underbrace{\left[{{\mathcal{T}}}(h)-I\right]u_{m}(t)}_{=:I_{0}(t,h)}+\underbrace{i\lambda\int_{t}^{t+h}{{\mathcal{T}}}(t+h-s)F(u_{m}(s))\,ds}_{=:I_{1}(t,h)}$
$\displaystyle\quad{}-\underbrace{i\int_{t}^{t+h}\int_{Z}{{\mathcal{T}}}(t+h-s)u_{m}(s)\,G(z)\tilde{\eta}_{m}(dz,ds)}_{=:I_{2}(t,h)}$
$\displaystyle\quad{}-\underbrace{i\int_{t}^{t+h}\int_{Z}{{\mathcal{T}}}(t+h-s)u_{m}(s)\,H(z)\nu_{m}(dz)\,ds}_{=:I_{3}(t,h)}.$

Secondly, note that

$\displaystyle\mathbb{E}\sup_{0\leq
h\leq\delta}|u_{m}(t+h)-u_{m}(t)|^{r}_{H^{\gamma}_{2}}$ $\displaystyle\leq$
$\displaystyle\mathbb{E}\sup_{0\leq
h\leq\delta}|u_{m}(t+h)-u_{m}(t)|_{L^{2}}^{r(1-\gamma)}|u_{m}(t+h)-u_{m}(t)|_{H^{1}_{2}}^{r\gamma}$
$\displaystyle\leq$ $\displaystyle\mathbb{E}\sup_{0\leq
h\leq\delta}|u_{m}(t+h)-u_{m}(t)|_{L^{2}}^{r(1-\gamma)}\left(|u_{m}(t+h)|_{H^{1}_{2}}+|u_{m}(t)|_{H^{1}_{2}}\right)^{r\gamma}$
$\displaystyle\leq$ $\displaystyle\mathbb{E}\sup_{0\leq
h\leq\delta}|u_{m}(t+h)-u_{m}(t)|_{L^{2}}^{r(1-\gamma)}\mathbb{E}\sup_{0\leq
h\leq
T}\left(|u_{m}(t+h)|_{H^{1}_{2}}+|u_{m}(t)|_{H^{1}_{2}}\right)^{r\gamma}$
$\displaystyle\leq$ $\displaystyle\left(\mathbb{E}\sup_{0\leq
h\leq\delta}|u_{m}(t+h)-u_{m}(t)|^{r}_{L^{2}}\right)^{1-\gamma}\left(\mathbb{E}\sup_{0\leq
h\leq
T}\left(|u_{m}(t+h)|_{H^{1}_{2}}+|u_{m}(t)|_{H^{1}_{2}}\right)^{r}\right)^{\gamma}.$

In order to estimate $I_{0}(t,h)$ we know for any $s\in{\mathbb{R}}$
$\left|\left[{{\mathcal{T}}}(h)-\mathcal{I}\right]u_{m}(t)\right|_{H^{s}_{2}}\leq
h\left|u_{m}(t)\right|_{H^{s+2}_{2}}$. Interpolation gives therefore

$\left|\left[{{\mathcal{T}}}(h)-\mathcal{I}\right]u_{m}(t)\right|_{L^{2}}\leq\sqrt{2h}\left|u_{m}(t)\right|_{H^{1}_{2}}.$

Since by Claim 5.1, $\mathbb{E}\,{{\mathcal{H}}}(u_{m}(t))$ is uniformly
bounded in $m$, there exists a constant $C=C(T)>0$ such that

$\mathbb{E}\sup_{0\leq
h\leq\delta}\left|\left[{{\mathcal{T}}}(h)-\mathcal{I}\right]u_{m}\right|^{2}_{L_{2}}\leq
C\sqrt{\delta}.$

Let $p=\alpha+1$, $p^{\prime}=(\alpha+1)/\alpha$, $r=4(1+\alpha)/(\alpha-1)d$
and $r^{\prime}=4(1+\alpha)/(4(1+\alpha)+d(1-\alpha))$. Applying the
Strichartz estimate, we get for $I_{1}(t,h)$

$\displaystyle\sup_{0\leq h\leq\delta}\left|I_{1}(t,h)\right|_{L^{2}}$
$\displaystyle\leq$ $\displaystyle\lambda\sup_{0\leq
h\leq\delta}\left|\int_{t}^{t+h}{{\mathcal{T}}}(t+h-r)F(u_{m}(r))\,dr\right|_{L^{2}}$
$\displaystyle\leq$ $\displaystyle
C\,\left(\int_{t}^{t+\delta}\left|F(u_{m}(s))\right|^{r^{\prime}}_{L^{p^{\prime}}}ds\right)^{1/r^{\prime}}\leq
C\left(\int_{t}^{t+\delta}\left|u_{m}(s)\right|^{r^{\prime}\alpha}_{L^{\alpha+1}}ds\right)^{1/r^{\prime}}.$

Sobolev embedding gives for $\alpha<(d+2)/(d-2)$

$\displaystyle\sup_{0\leq h\leq\delta}\left|I_{1}(t,h)\right|_{L^{2}}$
$\displaystyle\leq$ $\displaystyle
C\left(\int_{t}^{t+\delta}\left|u_{m}(s)\right|^{r^{\prime}\alpha}_{H^{1}_{2}}ds\right)^{1/r^{\prime}}.$

Now, taking the expectation, the Hölder inequality gives

$\displaystyle\mathbb{E}\sup_{0\leq
h\leq\delta}\left|I_{1}(t,h)\right|_{L^{2}}^{2\over\alpha}$
$\displaystyle\leq$ $\displaystyle
C\,\delta^{2/r^{\prime}\alpha}\,\mathbb{E}\sup_{t\leq h\leq
t+\delta}\left|u_{m}(h)\right|_{H^{1}_{2}}^{2}.$

By Hypothesis (1)-(i)-(a), it follows for $I_{2}(h,t)$

$\displaystyle\mathbb{E}\sup_{0\leq
h\leq\delta}\left|I_{2}(t,h)\right|_{L^{2}}^{2}$ $\displaystyle\leq$
$\displaystyle\mathbb{E}\sup_{0\leq
h\leq\delta}\left|\int_{t}^{t+h}\int_{Z}{{\mathcal{T}}}(t+h-s)u_{m}(s)G(z)\,\tilde{\eta}(dz,ds)\right|_{L^{2}}$
$\displaystyle\leq$
$\displaystyle\mathbb{E}\int_{t}^{t+\delta}\int_{Z}\left|{{\mathcal{T}}}(t+h-s)u_{m}(s)G(z)\,\right|^{2}_{L^{2}}\nu(dz)\,ds$
$\displaystyle\leq$ $\displaystyle C\,\delta\sup_{t\leq s\leq
t+\delta}\mathbb{E}\left|u_{m}(s)\right|^{2}_{L^{2}}.$

Again, by Hypothesis (1)-(i)-(a) and (iii), we get by Theorem A.1 for
$I_{3}(h,t)$

$\displaystyle\mathbb{E}\sup_{0\leq
h\leq\delta}\left|I_{3}(t,h)\right|_{L^{2}}$ $\displaystyle\leq$
$\displaystyle\mathbb{E}\sup_{0\leq
h\leq\delta}\left|\int_{t}^{t+h}\int_{Z}{{\mathcal{T}}}(t+h-s)u_{m}(s)H(z)\,\nu(dz)ds\right|_{L^{2}}$
$\displaystyle\leq$
$\displaystyle\mathbb{E}\int_{t}^{t+\delta}\int_{Z}\left|{{\mathcal{T}}}(t+h-s)u_{m}(s)H(z)\,\right|_{L^{2}}\nu(dz)\,ds$
$\displaystyle\leq$ $\displaystyle C\,\delta\sup_{t\leq s\leq
t+\delta}\mathbb{E}\left|u_{m}(s)\right|^{2}_{L^{2}}.$

Collecting altogether we arrive at

$\displaystyle\mathbb{E}\left(\sup_{t\leq s\leq
h}\left|u_{m}(t+s)-u_{m}(t)\right|_{L^{2}}^{2\over\alpha}\right)$
$\displaystyle\leq$ $\displaystyle\mathbb{E}\left(\sup_{0\leq s\leq
h}\left|\left[{{\mathcal{T}}}(s)-I\right]u_{m}(t)\right|_{L^{2}}\right)^{2\over\alpha}+\mathbb{E}\left(\left|I_{1}(t,h)\right|_{L^{2}}\right)^{2\over\alpha}+\mathbb{E}\left(\left|I_{2}(t,h)\right|_{L^{2}}\right)^{2\over\alpha}$
$\displaystyle\leq$ $\displaystyle
h^{\frac{1}{\alpha}}\left(1+\mathbb{E}\left|u_{m}(t)\right|_{H^{1}_{2}}^{2}\right),$

since $r^{\prime}\leq 2$. Applying Corollary B.1, we know the family of laws
of $\\{u_{m}:m\in\mathbb{N}\\}$ is tight in
$\mathbb{D}(0,T;H^{\gamma}_{2}({\mathbb{R}}^{d}))$.

### Step IV:

First, note, since $\nu_{m}\to\nu$ on $Z\setminus\\{0\\}$, hence
$\eta_{m}\to\eta$ in $M(Z\setminus\\{0\\}\times[0,T])$, so the family
$\\{\eta_{m}:m\in\mathbb{N}\\}$ is tight in $M(Z\setminus\\{0\\}\times[0,T])$.
Now, it follows from Step III, i.e. the fact that the sequence
$\\{u_{m}:m\in\mathbb{N}\\}$ is tight in
$\mathbb{D}(0,T;H^{\gamma}_{2}({\mathbb{R}}^{d}))$, that there exists a pair
$(u^{\ast},\eta^{\ast})$ of
$\mathbb{D}(0,T;H_{2}^{\gamma}({\mathbb{R}}^{d}))\times
M_{I}(Z\setminus\\{0\\}\times[0,T])$–valued random variables over
$\mathfrak{A}$ and a subsequence $\\{m_{k}:k\in\mathbb{N}\\}$ such that
$\\{(u_{m_{k}},\eta_{m_{k}}):k\in\mathbb{N}\\}$ converges to
$(u^{\ast},\eta^{\ast})$ weakly in
$M_{I}(Z\setminus\\{0\\}\times[0,T])\times\mathbb{D}(0,T;H_{2}^{\gamma}({\mathbb{R}}^{d}))$.
In fact, by the construction of $\eta_{m}$ we have even $\eta^{\ast}=\eta$.
For simplicity, we denote the subsequence
$\\{(u_{m_{k}},\eta_{m_{k}}):k\in\mathbb{N}\\}$ again by
$\\{(u_{m},\eta_{m}):m\in\mathbb{N}\\}$. By the modified version of the
Skorohod embedding Theorem, see Theorem D.1 [9], there exists a probability
space $(\bar{\Omega},\bar{{\mathcal{F}}},\bar{\mathbb{P}})$ and
$\mathbb{D}(0,T;H^{\gamma}_{2}({\mathbb{R}}^{d}))\times
M_{I}(Z\setminus\\{0\\}\times[0,T])$-valued random variables
$(\bar{u}_{1},\bar{\eta}_{1})$, $(\bar{u}_{2},\bar{\eta}_{2})$, $\ldots$,
having the same law as the random variables $(u_{1},\eta_{1})$,
$(u_{2},\eta_{2})$, $\ldots$, and a
$\mathbb{D}(0,T;H^{\gamma}_{2}({\mathbb{R}}^{d}))\times
M_{I}(Z\times{\mathbb{R}}_{+})$-valued random variable
$(\bar{u}^{\ast},\bar{\eta}^{\ast})$ on
$(\bar{\Omega},\bar{{\mathcal{F}}},\bar{\mathbb{P}})$ with
${{\mathcal{L}}}((\bar{u}^{\ast},\bar{\eta}^{\ast}))=\mu^{\ast}$ such that
${\mathbb{P}}$ a.s.

$\displaystyle\left(\bar{u}_{m},\bar{\eta}_{m}\right)\longrightarrow\left(\bar{u}^{\ast},\bar{\eta}^{\ast}\right)$
$\displaystyle\mbox{ in
}\mathbb{D}(0,T;{H^{\gamma}_{2}({\mathbb{R}}^{d})})\times
M_{I}(Z\setminus\\{0\\}\times{\mathbb{R}}_{+}).$

Before continuing, we will introduce the following notation. For a random
measure $\mu$ on $Z\setminus\\{0\\}\times[0,T]$ and for any
$U\in{\mathcal{B}(Z\setminus\\{0\\})}$ let us define an
$\bar{\mathbb{N}}$-valued process $(N_{\mu}(t,U))_{t\geq 0}$ by
$N_{\mu}(t,U):=\mu(U\times(0,t]),\;\;t\geq 0.$ In addition, we denote by
$(N_{\mu}(t))_{t\geq 0}$ the measure valued process defined by
$N_{\mu}(t)=\\{{\mathcal{S}}\ni U\mapsto N_{\mu}(t,U)\in\bar{\mathbb{N}}\\}$,
$t\in[0,T]$.

Now, let $\bar{\mathbb{F}}=(\bar{\mathcal{F}}_{t})_{t\geq 0}$ be the
filtration defined for any $t\in[0,T]$ by

(5.4)
$\bar{\mathcal{F}}_{t}=\sigma(\\{(N_{\bar{\eta}_{m}}1_{[0,s]},\bar{u}_{m}1_{[0,s]}),m\in{\mathbb{N}}\\},\,(N_{\bar{\eta}^{\ast}}1_{[0,s]},\bar{u}^{\ast}1_{[0,s]});0\leq
s\leq t).$

Since
$\sigma(N_{\bar{\eta}_{m}}1_{[0,s]})\subset\sigma(N_{\bar{\eta}_{m+1}}1_{[0,s]})$,
it is easy to show that the filtration obtained by deleting the family
$\\{\bar{\eta}_{m}:m\in\mathbb{N}\\}$ in (5.4) is the equal to
$\bar{\mathbb{F}}$.

######  Claim 5.3.

The following holds

  1. a.)

for every $m\in\mathbb{N}$, $\bar{\eta}_{m}$ is a time homogeneous Poisson
random measure on ${{\mathcal{B}}}(Z)\times{{\mathcal{B}}}({\mathbb{R}}_{+})$
over $(\bar{\Omega},\bar{{\mathcal{F}}},\bar{{\mathbb{F}}},\bar{\mathbb{P}})$
with intensity measure $\nu_{m}$;

  2. b.)

$\bar{\eta}^{\ast}$ is a time homogeneous Poisson random measure on
${{\mathcal{B}}}(Z)\times{{\mathcal{B}}}([0,T])$ over
$(\bar{\Omega},\bar{{\mathcal{F}}},\bar{{\mathbb{F}}},\bar{\mathbb{P}})$ with
intensity measure $\nu$;

Before starting with the actual proof, we cite the following Lemma.

######  Lemma 5.1.

A measurable mapping $\eta:\Omega\to M_{I}(Z\times R^{+}_{0})$ is a time
homogeneous Poisson random measure with intensity $\nu$ iff

  * (a)

for any ${U}\in{{\mathcal{Z}}}$ with $\nu(U)<\infty$, the random variable
$N_{\eta}(t,{U})$ is Poisson distributed with parameter $t\,\nu({U})$;

  * (b)

for any disjoint sets ${U}_{1},{U}_{2},\ldots,{U}_{n}\in{{\mathcal{Z}}}$, and
any $t\in[0,T]$ the random variables $N_{\eta}(t,{{U}_{1}})$,
$N_{\eta}(t,{{U}_{2}})$, …, $N_{\eta}(t,{{U}_{n}})$ are mutually independent;

  * (c)

the $M_{\bar{\mathbb{N}}}(Z)$-valued process $(N_{\eta}(t,\cdot))_{t\geq 0}$
is adapted to ${{\mathbb{F}}}$;

  * (d)

for any $t\in[0,T]$, ${U}\in{{\mathcal{Z}}}$, $\nu(U)<\infty$, and any
$r,s\geq t$, the random variables $N_{\eta}(r,{U})-N_{\eta}(s,{U})$ are
independent of ${{\mathcal{F}}}_{t}$.

###### Proof of Claim 5.3.

We have to show that for arbitrary $m\in\mathbb{N}$, $\bar{\eta}_{m}$
satisfies item (a), (b), (c) and (d). In order to show (a), let
${U}\in\mathcal{B}(Z)$. Then $N_{\eta_{m}}({U},t)$ is Poisson distributed with
parameter $t\nu_{m}({U})$. Since
${\mathcal{L}aw}(\eta_{m})={\mathcal{L}aw}(\bar{\eta}_{m})$, it follows (a).
In order to show (b), let $U_{1},\ldots,U_{k}\in{\mathcal{Z}}$, be $k$
disjoint sets and $t\geq 0$. Since $\eta_{m}$ is a time homogeneous Poisson
random measure, we have for all $k\geq 1$ and $\theta_{j}\in{\mathbb{R}}$,
$j=1,\ldots k$

(5.5)
$\displaystyle{\mathbb{E}}e^{i\left(\sum_{l=1}^{k}\theta_{l}N_{\eta_{m}}(t,U_{l})\right)}=\prod_{l=1}^{k}{\mathbb{E}}e^{i\;\theta_{l}N_{\eta_{m}}(t,U_{l})}.$

Since $\bar{\eta}_{m}$ and $\eta_{m}$ have the same laws, the random variables
$N_{\bar{\eta}_{m}}(t,U)$ and $N_{\eta_{m}}(t,U)$ have the same characteristic
functions for any $U\in\mathcal{Z}$ and $t\geq 0$. Therefore, it follows from
(5.5) that

$\displaystyle{\bar{\mathbb{E}}}e^{i\left(\sum_{l=1}^{k}\theta_{l}N_{\bar{\eta}_{m}}(t,U_{l})\right)}=\prod_{l=1}^{k}\bar{\mathbb{E}}e^{i\;\theta_{l}N_{\bar{\eta}_{m}}(t,U_{l})},$

which proves (b). Next, we have to show that $\bar{\eta}_{m}$ satisfies (c)
with the filtration defined in (5.4). For this purpose let us fix
$l\in\mathbb{N}$, $t_{0}\in[0,T]$ and $r\geq s\geq t_{0}$. It follows from the
definition of $\bar{{\mathbb{F}}}$ that $N_{\bar{\eta}_{m}}$ is
$\bar{{\mathbb{F}}}$-adapted. It remains to prove (d). In particular, it
remains to prove that the random variable
$\bar{X}=N_{\bar{\eta}_{m}}(r)-N_{\bar{\eta}_{m}}(s)$ is independent of
$\bar{\mathcal{F}}_{t_{0}}$. By Lemma 5.1 the random variable
$X=N_{\eta_{m}}(r)-N_{\eta_{m}}(s)$ is independent of $N_{\eta}(t_{0})$. Since
for any $k\geq 1$, the $\sigma$–algebra generated by $\eta$ until time $t_{0}$
is finer than the $\sigma$–algebra generated by $\eta_{k}$, we know $X$ is
independent from $N_{\eta_{k}}(t_{0})$ for all $k\in\mathbb{N}$. In
particular, for all $f,g:M_{I}(Z\times[0,T])\to{\mathbb{R}}$ we have for all
$k\geq 1$

$\mathbb{E}f(X)g(N_{\eta_{k}}(t_{0}))=\mathbb{E}f(X)\,\mathbb{E}g(N_{\eta_{k}}(t_{0})).$

Since for all $k\geq 1$, $\bar{\eta}_{k}$ have the same law as $\eta_{k}$, and
$\bar{\eta}_{m}$ have the same law as $\eta_{m}$, therefore $X$ has the same
law as $\bar{X}$. It follows that

$\bar{\mathbb{E}}f(\bar{X})g(N_{\bar{\eta}_{k}}(t_{0}))=\mathbb{E}f(\bar{X})\,\mathbb{E}g(N_{\bar{\eta}_{k}}(t_{0})).$

Hence, $\bar{X}$ is independent of the filtration
$\sigma\left(N_{\bar{\eta}_{k}}(t),\,t\leq t_{0}\right)$.

Next, we need to show that for any $k\geq 1$, $\bar{X}$ is independent of
$\bar{u}_{k}(t)$ for any $t\leq t_{0}$. In what follows we also fix
$t\in[0,t_{0}]$. Since
$\mathcal{L}(\bar{u}_{l},\bar{\eta}_{l})=\mathcal{L}(u_{l},\eta_{l})$, it
follows that

(5.6)
$\mathcal{L}(\bar{u}_{l}|_{[0,t]},\bar{X}_{l})=\mathcal{L}(u_{l}|_{[0,t]},X_{l}),$

where $X=N_{\eta_{m}}(r)-N_{\eta_{m}}(s)$. Now, we have to show that $\bar{X}$
is independent to the $\sigma(\\{\bar{u}_{k}|_{[0,t]}:t\leq
t_{0},k\in\mathbb{N}\\})$. Recall that $u_{k}$ is the unique solution to the
linear stochastic evolution equation (2.4), hence it is adapted to the
$\sigma$-algebra generated by $\eta_{k}$. Consequently, $u_{k}|_{[0,t]}$ is
independent of $X$ and we infer from this last remark and the equality of the
laws that $\bar{u}_{k}|_{[0,t]}$ is independent of $\bar{X}$ for all $t\leq
t_{0}$ and $k\in\mathbb{N}$.

It remains to prove that $\bar{X}$ is independent of
$\sigma(\\{N_{\bar{\eta}^{\ast}}(t),t\leq t_{0}\\})$ and $u^{\ast}|_{[0,t]}$,
but, this is the object of the next Lemma.

######  Lemma 5.2.

Let $Y$ be a Banach space, $z$ and $y_{\ast}$ be two $Y$-valued random
variables over $(\Omega,{\mathcal{F}},\mathbb{P})$. Let
$\\{y_{n}:n\in{\mathbb{N}}\\}$ be a family of $Y$-valued random variables over
a probability space $(\Omega,{\mathcal{F}},\mathbb{P})$ such that $y_{n}\to
y_{\ast}$ weakly, i.e. for all $\phi\in Y^{\ast}$,
$\mathbb{E}e^{i{\langle}\phi,y_{n}{\rangle}}\to\mathbb{E}e^{i{\langle}\phi,y{\rangle}}$.
If for all $n\geq 1$ the two random variables $y_{n}$ and $z$ are independent,
then $z$ is also independent of $y_{\ast}$.

###### Proof of Lemma 5.2.

The random variables $y_{\ast}$ and $z$ are independent iff

$\displaystyle{\mathbb{E}}e^{i({\langle}\theta_{1},z{\rangle}+{\langle}\theta_{2},y_{\ast}{\rangle})}={\mathbb{E}}e^{i{\langle}\theta_{1},z{\rangle}}\,{\mathbb{E}}e^{i{\langle}\theta_{2},y_{\ast}{\rangle}},\quad\theta_{1},\theta_{2}\in
Y^{\ast}.$

The weak convergence and the independence of $z$ and $y_{n}$ for all
$n\in{\mathbb{N}}$ justify the following chain of equalities.

$\displaystyle{\mathbb{E}}e^{i({\langle}\theta_{1},z{\rangle}+{\langle}\theta_{2},y_{\ast}{\rangle})}=\lim_{n\to\infty}{\mathbb{E}}e^{i({\langle}\theta_{1},z{\rangle}+{\langle}\theta_{2},y_{n}{\rangle})}=\lim_{n\to\infty}{\mathbb{E}}e^{i{\langle}\theta_{1},z{\rangle}}\,{\mathbb{E}}e^{i{\langle}\theta_{2},y_{n}{\rangle}}={\mathbb{E}}e^{i{\langle}\theta_{1},z{\rangle}}\,{\mathbb{E}}e^{i{\langle}\theta_{2},y_{\ast}{\rangle}}.$

∎

Fix $t\leq s$. Since $\bar{u}_{k}|_{[0,t]}$ is independent from $\bar{X}$,
Lemma 5.2 implies that $u^{\ast}|_{[0,t]}$ is independent from $\bar{X}$.
Similarly, $\bar{X}$ is independent from $N_{\eta^{\ast}}(t)$ for all $t\leq
t_{0}$.

Finally we have to show Claim 5.3-(b). In particular, we have to show that
$\eta_{\ast}\in{\mathcal{M}}_{I}(S\times{\mathbb{R}}^{+})$ is a time
homogeneous Poisson random measure with intensity $\nu$. Observe first that
$Z\setminus B(\varepsilon_{m})\uparrow Z\setminus\\{0\\}$ as $m\to\infty$. By
Theorem 4.3.4 [3] one knows $\eta_{m}\to\eta$ weakly. Using the fact that
$\eta$ is a Poisson random measure with intensity measure $\nu$ and
${\mathcal{L}aw}(\bar{\eta}_{m})={\mathcal{L}aw}(\eta_{m})$ and
${\mathcal{L}aw}(\eta^{\ast})={\mathcal{L}aw}(\eta)$, the assertion follows by
Lemma 5.2. ∎

######  Claim 5.4.

The following holds

  1. a.)

for all $m\in\mathbb{N}$, $\bar{u}_{m}$ is a
$\bar{{\mathbb{F}}}$-progressively measurable process;

  2. b.)

the process $\bar{u}^{\ast}$ is a $\bar{{\mathbb{F}}}$-progressively
measurable process.

###### Proof.

As we noted earlier, one can argue as in [8, Proposition B.5] and prove that
the random variables
$\bar{u}_{m},\bar{u}^{\ast}:\bar{\Omega}\to\mathbb{D}(0,T;H^{\gamma}_{2}({\mathbb{R}}^{d}))$
induce two $H^{\gamma}_{2}({\mathbb{R}}^{d})$-valued stochastic processes
still denoted with the same symbols. Here, we have to show that for each
$m\in{\mathbb{N}}$, $\bar{u}_{m}$ and $\bar{u}^{\ast}$ are
$\bar{\mathbb{F}}$-progressively measurable. By definition of
$\bar{\mathbb{F}}$, for fixed $m\in{\mathbb{N}}$ the process $\bar{u}_{m}$ is
adapted to $\bar{\mathbb{F}}$ by the definition of $\bar{\mathbb{F}}$. By Step
(III) the laws of the processes $\\{\bar{u}_{m}:m\in\mathbb{N}\\}$ are tight
in $\mathbb{D}(0,T;H^{\gamma}_{2}({\mathbb{R}}^{d}))$. Thus, for any
$\varepsilon>0$ there exists a compact set $K_{\varepsilon}\subset
H^{\gamma}_{2}({\mathbb{R}}^{d})$ such that
${\mathbb{P}}\left(\bar{u}_{m}\not\in K_{\varepsilon}\right)<\varepsilon$.
However, by Proposition B.5 the Haar projections $\pi_{n}$ converge uniformly
on $K_{\varepsilon}$. On the other side, $\pi_{n}u_{m}$ is progressively
measurable for any $m\in\mathbb{N}$ and $n\in\mathbb{N}$. Thus, we can choose
$\\{\bar{u}_{n}^{m},m\in{\mathbb{N}}\\}$ to be progressively measurable. Since
$\bar{u}_{n}^{m}\to\bar{u}_{n}$ as $m\to\infty$ weakly in
$\mathbb{D}(0,T;H^{\gamma}_{2}({\mathbb{R}}^{d}))$ and
$\bar{u}_{n}\to\bar{u}^{\ast}$ as $n\to\infty$ weakly also in
$\mathbb{D}(0,T;H^{\gamma}_{2}({\mathbb{R}}^{d}))$, it follows that
$\bar{u}^{\ast}$ is a limit in
$\mathbb{D}(0,T;H^{\gamma}_{2}({\mathbb{R}}^{d}))$ of some progressively
measurable step functions. In particular, $\bar{u}^{\ast}$ is also
progressively measurable, i.e. b.) holds. ∎

### Step V:

In the last step we will show that the process $\bar{u}^{\ast}$ is indeed a
mild solution to (2.4). In particular, we will show that for any $t\in[0,T]$
the identity (2.4) is satisfied. But before, we will state the following
proposition.

######  Proposition 5.3.

Let $H$ be a Hilbert space $({{\mathcal{T}}}(t))_{t\geq 0}$ be a group on $H$
and $\\{\xi_{m}:m\in\mathbb{N}\\}\subset{{\mathcal{M}}}^{2}([0,T];H)$ a
sequence of progressively measurable processes such that
$\mathbb{E}\int_{0}^{T}|\xi_{m}(s)-\xi(s)|_{H}^{2}\,ds\to 0$ and there exists
a constant $C>0$ such that
$\mathbb{E}\int_{0}^{T}|\xi_{m}(s)|_{H_{1}}^{2}\,ds\leq C$, $m\in\mathbb{N}$,
$H_{1}\hookrightarrow H$ compactly. Then the process

$[0,T]\ni
t\mapsto\mathfrak{S}(\xi_{m})(t)=\int_{0}^{t}{{\mathcal{T}}}(t-s)\xi_{m}(s)\tilde{\eta}(dz,ds)$

converges to $[0,T]\ni
t=\int_{0}^{t}{{\mathcal{T}}}(t-s)\xi(s)\,\tilde{\eta}(dz,ds)$ in
$\mathbb{D}(0,T;H)$.

###### Proof.

The proof follows by Theorem 7.8 [17, Chapter 3.7, p. 131] and the fact that
the set of the laws $\\{\mathfrak{S}(\xi_{m}):m\in\mathbb{N}\\}$ tight on
$\mathbb{D}([0,T];H)$ is.

∎

In order to show the identity (2.4) fix $t\in[0,T]$. The Lebesgue’s Dominated
Convergence Theorem gives

$\bar{\mathbb{E}}\phi(\bar{u}^{\ast}(t))=\bar{\mathbb{E}}\phi\left(\lim_{m\to\infty}\bar{u}_{m}(t)\right)=\lim_{m\to\infty}\bar{\mathbb{E}}\phi\left(\bar{u}_{m}(t)\right),$

for any $\phi\in{{\mathcal{C}}}_{b}(L^{2}({\mathbb{R}}^{d}))$. Since for any
$m\in\mathbb{N}$, the pairs $(\bar{u}_{m},\bar{\eta}_{m})$ have the same laws
as the random variables $(u_{m},\eta_{m})$, we can infer that

$\bar{\mathbb{E}}\phi(u^{\ast}(t))=\lim_{m\to\infty}\mathbb{E}\phi\left(u_{m}(t)\right).$

Since $u_{m}$ is a mild solution to (5),

$\bar{\mathbb{E}}\phi(u^{\ast}(t))=\lim_{m\to\infty}\mathbb{E}\phi\left((\mathfrak{T}u_{0})(t)+\mathfrak{F}_{F}(u_{m})(t)+\mathfrak{S}^{m}(u_{m})(t)+\mathfrak{F}_{z}(u_{m})(t)\right),$

where

$\displaystyle\mathfrak{S}^{m}(\xi)(t)=\int_{0}^{t}\int_{Z}{{\mathcal{T}}}(t-s)\xi(s)g(z)\;\tilde{\eta}_{m}(dz,ds),\quad
t\in[0,T].$

Again, since $(\bar{u}_{m},\bar{\eta}_{m})$, $m\in\mathbb{N}$, have the same
laws as the random variables $(u_{m},\eta_{m})$,

$\bar{\mathbb{E}}\phi(u^{\ast}(t))=\lim_{m\to\infty}\bar{\mathbb{E}}\phi\left((\mathfrak{T}u_{0})(t)+\mathfrak{F}_{F}(\bar{u}_{m})(t)+\bar{\mathfrak{S}}^{m}(\bar{u}_{m})(t)+\mathfrak{F}_{z}(\bar{u}_{m})(t)\right),$

where

$\displaystyle\bar{\mathfrak{S}}^{m}(\bar{u}_{m})(t)=\int_{0}^{t}\int_{Z}{{\mathcal{T}}}(t-s)\bar{u}_{m}(s)g(z)\;\tilde{\bar{\eta}}_{m}(dz,ds),\quad
t\in[0,T]$

and

$\displaystyle\bar{\mathfrak{F}}_{z}(\bar{u}_{m})(t)=\int_{0}^{t}\int_{Z}{{\mathcal{T}}}(t-s)\bar{u}_{m}(s)h(z){{\nu}}_{m}(dz)\,ds,\quad
t\in[0,T].$

It remains to show that $\bar{u}_{m}\to\bar{u}^{\ast}$ ${\mathbb{P}}$-a.s. in
$\mathbb{D}(0,T;H^{\gamma}_{2}({\mathbb{R}}^{d}))$ implies

$\displaystyle\lim_{m\to\infty}\bar{\mathbb{E}}\phi\left((\mathfrak{T}u_{0})(t)+\mathfrak{F}_{F}(\bar{u}_{m})(t)+\bar{\mathfrak{S}}^{m}(\bar{u}_{m})(t)+\mathfrak{F}_{z}(\bar{u}_{m})(t)\right)$
$\displaystyle=$
$\displaystyle\bar{\mathbb{E}}\phi\left((\mathfrak{T}u_{0})(t)+\mathfrak{F}_{F}(\bar{u}^{\ast})(t)+\mathfrak{S}_{\ast}(\bar{u}^{\ast})(t)+\mathfrak{F}_{z}(\bar{u}^{\ast})(t)\right)$

for any $\phi\in{{\mathcal{C}}}_{b}(L^{2}({\mathbb{R}}^{d}))$, where

$\displaystyle\mathfrak{S}_{\ast}(\bar{u}^{\ast})(t)=\int_{0}^{t}\int_{Z}{{\mathcal{T}}}(t-s)\bar{u}^{\ast}(s)g(z)\;\tilde{{\eta^{\ast}}}(dz,ds),\quad
t\in[0,T].$

First, we will show that
$\mathfrak{F}_{F}(\bar{u}_{m})\to\mathfrak{F}_{F}(\bar{u}^{\ast})$ in
$L^{2}({\mathbb{R}}^{d})$. Since we have for any $\gamma<1$

$\bar{u}_{m}\longrightarrow\bar{u}^{\ast}\quad\mbox{in}\quad\mathbb{D}(0,T;H_{2}^{\gamma}({\mathbb{R}}^{d}))\quad\mbox{as}\quad
m\to\infty,$

and $\alpha<{d+2\over d-2}$, we know by Sobolev embedding Theorems

$\bar{u}_{m}\longrightarrow\bar{u}^{\ast}\quad\mbox{in}\quad\mathbb{D}(0,T;L^{\alpha+1}({\mathbb{R}}^{d}))\quad\mbox{as}\quad
m\to\infty.$

Since $F:L^{\alpha+1}({\mathbb{R}}^{d})\to
L^{\alpha+1\over\alpha}({\mathbb{R}}^{d})$ is continuous, we obtain

$F(\bar{u}_{m})\longrightarrow F(\bar{u}^{\ast})\quad\mbox{in}\quad
L^{\infty}(0,T;L^{\alpha+1\over\alpha}({\mathbb{R}}^{d}))\quad\mbox{as}\quad
m\to\infty.$

Since $L^{\infty}(0,T)\hookrightarrow L^{q^{\prime}}(0,T)$,

$F(\bar{u}_{m})\longrightarrow F(\bar{u}^{\ast})\quad\mbox{in}\quad
L^{q^{\prime}}(0,T;L^{\alpha+1\over\alpha}({\mathbb{R}}^{d}))\quad\mbox{as}\quad
m\to\infty.$

By the Strichartz estimate we get

$\mathfrak{F}_{F}(\bar{u}_{m})\longrightarrow\mathfrak{F}_{F}(\bar{u}^{\ast})\quad\mbox{in}\quad
L^{\infty}(0,T;L^{2}({\mathbb{R}}^{d}))\quad\mbox{as}\quad m\to\infty.$

Next, we will investigate the third summand. By Step V we have
${\mathbb{P}}$-a.s.

$\bar{u}_{m}\longrightarrow\bar{u}^{\ast}\quad\mbox{in}\quad\mathbb{D}(0,T;H^{\gamma}_{2}({\mathbb{R}}^{d})),\quad\mbox{as}\quad
m\to\infty.$

By Hypothesis 1, we can infer by Theorem A.2 that

$\bar{u}_{m}g(z)\longrightarrow\bar{u}^{\ast}g(z)\quad\mbox{in}\quad\mathbb{D}(0,T;H^{\gamma}_{2}({\mathbb{R}}^{d}))\quad\mbox{as}\quad
m\to\infty.$

Since
$\int_{Z}1_{|z|>\varepsilon_{m}}|z|^{2}\nu(dz)\longrightarrow\int_{Z}|z|^{2}\nu(dz)$,
it follows $1_{|z|>\varepsilon_{m}}u_{m}z\longrightarrow u^{\ast}z$ in
${{\mathcal{M}}}^{2}(0,T,L^{2}({\mathbb{R}}^{d}))$ and therefore,

$1_{|z|>\varepsilon_{m}}\bar{u}_{m}z\longrightarrow\bar{u}^{\ast}z\quad\mbox{in}\quad{{\mathcal{M}}}^{2}(0,T;L^{2}({\mathbb{R}}^{d}))\quad\mbox{as}\quad
m\to\infty.$

Therefore it follows by Proposition 5.3

$\bar{\mathfrak{S}}^{m}(\bar{u}_{m}z)\longrightarrow\mathfrak{S}_{\ast}(\bar{u}^{\ast}z)\quad\mbox{in}\quad\mathbb{D}(0,T;L^{2}({\mathbb{R}}^{d}))\quad\mbox{as}\quad
m\to\infty.$

For the last term, one shows with the same kind of arguments that
$\bar{\mathfrak{F}}_{z}(\bar{u}_{m})$ converges to
$\bar{\mathfrak{F}}_{z}(\bar{u}^{\ast})$, as $m\to\infty$, in
$L^{\infty}(0,T;L^{2}({\mathbb{R}}^{d}))$. Hence, we have for all $t\in[0,T]$,

$\bar{\mathbb{E}}\phi(\bar{u}^{\ast}(t))=\bar{\mathbb{E}}\phi\left((\mathfrak{T}u_{0})(t)+\mathfrak{F}_{F}(\bar{u}^{\ast})(t)+\mathfrak{S}_{\ast}(\bar{u}^{\ast})(t)+\mathfrak{F}_{z}(\bar{u}^{\ast})(t)\right),$

Since $\bar{u}^{\ast}$ and $\bar{u}_{m}$ belong ${\mathbb{P}}$-a.s. to
$\mathbb{D}(0,T;L^{2}({\mathbb{R}}^{d}))$, it follows by Theorem 7.8 [17, p.
131] that $\bar{u}_{m}\to\bar{u}^{\ast}$ in
$\mathbb{D}(0,T;L^{2}({\mathbb{R}}^{d}))$. Hence, $\bar{u}^{\ast}$ is indeed a
solution to (2.4). ∎

##  Appendix A Multiplication

In the section we recall some well known facts concerning Nemytskii operators,
which are necessary to prove our main result. Most of the content is taken
from Runs and Sickel [23].

First let us introduce some functions spaces. Let $1\leq p\leq\infty$ and
$m\in\mathbb{N}_{0}$, then $W^{m}_{p}$ denotes the Sobolev spaces defined by

$W_{p}^{m}({\mathbb{R}}^{d}):=\left\\{f\in
L^{p}({\mathbb{R}}^{d}):|f|_{W^{p}_{m}}=|f|_{L^{p}}+\sum_{|\alpha|\leq
m}|D^{\alpha}f|_{L^{p}}<\infty\right\\}.$

Let $1\leq p<\infty$ and $s>0$, $s\not\in\mathbb{N}$, then $W^{s}_{p}$ denotes
the Slobodeckij spaces defined by

$\displaystyle W_{p}^{s}({\mathbb{R}}^{d}):=\left\\{f\in
L^{p}({\mathbb{R}}^{d}):|f|_{W^{p}_{m}}^{p}=|f|_{W^{[s]}_{p}}^{p}\right.$
$\displaystyle\left.{}+\sum_{|\alpha|=[s]}\int_{{\mathbb{R}}^{d}}\int_{{\mathbb{R}}^{d}}{|D^{\alpha}f(x)-D^{\alpha}f(y)|^{p}\over|x-y|^{n+(s-[s])p}}\,dx\,dy<\infty\right\\}.$

Let $1\leq p<\infty$ and $s\in{\mathbb{R}}$, then $H^{s}_{p}$ denotes the
Bessel Potential spaces or Sobolev spaces of fractional order defined by

$\displaystyle H_{p}^{s}({\mathbb{R}}^{d}):=\left\\{f\in
L^{p}({\mathbb{R}}^{d}):|f|_{H^{s}_{p}}=|{{\mathcal{F}}}^{-1}((1+\xi^{2})^{\frac{s}{2}}{{\mathcal{F}}}f)|_{L^{p}}<\infty\right\\}.$

Finally, let us introduce the Triebel Lizorkin spaces
$F_{p,q}^{s}({\mathbb{R}}^{d})$ and the Besov spaces
$B_{p,q}^{s}({\mathbb{R}}^{d})$ by

Let us shortly recall some known identities. The proof can be found e.g.in
[24].

  * •

$L_{p}({\mathbb{R}}^{d})=F_{p,2}^{0}({\mathbb{R}}^{d})$ for $1<p<\infty$,

  * •

$W_{p}^{m}({\mathbb{R}}^{d})=F_{p,2}^{m}({\mathbb{R}}^{d})$ for $1<p<\infty$,
$m\in\mathbb{N}$,

  * •

$W_{p}^{s}({\mathbb{R}}^{d})=F_{p,p}^{s}({\mathbb{R}}^{d})=B_{p,s}^{s}({\mathbb{R}}^{d})$
for $1<p<\infty$, $s>0$, $s\not\in\mathbb{N}$,

  * •

$H_{p}^{s}({\mathbb{R}}^{d})=F_{p,2}^{s}({\mathbb{R}}^{d})$ for $1<p<\infty$,
$s\in{\mathbb{R}}$.

In order to treat the nonlinearity, we list here some useful results. Assume
$s_{1}<0<s_{2}$.

######  Theorem A.1.

(see [23, p. 229]) Assume $s=s_{1}\leq s_{2}$,
$s_{1}+s_{2}>d\cdot\max(0,\frac{1}{p}-1)$, and $q\geq\max(q_{1},q_{2})$. Then

  * •

if $s_{2}>s_{1}$, then $F_{p,q_{1}}^{s_{1}}({\mathbb{R}}^{d})\cdot
B_{\infty,q_{2}}^{s_{2}}({\mathbb{R}}^{d})\hookrightarrow
F_{p,q_{1}}^{s_{1}}({\mathbb{R}}^{d})$;

  * •

if $s_{1}=s_{2}$ then $F_{p,q_{1}}^{s_{1}}({\mathbb{R}}^{d})\cdot
B_{\infty,q_{2}}^{s_{1}}({\mathbb{R}}^{d})\hookrightarrow
F_{p,q}^{s_{1}}({\mathbb{R}}^{d})$

######  Theorem A.2.

(see [23, p. 238]) Let $s>0$,

$\frac{1}{r_{1}}=\frac{1}{d}\left(\frac{d}{p_{1}}-s\right)>0,\quad\mbox{ and
}\quad\frac{1}{r_{2}}=\frac{1}{d}\left(\frac{d}{p_{2}}-s\right)>0,$

and

$\frac{1}{r_{1}}+\frac{1}{r_{2}}=\frac{1}{r}=\frac{1}{d}\left(\frac{d}{p}-s\right)<1.$

Then

$F_{p_{1},q_{1}}^{s}({\mathbb{R}}^{d})\cdot
F_{p_{2},q_{2}}^{s}({\mathbb{R}}^{d})\hookrightarrow
F_{p,q}^{s}({\mathbb{R}}^{d}),$

iff

$\max(q_{1},q_{2})\leq q\leq\infty.$

In addition,

$B_{p_{1},q_{1}}^{s}({\mathbb{R}}^{d})\cdot
B_{p_{2},q_{2}}^{s}({\mathbb{R}}^{d})\hookrightarrow
F_{p,q}^{s}({\mathbb{R}}^{d}),$

iff

$\max(q_{1},q_{2})\leq q\leq\infty,\quad\mbox{ and }\quad 0<q_{1}\leq
r_{1},\quad 0<q_{2}\leq r_{2}.$

##  Appendix B A Tightness criteria in $\mathbb{D}([0,T];Y)$

Let $Y$ be a separable and complete metric space and $T>0$. The space
$\mathbb{D}([0,T];Y)$ denotes the space of all right continuous functions
$x:[0,T]\to Y$ with left limits. The space of continuous functions is usually
equipped with the uniform topology. But, since $\mathbb{D}([0,T];Y)$ is
complete but not separable in the uniform topology, we equip
$\mathbb{D}([0,T];Y)$ with the Skorohod topology in which
$\mathbb{D}([0,T];Y)$ is both separable and complete. For more information
about Skorokhod space and topology we refer to Billingsley’s book [7] or
Ethier and Kurtz [17]. In this appendix we only state the following tightness
criterion which is necessary for our work. For this we denote by
${\mathcal{P}}\left(\mathbb{D}([0,T];Y)\right)$ the space of Borel probability
measures on $\mathbb{D}([0,T];Y)$.

######  Corollary B.1.

Let $\\{x_{n}:n\in{\mathbb{N}}\\}$ be a sequence of càdlàg processes, each of
the process defined on a probability space
$(\Omega_{n},{\mathcal{F}}_{n},\mathbb{P}_{n})$. Then the sequence of laws of
$\\{x_{n}:n\in{\mathbb{N}}\\}$ is tight on $\mathbb{D}([0,T];Y)$ if

  1. a.)

there exists a space $Y_{1}$, $Y_{1}\hookrightarrow Y$ compactly, such that
such that

$\mathbb{E}^{n}\left|x_{n}(t)\right|_{Y_{1}}^{r}\leq C,\,\forall
n\in{\mathbb{N}};$

  2. b.)

there exist two constants $c>0$ and $\gamma>0$ and a real number $r>0$ such
that for all $\theta>0$, $t\in[0,T-\theta]$, and $n\geq 0$

${\mathbb{E}}_{n}\sup_{t\leq s\leq t+\theta}|x_{n}(t)-x_{n}(s)|_{Y}^{r}\leq
c\,\theta^{\gamma}.$

###### Proof.

The inequality B.1-(a) and the Chebyscheff inequality gives the necessary
conditions for the compact containment condition. Next, comparing with [17,
Theorem 7.2, p. 128]. Now fix $t_{1}\leq t\leq t_{2}$. Then

$\displaystyle\mathbb{P}_{n}\left(|x_{n}(t)-x_{n}(t_{1})|_{Y}\geq\lambda,\,|x_{n}(t)-x_{n}(t_{2})|_{Y}\geq\lambda\right)$
$\displaystyle\leq\mathbb{P}_{n}\left(\sup_{t_{1}\leq s\leq
t_{2}}|x_{n}(s)-x_{n}(t_{1})|_{Y}\geq\lambda\right).$

Estimating the RHS by the Chebyshev inequality and using inequality B.1-(b)
leads to inequality the second condition in [17, Theorem 7.2, p. 128]. Thus
the assertion follows. ∎

##  Appendix C Compactness methods

######  Lemma C.1.

Let $1\leq p<\infty$. A set ${{\mathcal{A}}}\subset L^{p}({\mathbb{R}}^{d})$
is compact, if

  1. a.)

there exists a $\gamma>0$ such that ${{\mathcal{A}}}$ is bounded in
$L_{\gamma}^{p}({\mathbb{R}}^{d})$;

  2. b.)

there exist numbers $\delta\geq 0$, $s>0$, a sequence $R_{n}$ with $R_{n}\leq
R_{n+1}$ and $R_{n}\uparrow\infty$, and a constant $C>0$ such that

$\sup_{n\geq
1}\frac{1}{R_{n}^{\delta}}\left|v1_{B(R_{n})}\right|_{H^{s}_{2}(B(R_{n}))}\leq
C,\quad v\in{{\mathcal{A}}}.$

###### Proof of Lemma C.1:.

Let $\\{u_{n}:n\in\mathbb{N}\\}\subset{{\mathcal{A}}}$ be a sequence. Then, we
have to show that there exists a subsequence $\\{n_{k}:k\in\mathbb{N}\\}$ and
a $u^{\ast}$ such that

$u_{n_{k}}\longrightarrow u^{\ast},\quad\mbox{for}\quad
k\to\infty\quad\mbox{in}\quad L^{p}({\mathbb{R}}^{d}),$

or, that there exists a subsequence $\\{n_{k}:k\in\mathbb{N}\\}$ which is a
Cauchy sequence in $L^{p}({\mathbb{R}}^{d})$. The existence of a unique limit
$u^{\ast}$ follows.

In the next steps we will construct a subsequence, and then show that this
subsequence is a Cauchy sequence. Since $\\{u_{n}:n\in\mathbb{N}\\}$ is
bounded in $L^{p}({\mathbb{R}}^{d})$, there exists a subsequence
$\\{n_{k}:k\in\mathbb{N}\\}$ and $a^{\ast}\in[0,\infty)$ such that
$|u_{n_{k}}|^{p}_{L^{p}({\mathbb{R}}^{d})}\to a^{\ast}$ as $k\to\infty$. If
$a^{\ast}=0$, then $\\{u_{n_{k}}:k\in\mathbb{N}\\}$ converges to zero and we
are done. Let us assume $a^{\ast}>0$.

Next, we can assume, that
$|u_{n_{k}}|^{p}_{L^{p}}\in(\frac{1}{2}a^{\ast},\frac{3}{2}a^{\ast})$ for all
$k\in\mathbb{N}$. Let $\\{R_{m}:m\in\mathbb{N}\\}$ a sequences in
${\mathbb{R}}_{0}^{+}$ with $R_{m}\uparrow\infty$, such that

$\int_{{\mathbb{R}}^{d}\setminus
B(R_{m})}|u_{n}(x)|^{p}\,dx\leq\frac{1}{4}\,2^{-m}|u_{n}|^{p}_{L^{p}},\quad\forall\,n\in\mathbb{N},\,\,m\in\mathbb{N}.$

Due to the first condition on ${{\mathcal{A}}}$ such a sequence exists. In
particular, it follows by an application of the Chebyscheff inequality and the
fact that $|u_{n_{k}}|^{p}_{L^{p}}\geq\frac{1}{2}a^{\ast}$. Since
${{\mathcal{A}}}$ is bounded in $H^{s}_{p}(B(R_{1}))$ and
$H^{s}_{p}(B(R_{1}))\hookrightarrow L^{p}(B(R_{1}))$ compactly, there exists a
subsequence $\\{n^{1}_{k}:k\in\mathbb{N}\\}$ of $\\{n_{k}:k\in\mathbb{N}\\}$
such that $\\{u_{n^{1}_{k}}:k\in\mathbb{N}\\}$ is a Cauchy sequence in
$L^{p}(B(R_{1}))$.

Again, since ${{\mathcal{A}}}$ is bounded in $H^{\gamma}_{p}(B(R_{2}))$ and
$H^{\gamma}_{p}(B(R_{2}))\hookrightarrow L^{p}(B(R_{2}))$ compactly, there
exists a subsequence $\\{n^{2}_{k}:k\in\mathbb{N}\\}$ of
$\\{n^{1}_{k}:k\in\mathbb{N}\\}$ such that
$\\{u_{n^{2}_{k}}:k\in\mathbb{N}\\}$ is a Cauchy sequence in
$L^{p}(B(R_{2}))$.

Proceeding in this way we obtain subsequences
$\\{n^{m}_{k}:k\in\mathbb{N}\\}$, $m\in\mathbb{N}$, such that

  * •

$\\{n_{k}^{1}:k\in\mathbb{N}\\}\supset\\{n_{k}^{2}:k\in\mathbb{N}\\}\supset\\{n_{k}^{3}:k\in\mathbb{N}\\}\supset\ldots$;

  * •

for each fixed $m\in\mathbb{N}$, $\\{u_{n^{m}_{l}}:l\in\mathbb{N}\\}$ is a
Cauchy sequence in $L^{p}(B(R_{m}))$;

  * •

for each fixed $m\in\mathbb{N}$ we have $\int_{{\mathbb{R}}^{d}\setminus
B(R_{m})}|u_{n}(x)|^{p}\,dx\leq 2^{-m}|u_{n}|^{p}_{L^{p}},\,\forall
n\in\mathbb{N}$;

Let $\\{\tilde{n}_{k}:k\in\mathbb{N}\\}$ be the diagonal sequence defined for
$k\in\mathbb{N}$ by $\tilde{n}_{k}:=n_{k}^{k}$. Now, we claim that
$\\{u_{\tilde{n}_{k}}:k\in\mathbb{N}\\}$ is a Cauchy sequence in
$L^{p}({\mathbb{R}}^{d})$. In order to show it, fix $\varepsilon>0$. The task
is now to find an index $k_{1}\in\mathbb{N}$ such that

$\int_{{\mathbb{R}}^{d}}|u_{n_{k}}-u_{n_{l}}|^{p}\,dx\leq\varepsilon,\quad\forall\,l,k\geq
k_{1}.$

Let $m\in\mathbb{N}$ be the smallest integer such that
$m\geq\ln_{2}(a^{\ast})-\ln_{2}(\frac{\varepsilon}{6})$. Since
$\\{u_{n^{m}_{k}}:k\in\mathbb{N}\\}$ is a Cauchy sequence in $L^{p}(B(R_{m}))$
and $\\{\tilde{u}_{n_{k}}:k\geq
m\\}\subset\\{u_{n^{m}_{k}}:k\in\mathbb{N}\\}$, it follows that
$\\{\tilde{u}_{n_{k}}:k\geq m\\}$ is a Cauchy sequence in $L^{p}(B(R_{m}))$.
Therefore, there exists a $k_{1}\geq m$ such that

$\int_{B(R_{m})}|u_{n_{k}}-u_{n_{l}}|^{p}\,dx\leq\frac{\varepsilon}{2},\quad
k,l\geq k_{1}.$

Then we have for any $k,l\geq k_{1}$

$\displaystyle\int_{{\mathbb{R}}^{d}}|u_{n_{k}}-u_{n_{l}}|^{p}\,dx\leq\int_{B(R_{m})}|u_{n_{k}}-u_{n_{l}}|^{p}\,dx+\int_{{\mathbb{R}}^{d}\setminus
B(R_{m})}|u_{n_{k}}-u_{n_{l}}|^{p}\,dx.$

By the choice of $m$ we have

$\displaystyle\int_{{\mathbb{R}}^{d}\setminus
B(R_{m})}|u_{n_{k}}-u_{n_{l}}|^{p}\,dx$ $\displaystyle\leq$
$\displaystyle\int_{{\mathbb{R}}^{d}\setminus
B(R_{m})}|u_{n_{k}}|^{p}+\int_{{\mathbb{R}}^{d}\setminus
B(R_{m})}|u_{n_{l}}|^{p}\,dx$ $\displaystyle\leq
2^{-m}|u_{n_{k}}|^{p}+2^{-m}|u_{n_{l}}|^{p}$ $\displaystyle\leq$
$\displaystyle 2^{-m+1}\frac{3}{2}a^{\ast}\leq\varepsilon/2.$

Collecting altogether, we get

$\displaystyle\int_{{\mathbb{R}}^{d}}|u_{n_{k}}-u_{n_{l}}|^{p}\,dx\leq\frac{\varepsilon}{2}+\frac{\varepsilon}{2}=\varepsilon,$

and the assertion follows.

∎

## References

  * [1] F. Kh. Abdullaev, B.B. Baizakov, and V.V. Konotop.  Dynamics of a bose-einstein condensate in optical trap.  In Nonlinearity and Disorder: Theory and Applications, pages 69–78. Springer, 2001.
  * [2] G. P. Agrawal.  Nonlinear fiber optics.  Academic press, 2007.
  * [3] D. Applebaum.  Lévy processes and stochastic calculus. 2nd ed. Cambridge: Cambridge University Press, 2nd ed. edition, 2009.
  * [4] O. Bang, P. L Christiansen, K.Ø. Rasmussen, and Y.B. Gaididei.  Temperature effects in a nonlinear model of monolayer scheibe aggregates.  Physical Review E, 49(5):4627, 1994.
  * [5] V. Barbu, M. Röckner, and D. Zhang.  Stochastic nonlinear Schrödinger equations with linear multiplicative noise: rescaling approach.  J. Nonlinear Sci., 24(3):383–409, 2014.
  * [6] J. Bergh and J. Löfström.  Interpolation spaces. An introduction.  Springer-Verlag, Berlin-New York, 1976.  Grundlehren der Mathematischen Wissenschaften, No. 223.
  * [7] Patrick Billingsley.  Convergence of probability measures.  Wiley Series in Probability and Statistics: Probability and Statistics. John Wiley & Sons, Inc., New York, second edition, 1999.  A Wiley-Interscience Publication.
  * [8] Z. Brzeźniak and E. Hausenblas.  Uniqueness in law of the Itô integral with respect to Lévy noise.  In Seminar on Stochastic Analysis, Random Fields and Applications VI, volume 63 of Progr. Probab., pages 37–57. Birkhäuser/Springer Basel AG, Basel, 2011.
  * [9] Z. Brzeźniak, E. Hausenblas, and P. Razafimandimby.  Martingale solutions for stochastic equation of reaction diffusion type driven by lévy noise or poisson random measure, 2015, Archive.
  * [10] T. Cazenave.  Semilinear Schrödinger equations, volume 10 of Courant Lecture Notes in Mathematics.  New York University Courant Institute of Mathematical Sciences, New York, 2003.
  * [11] R. Cont and P. Tankov.  Financial modelling with jump processes.  Chapman & Hall/CRC Financial Mathematics Series. Chapman & Hall/CRC, Boca Raton, FL, 2004.
  * [12] A. de Bouard and A. Debussche.  A stochastic nonlinear Schrödinger equation with multiplicative noise.  Comm. Math. Phys., 205(1):161–181, 1999.
  * [13] A. de Bouard and A. Debussche.  The stochastic nonlinear Schrödinger equation in $H^{1}$.  Stochastic Anal. Appl., 21(1):97–126, 2003.
  * [14] A. de Bouard and A. Debussche.  Blow-up for the stochastic nonlinear Schrödinger equation with multiplicative noise.  Ann. Probab., 33(3):1078–1110, 2005.
  * [15] A. de Bouard and A. Debussche.  The nonlinear Schrödinger equation with white noise dispersion.  J. Funct. Anal., 259, 2010.
  * [16] W.-P. Düll, G. Schneider, and C. E. Wayne.  Justification of the nonlinear Schrödinger equation for the evolution of gravity driven 2D surface water waves in a canal of finite depth.  Arch. Ration. Mech. Anal., 220(2):543–602, 2016.
  * [17] S. N. Ethier and T. G. Kurtz.  Markov processes. Characterization and convergence. Hoboken, NJ: John Wiley & Sons, 2005.
  * [18] G.E. Falkovich, I. Kolokolov, V. Lebedev, and S.K. Turitsyn.  Statistics of soliton-bearing systems with additive noise.  Physical Review E, 63(2):025601, 2001.
  * [19] D. Haroske and H. Triebel.  Distributions, Sobolev spaces, elliptic equations.  EMS Textbooks in Mathematics. European Mathematical Society (EMS), Zürich, 2008.
  * [20] N. Ikeda and S. Watanabe.  Stochastic differential equations and diffusion processes, volume 24 of North-Holland Mathematical Library.  North-Holland Publishing Co., Amsterdam; Kodansha, Ltd., Tokyo, second edition, 1989.
  * [21] Y. Kodama and A. Hasegawa.  Amplification and reshaping of optical solitons in glass fiber iii. amplifiers with random gain.  Optics letters, 8(6):342–344, 1983.
  * [22] J. V. Moloney and A. C. Newell.  Nonlinear optics.  Westview Press, 2004.
  * [23] T. Runst and W. Sickel.  Sobolev spaces of fractional order, Nemytskij operators, and nonlinear partial differential equations, volume 3 of de Gruyter Series in Nonlinear Analysis and Applications.  Walter de Gruyter & Co., Berlin, 1996.
  * [24] H. Triebel.  Theory of function spaces, volume 78 of Monographs in Mathematics.  Birkhäuser Verlag, Basel, 1983.
  * [25] J. Villarroel and M. Montero.  On the effect of random inhomogeneities in kerr media modelled by a nonlinear schrödinger equation.  Journal of Physics B: Atomic, Molecular and Optical Physics, 43(13):135404, 2010.
  * [26] J. Villarroel and M. Montero.  On the integrability of the Poisson driven stochastic nonlinear Schrödinger equations.  Stud. Appl. Math., 127(4):372–393, 2011.
  * [27] V.E. Zakharov.  Stability of periodic waves of finite amplitude on the surface of a deep fluid.  Sov. Phys. J. Appl. Mech. Tech. Phys., 4:190–194, 1968.