# On nonlinear Schrödinger equations with random potentials: existence and
probabilistic properties

Leandro Cioletti
Departamento de Matemática, UnB, 70910-900 Brasília, Brazil.
<EMAIL_ADDRESS>
Lucas C. F. Ferreira
Universidade Estadual de Campinas, IMECC - Departamento de Matemática,
Rua Sérgio Buarque de Holanda, 651, CEP 13083-859, Campinas-SP, Brazil.
<EMAIL_ADDRESS>
Marcelo Furtado
Departamento de Matemática, UnB, 70910-900 Brasília, Brazil.
<EMAIL_ADDRESS>
###### Abstract

In this paper we are concerned with nonlinear Schrödinger equations with
random potentials. Our class includes continuum and discrete potentials.
Conditions on the potential $V_{\omega}$ are found for existence of solutions
almost sure $\omega$. We study probabilistic properties like central limit
theorem and law of larger numbers for the obtained solutions by independent
ensembles. We also give estimates on the expected value for the
$L^{\infty}$-norm of the solution showing how it depends on the size of the
potential.

AMS 2000 subject classification: 47B80, 60H25, 35J60, 35R60, 82B44, 47H10
Keywords: Random potentials; Random nonlinear equations; Schrödinger operators

##  1 Introduction

A class of models that appears naturally in a wide number of phenomena are the
random differential equations. This occurs because randomness is a powerful
tool and concept to control complex systems involving a large number of
variables and particles. The basic idea is describe complex systems by means
of their statistical properties. Another kind of phenomena are those governed
by quantum mechanics and uncertainty principle. In this direction, we have
Schrödinger equations, and their random versions, which are core in the study
of condensed matter.

In this paper we are concerned with a random version of the nonlinear
Schrödinger equation

$ih\dfrac{\partial\psi}{\partial
t}=-h^{2}\Delta\psi+V(x)\psi-|\psi|^{p-1}\psi,\leavevmode\nobreak\
\leavevmode\nobreak\ x\in\mathbb{R}^{n},$ (1.1)

where $t\in\mathbb{R}$, $n\geq 3$, $p>1$, $h$ is the Planck constant and $i$
is the imaginary unit. When looking for standing wave solutions, namely those
which have the special form $\psi(x,t):=e^{-i\frac{E}{h}t}u(x)$, with
$E\in\mathbb{R}$, we are leading to solve the following stationary equation

$-\Delta u+V(x)u=|u|^{p-1}u,\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ x\in\mathbb{R}^{N}.$

From the physical viewpoint, the function $V$ is the potential energy, and
therefore the force acting on the system is given by $F(x)=-\nabla V(x)$. In
the deterministic case, there are many papers concerning existence,
multiplicity and qualitative properties for the solution of the above equation
(see [19, 11, 2, 1] and references therein).

The main interest of this paper is to study situations where the potential $V$
is not deterministic. Worth to mention that during the last thirty years,
random Schrödinger operators, which originated in condensed matter physics,
have been studied intensively by physicists and mathematicians. The theory is
at the crossroads of a number of mathematical fields: the theory of operators,
partial differential equations, the theory of probabilities and also
stochastic process. This paper aims to prove the existence and probabilistic
properties of bounded solutions for the random equation

$\left\\{\begin{array}[]{rcll}-\Delta
u+V_{\omega}(x)u&=&b(x)u|u|^{p-1}+g(x),&\text{if}\ x\in U;\\\\[4.26773pt]
u&=&0,&\text{if}\ x\in\partial U,\end{array}\right.$ (1.2)

where $V_{\omega}$ is a random variable, $U\subset\mathbb{R}^{n}$ is a bounded
domain and the terms $b,\,g\in L^{\infty}(U)$ are deterministic. In fact, the
boundedness of $U$ is not essential and could be circumvented by working in
weighted $L^{\infty}$-spaces or Lebesgue spaces $L^{s}(\mathbb{R}^{n})$ with
$s\neq\infty$ (see [13, 14]). However, here this condition will simplify
matters a bit. The random potential $V_{\omega}$ is constructed via a
convolution with a realization of a random variable valued in the finite
random measure space. Precisely, given a continuous function
$f:\mathbb{R}^{N}\rightarrow\mathbb{R}$ we consider

$V_{\omega}(x):=\int_{U}f(x-y)\,d{\mu_{\omega}}(y)$ (1.3)

where $\mu_{\omega}$ is a $\mathcal{M}(U)$-valued random variable.

We present here some examples of (1.3) that have been treated in the
literature (see e.g. the review [17]). We first consider a model of an
unordered alloy, that is, a mixture of several materials with atoms located at
lattice positions. If we assume that the type of atom at the lattice
$i\in\mathbb{Z}^{n}$ is random we are leading to consider the following type
of potential

$V_{\omega}(x)=\sum_{i\in\mathbb{Z}^{n}}q_{i}(\omega)f(x-i),$ (1.4)

where the random variables $q_{i}$ describe the charge of the atom at the
position $i$ of the lattice. Other example can be obtained if we consider
materials like glass or rubber, where the position of the atoms of the
material are located at random points $\eta_{i}$ in space. By normalizing the
charge of the atoms, the suggested potential is formally

$V_{\omega}(x)=\sum_{i\in\mathbb{Z}^{n}}f(x-\eta_{i}(\omega)),$ (1.5)

where the $\eta_{i}(\omega)$ are random variables which localize the atoms in
the spaces.

The class of potentials allowed here is sufficient large to consider many
known models. For example, the case of glass considered in (1.5) can be
obtained if we take the random point measure
$\mu_{\omega}=\sum_{i}\delta_{\eta_{i}(\omega)}$. Actually, for this choice of
the measure we have that

$\sum_{i\in\mathbb{Z}^{n}\cap
U}f(x-\eta_{i}(\omega))=\int_{U}f(x-\eta)d\mu_{\omega}(\eta).$ (1.6)

Also, a combination of potentials like (1.4) and (1.5), namely
$\Sigma_{{}_{i\in\mathbb{Z}^{n}\cap U}}q_{i}(\omega)f(x-\eta_{i}(\omega))$
(see [8]), is also covered by (1.3) with
$\mu_{\omega}=\Sigma_{{}_{i\in\mathbb{Z}^{n}\cap
U}}q_{i}(\omega)\delta_{\eta_{i}(\omega)}$. It is not difficult to see that we
can also consider other models like, e.g., the Poisson model (see [17] for
more examples).

The models (1.4) and (1.5) correspond to discrete measures $\mu_{\omega}$ and
results for them about localization, spectral properties or decays can be
found in [5, 8, 15, 17, 20]. For Schrödinger equations defined in a lattice,
that is $x\in\mathbb{Z}^{n}$, we refer the reader to [4, 6]. Considering a
random time-dependent potential for (1.1), the authors of [3] studied
asymptotic behavior of solutions by showing convergence for stochastic
Gaussian limits when the two-point correlation function of the potential is
rapidly decaying. Still for time random potentials, scaling limits for
parabolic waves in random media were investigated in [12]. Despite important
progress in the last years, there is still a lack of result for random
equations, including Schrödinger ones (see [7]), mainly with respect to the
continuum case which seems to be harder-to-treating. Another type of random
equations are the parabolic ones, for which we refer the works [9, 10] and
their references.

In this paper we find conditions on the potential $V_{\omega}$ for the
nonlinear equation (1.2) having solutions almost sure $\omega.$ The solution
are understood in an integral sense coming from Green functions. From Theorem
3.5 we see how the expected value of the $L^{\infty}$-norm of solutions
depends on the size of potential. Our results also cover continuum random
potential like, among others, the examples given in Remark 3.3 and Theorem
3.4. Moreover, we study probabilistic properties like central limit theorem
and law of larger numbers for the obtained solutions by independent ensembles.
It is worthwhile to mention that, when dealing with the random variable
$\omega\mapsto u(x,\omega)$ which maps an element of $\Omega$ in the solution
of (1.2) associated with the random potential $V_{\omega}$, we need to extend
some known concepts of real random variables for that taking values in a more
general Banach space. We refer to Section 2 for more details.

As a further comment, we observe that the random potentials considered in this
paper are built from a very general probability space. In this setting does
not always make sense to ask what is the probability that the problem (1.2)
has an unique solution in $L^{\infty}(U)$. In order to give some sense to this
question we should restrict ourself to probability spaces
$(\Omega,\mathcal{F},\mathbb{P})$ and random potentials $V$ where the set

$\\{\omega\in\Omega:\ \text{the problem \eqref{eqdif-est1} has a unique
solution in}\ L^{\infty}(U)\\}$

is an event (measurable). Working in such probability spaces Theorem 3.2 give
us immediately a lower bound for the probability that the non-linear problem
(1.2) has a unique solution.

The manuscript is organized as follows. In the next section, we introduce some
notations, basic definitions and give some properties for an integral operator
associated with the random potential $V_{\omega}.$ The results are stated and
proved in Section 3.

##  2 Preliminaries and notation

Throughout this paper $(\Omega,\mathcal{F},\mathbb{P})$ denotes a given
complete probability space. If $(E,\mathcal{E})$ is a measurable space, any
$(\mathcal{F},\mathcal{E})$-measurable function $X:\Omega\rightarrow E$ will
be called a $E$-valued random variable. We use the abbreviation a.s. for
almost surely or almost sure.

Let $U\subset\mathbb{R}^{n}$ be a bounded domain. We adopt the standard
notation $\mathcal{M}(U)$ to denote the set of all Random measures over $U$
having finite variation and we call $\mathscr{B}(\mathcal{M}(U))$ the
$\sigma$-algebra of the borelians of $\mathcal{M}(U)$ generated by the total
variation norm. The space of all bounded continuous real-valued functions
defined on $U$ will be denoted by $BC(U)$. Since $BC(U)$ is a metric space
with the supremum norm, when we refer to a $BC(U)$-valued random variable, the
$\sigma$-algebra we are considering is always the one generated by the
borelians. Similarly to a $\mathcal{X}$-valued Borel random variable
$X:\Omega\rightarrow\mathcal{X},$ where $\mathcal{X}$ is an arbitrary metric
space.

The random potentials considered here are the $BC(U)$-valued random variables
defined as follows. Take any random variable
$X:\Omega\rightarrow\mathcal{M}(U)$ (which is simply a random measure in
$\mathcal{M}(U)$) and a fixed function $f\in BC(\mathbb{R}^{n})$. Then, for
$\mu_{\omega}=X(\omega)$, the function $V:\Omega\rightarrow BC(U)$ defined by

$V_{\omega}(x):=\int_{U}f(x-y)\,d\mu_{\omega}(y),\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ x\in U,$

is a $BC(U)$-valued random variable that will be called a random potential. To
see that $V$ is a well-defined $BC(U)$-valued random variable, is enough to
consider the mapping $T_{f}:\mathcal{M}(U)\rightarrow BC(U)$ given by

$T_{f}(\mu)(x)=\int_{U}f(x-y)\,d\mu(y),\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ x\in U,$

and to observe that $V=T_{f}\circ X$. In fact, if we denote by
$\rule[-2.27621pt]{1.13809pt}{9.6739pt}\,\mu\,\rule[-2.27621pt]{1.13809pt}{9.6739pt}$
the total variation of the measure $\mu$, the inequality

$\|T_{f}(\mu)\|_{\infty}:=\sup_{x\in
U}|T_{f}(\mu)(x)|\leq\left(\sup_{x\in\mathbb{R}^{n}}\left|f(x)\right|\right)\,\rule[-2.27621pt]{1.13809pt}{9.6739pt}\,\mu\,\rule[-2.27621pt]{1.13809pt}{9.6739pt}$
(2.1)

implies that $T_{f}$ is a continuous and Borel measurable function. Since $V$
is a composition of two Borel measurable functions, $V$ is a $BC(U)$-valued
random variable.

As usual, if $(U,\mathscr{B},\mu)$ is a measure space, we define

$\|f\|_{L^{\infty}(U,d\mu)}=\inf\left\\{a\geq
0:\mu(\\{x:|f(x)|>a\\})=0\right\\}$

and the space $L^{\infty}(U,\mathscr{B}(U),\mu)$ as being the set

$\\{f:U\rightarrow\mathbb{R}:f\ \text{is Borel measurable and}\
\|f\|_{L^{\infty}(U,d\mu)}<\infty\\}.$

When $d\mu=dx$ is the Lebesgue measure in $U\subset\mathbb{R}^{n},$ we simply
denote $L^{\infty}(U)=L^{\infty}(U,\mathscr{B}(U),dx)$. Although we are
assuming that $f\in BC(\mathbb{R}^{n})$, most of the results presented here
are also valid if we suppose only the weaker condition
$f\in\cap_{\mu\in\mathcal{M}(U-U)}L^{\infty}(U-U,\mathscr{B}(U-U),\mu)$.

In order to state some convergence results obtained in this paper we need to
use the notion of Bochner integrals. Let
$(\mathcal{X},\|\cdot\|_{\mathcal{X}})$ be a Banach space and
$(\Omega,\mathcal{F},\mathbb{P})$ be a probability space. If
$X:\Omega\rightarrow\mathcal{X}$ is a $\mathcal{X}$-valued Borel random
variable such that $X=Y$ a.s. in $\Omega,$ where
$Y:\Omega\rightarrow\mathcal{X}$ is a $\mathcal{X}$-valued Borel random
variable with $Y(\Omega)\subset\mathcal{X}$ separable, and

$\int_{\Omega}\|X(\omega)\|_{\mathcal{X}}\,d\mathbb{P}(\omega)<\infty,$

then there exist a unique element $\mathbb{E}[X]\in\mathcal{X}$ with the
property

$\ell(\mathbb{E}[X])=\int_{\Omega}\ell(X(\omega))\,d\mathbb{P}(\omega)$

for all $\ell\in\mathcal{X}^{\ast}$, where $\mathcal{X}^{\ast}$ is the dual of
$\mathcal{X}$. Following the standard notation we write

$\mathbb{E}[X]=\int_{\Omega}X(\omega)\,d\mathbb{P}(\omega).$

We call $\mathbb{E}[X]$ the Bochner integral of $X$ with respect to
$\mathbb{P}$. More details about the existence and some properties of this
integral can be found in [16, 18]. For these $\mathcal{X}$-valued random
variables we define the convergence in probability similarly to the real-
valued case, that is, if $\\{X_{j}\\}$ is a sequence of $\mathcal{X}$-valued
random variable we say that $X_{j}$ converges to a $\mathcal{X}$-valued random
variable $X$ in probability if for all $\varepsilon>0$, we have

$\lim_{j\rightarrow\infty}\mathbb{P}(\\{\omega\in\Omega:\|X_{j}(\omega)-X(\omega)\|_{\mathcal{X}}\geq\varepsilon\\})=0.$
(2.2)

When $X$ is real-valued random variable, we use the usual notation and denote
the expected value of $X$ and its variance by

$\mathbb{E}[X]:=\int_{\Omega}X(\omega)\,d\mathbb{P}(\omega)\text{ \ \ and \
Var}\,X:=\mathbb{E}[(\mathbb{E}[X]-X)^{2}],$

respectively. For the both senses of expectation presented above we also use
the notation

$\mathbb{E}_{A}[X]=\int_{A}X(\omega)\,d\mathbb{P}(\omega),$

whenever $A\subset\Omega$ is measurable and the right-hand-side of the
expression makes sense.

Let $X$ and $Y$ be two $E$-valued random variable in the same probability
space. We say that they are identically distributed if for all
$A\in\mathcal{E}$ we have $\mathbb{P}(X^{-1}(A))=\mathbb{P}(Y^{-1}(A))$. Now
we introduce the notion of independence. Given a finite set of random
variables $X_{1},\ldots X_{j}$ we say they are independent if for all
$A_{i}\in\mathcal{E},1\leq i\leq j$, we have

$\mathbb{P}(\cap_{i=1}^{j}X_{i}\in A_{i})=\prod_{i=1}^{j}\mathbb{P}(X_{i}\in
A_{i}).$

Finally a sequence of random variables $\\{X_{1},X_{2}\ldots\\}$ is said
independent if all finite collection of this sequence form a set of
independent random variables. If $X_{1},X_{2},\ldots$ is a sequence of
independent and identically distributed random variables we say that
$X_{1},X_{2},\ldots$ are i.i.d. random variables.

##  3 Main results and proofs

Let $G$ be the Green function of the laplacian operator $-\Delta$ in the
bounded domain $U\subset\mathbb{R}^{n}$ with $n\geq 3.$ It is known that, for
all $x,\,y\in U$, there holds

$0\leq
G(x,y)\leq\frac{1}{n\alpha_{n}(n-2)}\frac{1}{|x-y|^{n-2}},\leavevmode\nobreak\
$

where $\alpha_{n}$ stands for the volume of the unit ball in $\mathbb{R}^{n}$.
Hence, if we denote by $d_{U}$ the the diameter of $U$, namely

$d_{U}:=\sup_{x_{1},\,x_{2}\in U}{|}x_{1}-x_{2}{|},$

and $B_{d_{U}}(x)=\\{x\in\mathbb{R}^{n};\left|x\right|<d_{U}\\},$ a
straightforward calculation provides

$\begin{array}[]{lcl}\displaystyle\int_{U}G(x,y)dy&\leq&\dfrac{1}{n\alpha_{n}(n-2)}\displaystyle\int_{B_{d_{U}}(x)}\dfrac{1}{|x-y|^{n-2}}dy\vspace{0.2cm}\\\
&=&\dfrac{1}{n\alpha_{n}(n-2)}\dfrac{n\alpha_{n}d_{U}^{2}}{2}=\dfrac{d_{U}^{2}}{2(n-2)},\end{array}$
(3.1)

for all $x\in U$. From now on we write only $l_{0}=l_{0}(n,U)$ to denote the
following quantity

$l_{0}:=\dfrac{d_{U}^{2}}{2(n-2)}.$ (3.2)

Inequality (3.1) implies that is well defined the map
$H:L^{\infty}(U)\rightarrow L^{\infty}(U)$ given by

$H(\varphi)(x):=\int_{U}G(x,y)\varphi(y)dy,\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ x\in U.$

More specifically, for any $\varphi\in L^{\infty}(U)$, there holds

$|H(\varphi)(x)|\leq\int_{U}G(x,y)|\varphi(y)|dy\leq\|\varphi\|_{\infty}\int_{U}G(x,y)dy$

and therefore

$\|H(\varphi)\|_{\infty}\leq l_{0}\|\varphi\|_{\infty}.$ (3.3)

Standard calculations show that the problem (1.2) is formally equivalent to
the integral equation

$u(x)=H(g)+H(V_{\omega}u)+H(bu|u|^{p-1}).$ (3.4)

In what follows we make suitable estimates on the terms of the integral
equation in order to be able to apply a fixed point argument. We first set
$\mathcal{X}:=L^{\infty}(U)$ and define, for any fixed $\omega\in\Omega$, the
linear function $T:\mathcal{X}\rightarrow\mathcal{X}$ by

$T(u):=H(V_{\omega}u),\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \forall\,u\in\mathcal{X}.$

It follows from (3.3) and (2.1) that, for any $u\in\mathcal{X}$, there holds

$\|T(u)\|_{\infty}\leq l_{0}\|V_{\omega}u\|_{\infty}\leq
l_{0}\|f\|_{\infty}\rule[-2.27621pt]{1.13809pt}{9.6739pt}\,\mu_{\omega}\,\rule[-2.27621pt]{1.13809pt}{9.6739pt}\text{
}\|u\|_{\infty},$ (3.5)

and therefore

$\|T\|_{\infty}\leq
l_{0}\|f\|_{\infty}\rule[-2.27621pt]{1.13809pt}{9.6739pt}\,\mu_{\omega}\,\rule[-2.27621pt]{1.13809pt}{9.6739pt}.$

For the nonlinear term we define $B:\mathcal{X}\rightarrow\mathcal{X}$ by
setting

$B(u):=H(b|u|^{p-1}u),\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \forall\,u\in\mathcal{X}.$

If $a_{1},\,a_{2}\in\mathbb{R}$ there holds

$\left|a_{1}|a_{1}|^{p-1}-a_{2}|a_{2}|^{p-1}\right|\leq
p|a_{1}-a_{2}|\left(|a_{1}|^{p-1}-|a_{2}|^{p-1}\right),$

and therefore it follows that

$\|b(\cdot)\left(u|u|^{p-1}-\tilde{u}|\tilde{u}|^{p-1}\right)\|_{\infty}\leq\|b\|_{\infty}\|u-\tilde{u}\|_{\infty}\left(\|u\|_{\infty}^{p-1}-\|\tilde{u}\|_{\infty}^{p-1}\right).$

This inequality and the same argument used in (3.5) imply that

$\|B(u)-B(\tilde{u})\|_{\infty}\leq
l_{0}p\|b\|_{\infty}\|u-\tilde{u}\|_{\infty}\left(\|u\|_{\infty}^{p-1}-\|\tilde{u}\|_{\infty}^{p-1}\right),$
(3.6)

for any $u,\,\tilde{u}\in L^{\infty}(U)$.

All together, the above estimates enable us to solve the random equation (1.2)
as follows.

######  Proposition 3.1.

Given $f,\,b,g\in L^{\infty}(U)$ and $\omega\in\Omega$, we consider the
potential $V_{\omega}$ induced by the random measure
$\mu_{\omega}:=X(\omega)$. Let $l_{0}$ be the quantity introduced in (3.2) and
set

$\tau_{\omega}:=l_{0}\|f\|_{\infty}\rule[-2.27621pt]{1.13809pt}{9.6739pt}\,\mu_{\omega}\,\rule[-2.27621pt]{1.13809pt}{9.6739pt}\text{
\ \ and}\leavevmode\nobreak\ \leavevmode\nobreak\ K:=l_{0}p\|b\|_{\infty}.$
(3.7)

If $\varepsilon>0$ and $\omega\in\Omega$ are such that

$0\leq\tau_{\omega}<1,\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\frac{2^{p}K\varepsilon^{p-1}}{(1-\tau_{\omega})^{p-1}}+\tau_{\omega}<1,$
(3.8)

and $\|g\|_{\infty}\leq\varepsilon/l_{0}$, then the equation (1.2) has a
unique integral solution

$u_{\omega}=u(\cdot,\omega)\in L^{\infty}(U)\text{ such that
}\|u_{\omega}\|_{\infty}\leq\frac{2\varepsilon}{1-\tau_{\omega}}.$ (3.9)

Proof. For each $\omega\in\Omega,$ we consider the closed ball

$\mathcal{B}_{\varepsilon}=\left\\{u\in
L^{\infty}(U);\|u\|_{\infty}\leq\frac{2\varepsilon}{(1-\tau_{\omega})}\right\\}$

endowed with the metric $d(u,v):=\|u-v\|_{\infty}.$ We are going to show that
the map

$\Phi(u):=H(g)+H(V_{\omega}u)+H(bu\left|u\right|^{p-1})=H(g)+T(u)+B(u)$ (3.10)

is a contraction on the complete metric space $(\mathcal{B}_{\varepsilon},d).$
Using the estimates (3.3), (3.5), and (3.6) with $\tilde{u}=0,$ we obtain

$\displaystyle\left\|\Phi(u)\right\|_{\infty}$
$\displaystyle\leq\|H(g)\|_{\infty}+\|T(u)\|_{\infty}+\|B(u)\|_{\infty}$
$\displaystyle\leq
l_{0}\left\|g\right\|_{\infty}+\tau_{\omega}\|u\|_{\infty}+K\|u\|_{\infty}^{p}$
$\displaystyle\leq\varepsilon+\tau_{\omega}\frac{2\varepsilon}{1-\tau_{\omega}}+\frac{2^{p}K\varepsilon^{p}}{(1-\tau_{\omega})^{p}}$
$\displaystyle=\left(1+\tau_{\omega}+\frac{2^{p}K\varepsilon^{p-1}}{(1-\tau_{\omega})^{p-1}}\right)\frac{\varepsilon}{1-\tau_{\omega}}$

for all $u\in\mathcal{B}_{\varepsilon}$ and $\omega\in\Omega$. Hence, it
follows from (3.8) that

$\left\|\Phi(u)\right\|_{\infty}\leq\frac{2\varepsilon}{1-\tau_{\omega}}.$

This shows that $\Phi$ maps $\mathcal{B}_{\varepsilon}$ into
$\mathcal{B}_{\varepsilon}$.

For any $u,\widetilde{u}\in\mathcal{B}_{\varepsilon},$ it follows from (3.5)
and (3.6) that

$\displaystyle\|\Phi(u)-\Phi(\widetilde{u})\|_{\infty}$
$\displaystyle=\|T(u-\widetilde{u})\|_{\infty}+\|B(u)-B(\widetilde{u})\|_{\infty}$
$\displaystyle\leq\tau_{\omega}\|u-\widetilde{u}\|_{\infty}+K\|u-\widetilde{u}\|_{\infty}\left(\|u\|_{\infty}^{p-1}+\|\widetilde{u}\|_{\infty}^{p-1}\right)$
$\displaystyle\leq\left(\tau_{\omega}+\frac{2^{p}K\varepsilon^{p-1}}{(1-\tau_{\omega})^{p-1}}\right)\|u-\widetilde{u}\|_{\infty}.$

Recalling (3.8), the above estimate implies that the map $\Phi$ is a
contraction. The Banach fixed point theorem assures that there is a unique
solution $u$ for the integral equation (3.4) such that
$\|u\|_{\infty}\leq(2\varepsilon)/(1-\tau_{\omega}).$

The next results are related to the randomness introduced by the random
potential $V$ and the existence and uniqueness of solutions for the problem
(1.2). Roughly speaking, we first obtain the probability of (1.2) having a
solution given by the method discussed above. In the sequel we study two
important limit theorems in probability theory, namely, the central limit
theorem and the law of large numbers for a sequence of random potentials.

######  Theorem 3.2.

Let $\nu$ be the probability measure induced on $\mathbb{R}$ by the random
variable
$\omega\mapsto\rule[-2.27621pt]{1.13809pt}{9.6739pt}\,\mu_{\omega}\rule[-2.27621pt]{1.13809pt}{9.6739pt}\,\,.$
Let $g\in L^{\infty}(U)$ be such that
$\left\|g\right\|_{\infty}<\frac{1}{l_{0}}(\frac{1}{2^{p}K})^{\frac{1}{p-1}}$,
where $K=l_{0}p\|b\|_{\infty}$. Choose $0<c_{0}<1$and set

$\varepsilon_{0}:=\left(\frac{(1-c_{0})^{p}}{2^{p}K}\right)^{\frac{1}{p-1}}.$

Let $\mathcal{A}$ be the set of $\omega\in\Omega$ such that (1.2) has a unique
solution $u(\cdot,\omega)$ given by Proposition 3.1 with
$\varepsilon=\varepsilon_{0}$. The set $\mathcal{A}$ is called the admissible
one for the random variable $X.$

  * (i)

The set $\mathcal{A}$ is $\mathcal{F}$-measurable and the probability of (1.2)
having a solution is

$\mathbb{P(\mathcal{A})}=\nu\left(\left[0,\frac{1}{l_{0}\|f\|_{\infty}}\right)\right).$

  * (ii)

Let $u_{\omega},\,\tilde{u}_{\omega}$ be two solutions of (1.2) corresponding,
respectively, to $\mu_{\omega},g,\mathcal{A}$ and
$\tilde{\mu}_{\omega},\tilde{g},\widetilde{\mathcal{A}}$. Assume that
$\mathcal{A\cap}\widetilde{\mathcal{A}}\neq\varnothing$ and define, for
$\omega\in\mathcal{A\cap}\widetilde{\mathcal{A}}$,

$\eta_{\omega}:=l_{0}\|f\|_{\infty}\max\\{\rule[-2.27621pt]{1.13809pt}{9.6739pt}\,\mu_{\omega}\,\rule[-2.27621pt]{1.13809pt}{9.6739pt},\rule[-2.27621pt]{1.13809pt}{9.6739pt}\,\widetilde{\mu}_{\omega}\,\rule[-2.27621pt]{1.13809pt}{9.6739pt}\\}.$

We have that

$\|u(\cdot,\omega)-\tilde{u}(\cdot,\omega)\|_{\infty}\leq\frac{l_{0}\left(\|g-\tilde{g}\|_{\infty}+\dfrac{2\varepsilon_{0}}{1-\eta_{\omega}}\|f\|_{\infty}\rule[-2.27621pt]{1.13809pt}{9.6739pt}\,\mu_{\omega}-\tilde{\mu}_{\omega}\,\rule[-2.27621pt]{1.13809pt}{9.6739pt}\right)}{1-\eta_{\omega}-\dfrac{2^{p}K\varepsilon_{0}^{p-1}}{(1-\eta_{\omega})^{p-1}}}$
(3.11)

for all $\omega\in\mathcal{A\cap}\widetilde{\mathcal{A}}$.

  * (iii)

The map $\ \mathcal{U}:\mathcal{A}\rightarrow L^{\infty}(U)$ given by
$\mathcal{U}(\omega):=u(\cdot,\omega)$ is a random variable and there holds

$\|u(\cdot,\omega)\|_{\infty}\leq\frac{2\varepsilon_{0}}{1-\tau_{\omega}}=2\varepsilon_{0}\sum_{j=0}^{\infty}\tau_{\omega}^{j},$
(3.12)

for all $\omega\in\mathcal{A}.$

Proof. We first notice that the choice of $\varepsilon_{0}$ implies that
$\|g\|_{\infty}\leq\varepsilon_{0}/l_{0}.$ Moreover, $\omega\in\mathcal{A}$ if
only if
$\tau_{\omega}=l_{0}\|f\|_{\infty}\rule[-2.27621pt]{1.13809pt}{9.6739pt}\,\mu_{\omega}\,\rule[-2.27621pt]{1.13809pt}{9.6739pt}$
verifies (3.8) with $\varepsilon=\varepsilon_{0}.$ Then, if
$Y(\omega)=\rule[-2.27621pt]{1.13809pt}{9.6739pt}\,X(\omega)\rule[-2.27621pt]{1.13809pt}{9.6739pt}\,=\rule[-2.27621pt]{1.13809pt}{9.6739pt}\,\,\mu_{\omega}\,\rule[-2.27621pt]{1.13809pt}{9.6739pt},$
it follows that
$\mathcal{A}=\left\\{Y\in\left[0,\frac{1}{l_{0}\|f\|_{\infty}}\right)\right\\}$
is measurable and

$\begin{array}[]{lcl}\mathbb{P}(\mathcal{A})&=&\mathbb{P}\left(Y\in\left[0,\dfrac{1}{l_{0}\|f\|_{\infty}}\right)\right)=\mathbb{P}_{Y}\left(\left[0,\dfrac{1}{l_{0}\|f\|_{\infty}}\right)\right)\vspace{0.2cm}\\\
&=&\nu\left(\left[0,\dfrac{1}{l_{0}\|f\|_{\infty}}\right)\right).\end{array}$

This establishes (i).

Now we deal with item (ii). Firstly, observe that
$\eta_{\omega}=\max\\{\tau_{\omega},\widetilde{\tau}_{\omega}\\},$ where

$\tau_{\omega}=l_{0}\|f\|_{\infty}\rule[-2.27621pt]{1.13809pt}{9.6739pt}\mu_{\omega}\,\rule[-2.27621pt]{1.13809pt}{9.6739pt}\text{
and
}\widetilde{\tau}_{\omega}=l_{0}\|f\|_{\infty}\rule[-2.27621pt]{1.13809pt}{9.6739pt}\tilde{\mu}_{\omega}\,\rule[-2.27621pt]{1.13809pt}{9.6739pt}.$

Subtracting the integral equations verified by $u_{\omega}$ and
$\,\tilde{u}_{\omega},$ and afterwards computing $\|\cdot\|_{\infty}$, we
obtain

$\begin{array}[]{lcl}\left\|u_{\omega}-\tilde{u}_{\omega}\right\|_{\infty}&\leq&\left\|H(g-\tilde{g})\right\|_{\infty}+\left\|H(V_{\omega}(u-\tilde{u}_{\omega}))\right\|_{\infty}\vspace{0.2cm}\\\
&&+\|H((V_{\omega}-\widetilde{V}_{\omega})\tilde{u}_{\omega})\|_{\infty}\vspace{0.2cm}\\\
&&+\left\|H(b\left(u_{\omega}|u_{\omega}|^{p-1}-\tilde{u}_{\omega}|\tilde{u}_{\omega}|^{p-1}\right))\right\|_{\infty}\vspace{0.2cm}\\\
&\leq&l_{0}\|g-\tilde{g}\|_{\infty}+l_{0}\|f\|_{\infty}\rule[-2.27621pt]{1.13809pt}{9.6739pt}\,\mu_{\omega}\,\rule[-2.27621pt]{1.13809pt}{9.6739pt}\|u_{\omega}-\tilde{u}_{\omega}\|_{\infty}\vspace{0.2cm}\\\
&&+l_{0}\|f\|_{\infty}\rule[-2.27621pt]{1.13809pt}{9.6739pt}\,\mu_{\omega}-\tilde{\mu}_{\omega}\,\rule[-2.27621pt]{1.13809pt}{9.6739pt}\|\tilde{u}_{\omega}\|_{\infty}\vspace{0.2cm}\\\
&&+l_{0}p\|b\|_{\infty}\|u_{\omega}-\tilde{u}_{\omega}\|_{\infty}(\|u_{\omega}\|_{\infty}^{p-1}-\|\tilde{u}_{\omega}\|_{\infty}^{p-1}).\end{array}$

It follows from (3.9) that

$\|u_{\omega}\|_{\infty}\leq\frac{2\varepsilon_{0}}{1-\tau_{\omega}}\leq\frac{2\varepsilon_{0}}{1-\eta_{\omega}}\text{
and
}\|\tilde{u}\|_{\infty}\leq\frac{2\varepsilon_{0}}{1-\widetilde{\tau}_{\omega}}\leq\frac{2\varepsilon_{0}}{1-\eta_{\omega}}.$

The two above expressions give us

$\displaystyle\left\|u_{\omega}-\tilde{u}_{\omega}\right\|_{\infty}$
$\displaystyle\leq$ $\displaystyle
l_{0}\|g-\tilde{g}\|_{\infty}+l_{0}\|f\|_{\infty}\rule[-2.27621pt]{1.13809pt}{9.6739pt}\,\mu_{\omega}\,\rule[-2.27621pt]{1.13809pt}{9.6739pt}\left\|u_{\omega}-\tilde{u}_{\omega}\right\|_{\infty}$
$\displaystyle+\,l_{0}\frac{2\varepsilon_{0}}{1-\eta_{\omega}}\|f\|_{\infty}\rule[-2.27621pt]{1.13809pt}{9.6739pt}\,\mu_{\omega}-\tilde{\mu}_{\omega}\,\rule[-2.27621pt]{1.13809pt}{9.6739pt}+\frac{2^{p}K\varepsilon_{0}^{p-1}}{(1-\eta_{\omega})^{p-1}}\|u_{\omega}-\tilde{u}_{\omega}\|_{\infty}$
$\displaystyle=$ $\displaystyle
l_{0}\|g-\tilde{g}\|_{\infty}+l_{0}\frac{2\varepsilon_{0}}{1-\eta_{\omega}}\|f\|_{\infty}\rule[-2.27621pt]{1.13809pt}{9.6739pt}\,\mu_{\omega}-\tilde{\mu}_{\omega}\,\rule[-2.27621pt]{1.13809pt}{9.6739pt}$
$\displaystyle+\,\left[\eta_{\omega}+\frac{2^{p}K\varepsilon_{0}^{p-1}}{(1-\eta_{\omega})^{p-1}}\right]\left\|u_{\omega}-\tilde{u}_{\omega}\right\|_{\infty},$

which yields (3.11).

Taking $\mu_{\omega},\tilde{\mu}_{\omega}$ independent of $\omega,$ i.e.
$\mu_{\omega}=\mu$ and $\tilde{\mu}_{\omega}=\tilde{\mu},$ for all
$\omega\in\Omega,$ we see from (3.7) and (3.11) that the data-map solution
$\mathcal{L}(\mu,g)=u$ is continuous from

$\left\\{(\mu,g)\in\mathcal{M}(U)\times L^{\infty}(U);\text{
}\rule[-2.27621pt]{1.13809pt}{9.6739pt}\,\mu\,\rule[-2.27621pt]{1.13809pt}{9.6739pt}<\frac{1}{l_{0}\|f\|_{\infty}},\left\|g\right\|_{\infty}<\frac{1}{l_{0}}\left(\frac{1}{2^{p}K}\right)^{\frac{1}{p-1}}\right\\}\text{to
}L^{\infty}(U),$ (3.13)

where $u$ is the deterministic solution of (1.2) corresponding to the data
$(\mu,g).$ From this, and because $X|_{\mathcal{A}}$ given by
$X(\omega)=\mu_{\omega}$ is measurable, it follows that the composition
$\mathcal{U}(\omega)=\mathcal{L}(\mu_{\omega},g)=\mathcal{L}(X(\omega),g)\,$
from $\mathcal{A}$ to $L^{\infty}(U)$ is measurable.

In view of the series $\frac{1}{1-z}=\sum_{j=0}^{\infty}z^{j}$ for
$\left|z\right|<1,$ we finish by observing that (3.12) follows at once from
(3.9) with $\varepsilon=\varepsilon_{0}$ and $\omega\in\mathcal{A}.$

######  Remark 3.3.

Here we give examples of random potentials for which there exists solution
almost surely in $\Omega$. The first setting occurs if we suppose that the
measure $\nu$ has compact support contained in the interval $[0,a]$, with
$a<\frac{1}{l_{0}\|f\|_{\infty}}$. In this case it follows from the first item
of the above theorem that $\mathbb{P(\mathcal{A})}=1$, i.e., the solution
exists almost surely in $\Omega.$ Secondly, we take
$\\{\mu_{j}\\}_{j\in\mathbb{N}}$ a sequence in $\mathcal{M}(U)$ and let
$\\{a_{j}(\omega)\\}_{j\in\mathbb{N}}$ be a sequence of random variables from
$\Omega$ to $\mathbb{R}.$ Consider the random variable $\mu_{\omega}$ defined
by

$\mu_{\omega}=\sum_{j=1}^{\infty}a_{j}(\omega)\mu_{j}.$

For $q>1,$ suppose that

$|a_{j}(\omega)|<\frac{(\sum_{k=1}^{\infty}\frac{1}{k^{q}})^{-1}}{l_{0}\rule[-2.27621pt]{1.13809pt}{9.6739pt}\,\mu_{j}\rule[-2.27621pt]{1.13809pt}{9.6739pt}\,\|f\|_{\infty}}\cdot\frac{1}{j^{q}}\text{
a.s. in }\Omega,$

for all $j\in\mathbb{N}.$ Then

$\rule[-2.27621pt]{1.13809pt}{9.6739pt}\,\mu_{\omega}\rule[-2.27621pt]{1.13809pt}{9.6739pt}\,\leq\sum_{j=1}^{\infty}|a_{j}(\omega)|\rule[-2.27621pt]{1.13809pt}{9.6739pt}\,\mu_{j}\rule[-2.27621pt]{1.13809pt}{9.6739pt}\,<\frac{1}{l_{0}\|f\|_{\infty}}\text{
a.s. in }\Omega,$

and Theorem 3.2 assures that there is an integral solution for (1.2) a.s. in
$\Omega.$

In the sequel we show how the Borel-Cantelli’s Lemma can be used to give a
sufficient condition for the existence of solution a.s. in $\Omega$.

######  Theorem 3.4.

Let $\\{\mu_{j}\\}_{j\in\mathbb{N}}$ be a sequence in $\mathcal{M}(U)$ and let
$\\{a_{j}(\omega)\\}_{j\in\mathbb{N}}$ be a sequence of random variables from
$\Omega$ to $\mathbb{R}.$ Assume that the following series is convergent in
$\mathcal{M}(U)$

$\mu_{\omega}=\sum_{j=1}^{\infty}a_{j}(\omega)\mu_{j}.$

For any $k\in\mathbb{N}$ define

$S_{k}(\omega)=\sum_{j=1}^{k}a_{j}(\omega)\mu_{j}$

and
$L_{k}=\\{\omega\in\Omega:\rule[-2.27621pt]{1.13809pt}{9.6739pt}\,S_{k}\rule[-2.27621pt]{1.13809pt}{9.6739pt}\,\geq\tilde{c}\\}$,
with $0<\tilde{c}<1/(l_{0}\|f\|_{\infty})$. If

$\sum_{k=1}^{\infty}\mathbb{P}(L_{k})<\infty$

then there is an integral solution for (1.2) almost surely in $\Omega$.

Proof. By the Borel-Cantelli’s Lemma we get that $\mathbb{P}(\limsup
L_{k})=0$, that is,

$\mathbb{P}\left(\cup_{j=1}^{\infty}\cap_{k=j}^{\infty}\\{\rule[-2.27621pt]{1.13809pt}{9.6739pt}\,S_{k}\,\rule[-2.27621pt]{1.13809pt}{9.6739pt}<\tilde{c}\\}\right)=1$

It follows that, for almost sure $\omega,$ there is $j_{0}=$ $j_{0}(\omega)$
such that for all $j>j_{0}$, we have

$\rule[-2.27621pt]{1.13809pt}{9.6739pt}\,S_{k}(\omega)\,\rule[-2.27621pt]{1.13809pt}{9.6739pt}<\tilde{c}.$

Therefore by taking the limit when $k$ goes to infinity, we obtain

$\rule[-2.27621pt]{1.13809pt}{9.6739pt}\,\mu_{\omega}\,\rule[-2.27621pt]{1.13809pt}{9.6739pt}=\lim_{j\rightarrow\infty}\rule[-2.27621pt]{1.13809pt}{9.6739pt}\,S_{j}(\omega)\,\rule[-2.27621pt]{1.13809pt}{9.6739pt}\leq\tilde{c}<\frac{1}{l_{0}\|f\|_{\infty}}\quad\mbox{a.s.
in }\Omega.$

This inequality and Theorem 3.2 imply that there is an integral solution
$u(x,\omega)$ for (1.2) almost surely in $\Omega$.

A straightforward calculation shows that in general
$\mathbb{E}_{\Omega}(u(x,\omega))$ does not satisfies the equation (1.2), even
if we replace the random potential by its mean. However, we are able to obtain
some information on the average and moments of the random solution
$u_{\omega}$ previously obtained. It is worthwhile to mention that, when
dealing with the random variable $\omega\mapsto u_{\omega}$, the expectation
has to be understood in the Bochner sense (see Section 2). Note also that a
solution $u_{\omega}\in L^{\infty}(U)$ for (3.4) in fact belongs to the
separable subspace $C(\overline{U}).$

######  Theorem 3.5.

Under hypotheses of Theorem 3.2 let us denote by
$u_{\omega}(x)=u(x,\omega)\in\mathcal{A}$ the solution of (1.2). Let
$m\in\mathbb{N}$ and suppose that

$\sum_{j=1}^{\infty}\frac{(m+j-1)!}{(m-1)!j!}(l_{0}\|f\|_{\infty})^{j}\
\mathbb{E}_{\mathcal{A}}[\
\rule[-2.27621pt]{1.13809pt}{9.6739pt}\,\mu_{\omega}\,\rule[-2.27621pt]{1.13809pt}{9.6739pt}^{\,j}\,]<+\infty.$
(3.14)

Then $\mathbb{E}_{\mathcal{A}}[|u|^{m}(x,\omega)]\in L^{\infty}(U)$ and

$\mathbb{E}_{\mathcal{A}}\left[\left\||u|^{m}(\cdot,\omega)\right\|_{L^{\infty}(U)}\right]<\infty.$
(3.15)

In particular, $\mathbb{E}_{\mathcal{A}}[u(x,\omega)]\in L^{\infty}(U).$

Proof. It follows from (3.12) that

$\||u|^{m}(\cdot,\omega)\|_{L^{\infty}(U)}\leq\|u(\cdot,\omega)\|_{L^{\infty}(U)}^{m}\leq\frac{(2\varepsilon_{0})^{m}}{(1-\tau_{\omega})^{m}}.$
(3.16)

Recalling that
$\tau_{\omega}=l_{0}\|f\|_{\infty}\rule[-2.27621pt]{1.13809pt}{9.6739pt}\,\mu_{\omega}\,\rule[-2.27621pt]{1.13809pt}{9.6739pt}$
and computing $\mathbb{E}_{\mathcal{A}}$ in (3.16), we obtain

$\displaystyle\left\|\mathbb{E}_{\mathcal{A}}\left[|u|^{m}(x,\omega)\right]\right\|_{L^{\infty}(U)}$
$\displaystyle\leq$
$\displaystyle\mathbb{E}_{\mathcal{A}}\left[\left\||u|^{m}(x,\omega)\right\|_{L^{\infty}(U)}\right]$
$\displaystyle\leq$
$\displaystyle(2\varepsilon_{0})^{m}\mathbb{E}_{\mathcal{A}}\left[\left(1+\sum_{j=1}^{\infty}\frac{(m+j-1)!}{(m-1)!j!}\tau_{\omega}^{j}\right)\right]$

By using the linearity of the expectation and definition of $\tau_{\omega}$ we
get the following upper bound for the right hand side above

$(2\varepsilon_{0})^{m}+(2\varepsilon_{0})^{m}\sum_{j=1}^{\infty}\frac{(m+j-1)!}{(m-1)!j!}\left(l_{0}\|f\|_{\infty}\right)^{j}\mathbb{E}_{\mathcal{A}}\left[\
\rule[-2.27621pt]{1.13809pt}{9.6739pt}\,\mu_{\omega}\,\rule[-2.27621pt]{1.13809pt}{9.6739pt}^{\,j}\
\right],$

which is finite due to (3.14). The last assertion of the statement follows
from (3.15) with $m=1$ and the easy estimate

$\left\|\mathbb{E}_{\mathcal{A}}\left[u(x,\omega)\right]\right\|_{L^{\infty}(U)}\leq\mathbb{E}_{\mathcal{A}}\left[\left\||u|(x,\omega)\right\|_{L^{\infty}(U)}\right].$

###  3.1 Classical Probability Limit Theorems

We start this section by recalling basic background concerning to some main
limit theorems in probability. A real-valued random variable
$X:\Omega\to\mathbb{R}$ in a probability space
$(\Omega,\mathcal{F},\mathbb{P})$ has standard normal distribution, notation
$X\sim N(0,1)$, if for all $x\in\mathbb{R}$ its cumulative distribution
function verifies

$\mathbb{P}(X\leq
x)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{x}e^{-\frac{1}{2}t^{2}}dt.$

A sequence of real-valued random variable $\\{Y_{j}\\}_{j\in\mathbb{N}}$ in a
probability space $(\Omega,\mathcal{F},\mathbb{P})$ is said to converge in
distribution to a standard normal random variable, notation $Y_{j}\to N(0,1)$,
if for all $x\in\mathbb{R}$ we have

$\lim_{j\to\infty}\mathbb{P}(Y_{j}\leq
x)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{x}e^{-\frac{1}{2}t^{2}}dt.$

In the sequel we show versions of the central limit theorem and a weak law of
large numbers for the random $L^{\infty}(U)$-solutions obtained in Section 2.

######  Theorem 3.6.

Let $\\{X_{j}\\}_{j\in\mathbb{N}}$ be an independent identically distributed
(i.i.d.) sequence of random variables $X_{j}:\Omega\rightarrow\mathcal{M}(U)$.
Assume that the admissible set $\mathcal{A}_{j}=\Omega$ for all $j$, and let
$u_{j}(\cdot,\omega)\in L^{\infty}(U)$ be the solution given by Theorem 3.2
with respect to $X_{j}(\omega)=\mu_{\omega,j}$ and $g.$ We have that
$\\{Z_{j}\\}_{j\in\mathbb{N}}$ given by
$Z_{j}(\omega):=\|u_{j}(\cdot,\omega)\|_{\infty}$ is a i.i.d. sequence of
random variables, and if
$m=\mathbb{E}[\|u_{j}(\cdot,\omega)\|_{\infty}]<\infty$ and
$\sigma^{2}:=\text{Var}\,Z_{j}<\infty$ then following holds as
$k\rightarrow+\infty$

$\sum_{j=1}^{k}\frac{(Z_{j}-m)}{\sigma\sqrt{k}}\rightarrow N(0,1).$

Proof. Recall the data-solution map $\mathcal{L}(\mu,g)$ defined in the proof
of Theorem 3.2 (see (3.13)). Fixed $g$ such that
$\left\|g\right\|_{\infty}<\frac{1}{l_{0}}(\frac{1}{2^{p}K})^{\frac{1}{p-1}},$
consider

$S_{g}(\mu)=\mathcal{L}(\mu,g)$ (3.17)

defined from $D$ to $L^{\infty}(U),$ where
$D=\left\\{\mu\in\mathcal{M}(U):\rule[-2.27621pt]{1.13809pt}{9.6739pt}\,\mu\,\rule[-2.27621pt]{1.13809pt}{9.6739pt}<\frac{1}{l_{0}\|f\|_{\infty}}\right\\}.$
Since $\|\cdot\|_{\infty}$ is continuous from $L^{\infty}(U)$ to $\mathbb{R}$
and

$Z_{j}(\omega)=\|u_{j}(\cdot,\omega)\|_{\infty}=\|S_{g}\circ
X_{j}(\omega)\|_{\infty},$

we get that $\\{Z_{j}\\}_{j\in\mathbb{N}}$ is a i.i.d. sequence. The
convergence stated in the theorem follows from the central limit theorem.

######  Theorem 3.7.

Let $\\{X_{j}\\}_{j\in\mathbb{N}}$ be an independent sequence of random
variables $X_{j}:\Omega\rightarrow\mathcal{M}(U)$. Assume that the admissible
set $\mathcal{A}_{j}=\Omega$ for all $j$, and let $u_{j}(\cdot,\omega)\in
L^{\infty}(U)$ be the solution given by Theorem 3.2 with respect to
$X_{j}(\omega)=\mu_{\omega,j}$ and $g.$ If $X_{j}\rightarrow X$ a.s. and

$L=\sup_{j\in\mathbb{N}}\left(\mathrm{ess}\sup_{\omega\in\Omega}\rule[-2.27621pt]{1.13809pt}{9.6739pt}\,\mu_{\omega,j}\,\rule[-2.27621pt]{1.13809pt}{9.6739pt}\right)<\frac{1}{l_{0}\|f\|_{\infty}},$
(3.18)

then

$\sum_{j=1}^{k}\frac{u_{j}(x,\omega)-\mathbb{E}_{\Omega}[u_{j}(x,\omega)]}{k}\rightarrow
0$ (3.19)

and

$\sum_{j=1}^{k}\frac{\|u_{j}(\cdot,\omega)\|_{\infty}-\mathbb{E}_{\Omega}[\|u_{j}(\cdot,\omega)\|_{\infty}]}{k}\rightarrow
0,$ (3.20)

when $k\rightarrow\infty$, where the convergence in (3.19) and (3.20) are in
probability sense (see (2.2)).

Proof. Notice that $X_{j}\rightarrow X$ a.s. is equivalent to
$\mu_{\omega,j}\rightarrow\mu_{\omega}=X(\omega)$ in $\mathcal{M}(U)$ almost
surely. From this and the continuity of data-solution map
$\mathcal{L}(\cdot,\cdot)$ (see (3.13)), it follows that

$\|u_{j}(\cdot,\omega)-u(\cdot,\omega)\|_{\infty}=\|\mathcal{L}(\mu_{\omega,j},g)-\mathcal{L}(\mu,g)\|_{\infty}\rightarrow
0,$

when $j\rightarrow\infty$. Recalling (3.12) and afterwards using (3.18), we
obtain

$\displaystyle\|u_{j}(\cdot,\omega)\|_{\infty}$ $\displaystyle\leq$
$\displaystyle\frac{2\varepsilon_{0}}{1-l_{0}\|f\|_{\infty}(\mathrm{ess}\sup_{\omega\in\Omega}\rule[-2.27621pt]{1.13809pt}{9.6739pt}\,\mu_{\omega},_{j}\,\rule[-2.27621pt]{1.13809pt}{9.6739pt})}$
(3.21) $\displaystyle\leq$
$\displaystyle\frac{2\varepsilon_{0}}{1-L}=Q_{0},\text{ a.s. in }\Omega.$

Since $X_{j}$’s are independent, it follows that
$\\{Y_{j}\\}_{j\in\mathbb{N}}$ defined by
$Y_{j}=\left\|u_{j}(\cdot,\omega)\right\|_{\infty}=\|S_{g}\circ
X_{j}(\omega)\|_{\infty}$ are also independent, where $S_{g}$ is as in (3.17).
So, from Chebyshev’s inequality and the independence of
$\\{Y_{j}\\}_{j\in\mathbb{N}}$, we have that

$\displaystyle\mathbb{P}\left(\left|k^{-1}\sum_{j=1}^{k}(\|u_{j}(\cdot,\omega)\|_{\infty}-\mathbb{E}_{\Omega}[\|u_{j}(\cdot,\omega)\|_{\infty}])\right|\geq\delta\right)$
$\displaystyle\leq$
$\displaystyle\frac{1}{(k\delta)^{2}}\mathbb{E}_{\Omega}\left[\left|\sum_{j=1}^{k}\left(\|u_{j}(\cdot,\omega)\|_{\infty}-\mathbb{E}_{\Omega}[\|u_{j}(\cdot,\omega)\|_{\infty}]\text{
}\right)\right|^{2}\right]$ $\displaystyle=$
$\displaystyle\frac{1}{(k\delta)^{2}}\sum_{j=1}^{k}\mathbb{E}_{\Omega}\left[\left|\left(\|u_{j}(\cdot,\omega)\|_{\infty}-\mathbb{E}_{\Omega}[\|u_{j}(\cdot,\omega)\|_{\infty}]\text{
}\right)\right|^{2}\right]$ $\displaystyle\leq$
$\displaystyle\frac{1}{(k\delta)^{2}}\sum_{j=1}^{k}\mathbb{E}_{\Omega}\left[\left|2Q_{0}\right|^{2}\right]\leq\frac{4Q_{0}^{2}}{\delta^{2}}\frac{1}{k},$

where we have used (3.21). Letting $k\rightarrow+\infty$ in the above
expression we get (3.20). The convergence (3.19) can be proved with similar
arguments.

## Acknowledgments

L.C.F. Ferreira was supported by FAPESP-SP and CNPq, Brazil. M. Furtado was
supported by CNPq, Brazil.

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