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SPDEs with-stable Levy noise:a random field approach

Raluca M. Balan

November 25, 2013

Abstract

This article is dedicated to the study of an SPDE of the form

Lu(t, x) =(u(t, x))Z(t, x) t >0, x Owith zero initial conditions and Dirichlet boundary conditions, where is a Lipschitz function, L is a second-order pseudo-differential oper-ator,O is a bounded domain in Rd, and Z is an -stable Levy noisewith (0, 2), = 1 and possibly non-symmetric tails. To give ameaning to the concept of solution, we develop a theory of stochasticintegration with respect to Z, by generalizing the method of [11] to

higher dimensions and non-symmetric tails. The idea is to first solvethe equation with truncated noise ZK (obtained by removing fromZ the jumps which exceed a fixed value K), yielding a solution uK,and then show that the solutions uL, L > K coincide on the eventtK, for some stopping times K a.s. A similar idea was usedin [22] in the setting of Hilbert-space valued processes. A major stepis to show that the stochastic integral with respect to ZK satisfies ap-th moment inequality, for p(, 1) if 1.This inequality plays the same role as the Burkholder-Davis-Gundyinequality in the theory of integration with respect to continuous mar-tingales.

MSC 2000 subject classification: Primary 60H15; secondary 60H05, 60G60Department of Mathematics and Statistics, University of Ottawa, 585 King Edward

Avenue, Ottawa, ON, K1N 6N5, Canada. E-mail address: rbalan@uottawa.caResearch supported by a grant from the Natural Sciences and Engineering Research

Council of Canada.

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1 Introduction

Modeling phenomena which evolve in time or space-time and are subjectto random perturbations is a fundamental problem in stochastic analysis.When these perturbations are known to exhibit an extreme behavior, asseen frequently in finance or environmental studies, a model relying on theGaussian distribution is not appropriate. A suitable alternative could be amodel based on a heavy-tailed distribution, like the stable distribution. Insuch a model, these perturbations are allowed to have extreme values with aprobability which is significantly higher than in a Gaussian-based model.

In the present article, we introduce precisely such a model, given rigor-ously by a stochastic partial differential equation (SPDE) driven by a noise

term which has a stable distribution over any space-time region, and hasindependent values over disjoint space-time regions (i.e. it is a Levy noise).More precisely, we consider the SPDE:

Lu(t, x) =(u(t, x))Z(t, x), t >0, x O (1)

with zero initial conditions and Dirichlet boundary conditions, where isa Lipschitz function, L is a second-order pseudo-differential operator on abounded domainO Rd, and Z(t, x) = d+1Ztx1...xd is the formal derivative ofan -stable Levy noise with (0, 2), = 1. The goal is to find sufficientconditions on the fundamental solution G(t,x,y) of the equation Lu = 0 on

R+ O, which will ensure the existence of a mild solution of equation (1).We say that a predictable process u ={u(t, x); t 0, x O} is a mildsolutionof (1) if for any t >0, x O,

u(t, x) =

t0

O

G(t s,x,y)(u(s, y))Z(ds,dy) a.s. (2)

We assume thatG(t,x,y) is a function int, which excludes from our analysisthe case of the wave equation with d3.

To explain the connections with other works, we describe briefly the con-struction of the noise (the details are given in Section 2 below). This con-

struction is similar to that of a classical-stable Levy process, and is basedon a Poisson random measure (PRM) N on R+ Rd (R\{0}) of intensitydtdx(dz), where

(dz) = [pz11(0,)(z) +q(z)11(,0)(z)]dz (3)

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for somep, q0 withp+q= 1. More precisely, for any set B Bb(R+Rd),

Z(B) =B{|z|1}

zN(ds,dx,dz) + B{|z|>1}

zN(ds,dx,dz) |B|, (4)

whereN(B ) =N(B ) |B|() is the compensated process and isa constant (specified by Lemma 2.3 below). Here,Bb(R+ Rd) is the classof bounded Borel sets in R+ Rd and|B| is the Lebesque measure ofB .

As the term on the right-hand side of (2) is a stochastic integral withrespect toZ, such an integral should be constructed first. Our constructionof the integral is an extension to random fields of the construction provided byGine and Marcus in [11] in the case of an-stable Levy process{Z(t)}t[0,1].Unlike these authors, we do not assume that the measure is symmetric.

Since any Levy noise is related to a PRM, in a broad sense, one couldsay that this problem originates in Itos papers [12] and [13] regarding thestochastic integral with respect to a Poisson noise. SPDEs driven by a com-pensated PRM were considered for the first time in [14], using the approachbased on Hilbert-space-valued solutions. This study was motivated by anapplication to neurophysiology leading to the cable equation. In the caseof the heat equation, a similar problem was considered in [1], [26] and [3]using the approach based on random-field solutions. One of the results of[26] shows that the heat equation:

ut

(t, x) =12

u(t, x) + U

f(t,x,u(t, x); z)N(t,x,dz) +g(t,x,u(t, x))has a unique solution in the space of predictable processes u satisfyingsup(t,x)[0,T]RdE|u(t, x)|p 1, assumingthat the noise has only positive jumps (i.e. q= 0). The methods used by

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these authors are different from those presented here, since they investigate

the more difficult case of a non-Lipschitz function (u) =u

with >0. In[16], Mueller removes the atoms ofZof mass smaller than 2n and solvesthe equation driven by the noise obtained in this way; here we remove theatoms ofZof mass larger than Kand solve the resulting equation. In [18],Mytnik uses a martingale problem approach and gives the existence of a pair(u, Z) which satisfies the equation (the so-called weak solution), whereasin the present article we obtain the existence of a solution u for agivennoiseZ(the so-called strong solution). In particular, when >1 and= 1/,the existence of a weak solution of the heat equation with -stable Levynoise is obtained in [18] under the condition

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In Section 5, we introduce the process ZK obtained by removing fromZ the jumps exceeding a fixed value K, and we develop a theory ofintegration with respect to this process. For this, we need to treatseparately the cases < 1 and > 1. In both cases, we obtain a

p-th moment inequality for the integral process for p(, 1) if 1. This inequality plays the same role as theBurkholder-Davis-Gundy inequality in the theory of integration withrespect to continuous martingales.

In Section 6 we prove the main result about the existence of the mildsolution of equation (1). For this, we first solve the equation withtruncated noise ZKusing a Picard iteration scheme, yielding a solu-

tionuK. We then introduce a sequence (K)K1of stopping times withK a.s. and we show that the solutions uL, L > K coincide onthe event tK. For the definition of the stopping times K, we needagain to consider separately the cases 1.

Appendix A contains some results about the tail of a non-symmetricstable random variable, and the tail of an infinite sum of random vari-ables. Appendix B gives an estimate for the Green function associatedto the fractional power of the Laplacian. Appendix C gives a localproperty of the stochastic integral with respect to Z (or ZK).

2 Definition of the noise

In this section we review the construction of the -stable Levy noise onR+ Rd and investigate some of its properties.

Let N =

i1 (Ti,Xi,Zi) be a Poisson random measure on R+ Rd (R\{0}), defined on a probability space (, F, P), with intensity measuredtdx(dz), where is given by (3). Let (j)j0 be a sequence of positivereal numbers such that j0 as j and 1 =0> 1 > 2 > . . .. Let

j ={zR; j 1}.For any set B Bb(R+ Rd), we define

Lj(B) =

Bj

zN(dt,dx,dz) =

(Ti,Xi)B

Zi1{Zij}, j0.

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Remark 2.1 The variable L0(B) is finite since the sum above contains

finitely many terms. To see this, we note that E[N(B 0)] =|B|(0)

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For any disjoint sets B1, . . . , Bn and for any u1, . . . , unR, we have:

E[exp(ink=1

ukY(Bk))] =E[exp(iY(nk=1

uk1Bk))]

= exp

R+RdR

(eizn

k=1 uk1Bk (t,x) 1 iz1{|z|1}nk=1

uk1Bk(t, x))dtdx(dz)

= exp

nk=1

|Bk|R

(eiukz 1 iukz1{|z|1})(dz)

(10)

=n

k=1E[exp(iukY(Bk))],

using (9) with =n

k=1 uk1Bk for the second equality, and (6) for the lastequality. This proves that Y(B1), . . . , Y (Bn) are independent.

(b) Let Sn =n

k=1 Y(Bk) and S = Y(B), where B =n1 Bn. By

Levys equivalence theorem, (Sn)n1converges a.s. if and only if it convergesin distribution. By (10), withui= u for all i = 1, . . . , k, we have:

E(eiuSn) = exp

|nk=1

Bk|R

(eiuz 1 iuz1{|z|1})(dz)

.

This clearly converges to E(eiuS) = exp|B| R(eiuz 1 iuz1{|z|1})(dz),and hence (Sn)n1 converges in distribution to S. Recall that a random variableXhas an-stable distributionwith param-

eters(0, 2), [0, ), [1, 1], R if for anyu R,E(eiuX) = exp

|u|

1 isgn(u)tan

2

+iu

if = 1, or

E(eiuX) = exp

|u|

1 +isgn(u)

2

ln |u|

+iu

if = 1

(see Definition 1.1.6 of [27]). We denote this distribution by S