Additive and Multiplicative Noise Excitability of Stochastic Partial Differential Equations

GUO Zhongkai(郭仲凯),CHENG Shuilin(程水林),WANG Weifeng(王维峰)

( 1.School of Mathematics and Statistics,South-Central University for Nationalities,Wuhan 430074,China; 2.School of Statistics and Mathematics,Zhongnan University of Economics and Law,Wuhan 430073,China )

Abstract: In this paper,noise excitability of energy solution to stochastic partial differential equation with additive and multiplicative cases are considered.By using Itˆo’s formula and energy estimate method,we obtain excitation indices of u at different cases with different results.Thus,from this point of view,the effects of additive noise and multiplicative noise on the system are different.

Key words: Itˆo’s formula; Stochastic partial differential equation; Energy estimate method; Noise excitability

1.Introduction

In recent years,the impact of noise on the system has been extensively studied.In some systems,noise is not negligible,whereas in some other situations,noise could even be beneficial.Thus,it is desirable to better understand the impact of noise on the dynamical evolution of complex systems.Many authors concerned this topic by different approaching[1,5−7,9−10].

Stochastic partial differential equations with additive and multiplicative have been studied by many authors from the stochastic dynamic view.By converting an stochastic differential equations system to a system of random differential equations,stochastic invariant manifolds,stochastic attractor and other topics have been investigated[3−4].The main objective of this paper is to study the effect of noise on the solutions to stochastic partial differential equation with additive noise and multiplicative noise case by the noise excitability index approaching.Forl >0,consider the additive noise case and multiplicative noise case as follows:

and

whereW(t) is a Wiener process,with covarianceQ,taking values in the Hilbert spaceH=L2(0,l).We assume that Tr(Q)< ∞,that means the operatorQis a trace class operator,νis a positive number,˙W(t) is a noise,colored in space but white in time,λis a positive number called the noise intensity,andB(t) is a scalar Brownian motion.

Hereandrepresent the lower and upper excitation indices ofuat timet.In many cases of interest,andare equal and do not depend on the time variablet.In such case we denotee(t)=(t)=(t).

Recently,many authors have considered similarity problems[8,10,11].All the authors considered the case that stochastic heat equation without nonlinear term under Dirichlet boundary condition and Neumann boundary condition.In [8],the authors tooku(t) as a mild solution for stochastic differential equation inEt(λ).Using heat kernel estimate,the authors derived the lower excitation index and upper excitation index ofuat timet.In [10],the authors tooku(t)as an energy solution for stochastic differential equation inEt(λ).Using the energy estimate for the upper excitation index ofuat timet,for the lower excitation index,they considered the corresponding eigenvalue problem for the elliptic equation

And using the Ito’s formula to=(u,φ1)2,they derived the lower excitation index.Similar method will be used in our paper.

In this paper,we study the noise excitability of energy solution for stochastic parabolic equations with nonlinear term.Under different noise cases,we derive different results,from which another point of view reflects the difference between the additive noise and multiplicative noise.This paper is organized as follows: In Section 2,basic notation and main results are presented.In Section 3,we concern with the proof of the main results.

2.Preliminaries and Main Results

In this section,we first give the assumption of the nonlinear term off(u) and recall some known results about the existence and unique solution for stochastic partial differential equation,and then state out the main results.

Hypothesis H1There exists positive constantsL,K,κsuch thatfsatisfies the following conditions

Theorem 2.1[2,7]Suppose Hypothesis H1 holds,andu0is aF0-measurable random field such that E∥u0∥2< ∞.Then the initial-boundary value problem for Eqs.(1.1) and(1.2) has a unique mild solutionu(t) which is a continuous adapted process inHsuch thatu ∈L2(Ω;C([0,T];H)) and

for some constantC >0.

Theorem 2.2Suppose the condition in Hypothesis H1 holds.Then the nose excitation index of the energy solution to (1.1) is 0.

Theorem 2.3Suppose the condition in Hypothesis H1 holds.Then nose excitation index of the energy solution to (1.2) is 2.

3.Proof of Main Results

In this section,we will deduce the excitation index ofuat timetfor stochastic partial differential equation with additive and multiplicative noise cases separate.

Proof of Theorem 2.2(Additive noise case) Consider the following equation

To compute the excitation index of the solution at timet,we divide it into two steps.

Thus,taking expectations on both sides,we have

Then,using Gronwall’s inequality,we get

Using the definition of noise upper excitation index of the solution at timet,we have

Step 2(t)≥0

On the other hand,in order to get the lower bounded,we will consider the eigenvalue problem for the elliptic equation

Then,all the eigenvalues are strictly positive,which are denoted byand the corresponding eigenfunctions areConsiderheren=1,2,3,···.By applying Itˆo’s formula toand making use of (3.2),we have

Then we have

and

By the comparison principle,we know that

From the definition of noise excitation index of the solution at timet,we have

Remark 3.1In [8],the definition of the noise excitation index of the solution at timetis required to be strictly positive for mild solution.However,if we consider the energy solution,the definition of the noise excitation index of the solution at timetcan be extended to non-negative.

Proof of Theorem 2.3(Multiplicative noise case) Considering the following equation

we have the result ofe(t)=2.

By Itˆo’s formula,we have

Thus,

Step 2(t)≥2

In order to get the lower bounded,similarly,we will consider the eigenvalue problem for the elliptic equation

Then,we have

and

Taking expectations on both sides yields that

By the comparison principle,we have