Precise Moment Asymptotics For A Class Of Stochastic Parabolic Anderson Model

Stochastic partial differential equation is the partial differential equation with ran-dom terms or random coefficients,and it is an important tool to simulate the reality problem.Stochastic partial differential equations have widely applications in many fields,such as physics,chemistry,financial mathematics,life sciences,control problem,and so on.At present,stochastic partial differential equation has become an impor-tant branch in the field of probability theory.As a kind of very important stochastic partial differential parabolic equations,parabolic Anderson model has attracted much attention and has been well studied and developed.In recent years,a series of research achievements have been obtained on the properties of the solution of parabolic Ander-son equations,the intermittency of the solution has been most concerned.In fact,from a mathematical point of view,to study the intermittency of the solution of parabolic Anderson equations is actually to study the moment asymptotics of the solution.In this article,we consider the following parabolic Anderson equation with the Gaussian noise(?)w/(?)t(t,x)that is formally given as the time derivative of the mean-zero Gaussian field W(t,x)with the covariance function Cov(W(t,x),W(x,y))=1/2(t2H0+s2H0-|t-5|2HO)Γ(X,y),(t,x),(s,y)∈R+×Rd,where the time Hurst parameter H0∈(0,1)Γ(x,y)is the space covariance.In Chapter 3,to the generalized Gaussian noise(?)/(?)tW(t,x)in the above equation,we assume that the space covariance function Γ(x,y)has the homogeneity in the sense that for any x,y ∈Rd and C ∈R,where the constant H ∈(0,1).By using Varadhan’s integral lemma,Brown motion sample path large deviations and Some inequalities in probability theory,We proved the following conclusion:Theorem 1 Assume that 0<HO,H<1 and 1-2H0<H<1,Γ(t,x)is locally bounded and satisfy(2).We have that for every x ∈ Rd,where ε(H0)expressed respectively as follows:when 1/2<H0<1,when 0<H0<1/2,where CH0=H0(2H0-1)and Hd is Cameron-Martin Space.In fact,all the variations can be unified into the following form It should be noticed that the following two cases are covered by the main theorem in Chapter 3:Reamrk 1 When W(t,x)is a fractional Brownian sheet with Hurst parameter(H0,H1,...,Hd),the covariance function is where RHj(xj,yi)=1/2{|xj|2Hj+|yi|2Hi-|xj-yj|2Hj},j=1,…,d one can verify the homogenous assumption in Theorem 1 with H=H1+…+Hd.Reamrk 2 When W(t,x)is a spatial radial fractional Brownian sheet with Hurst parameter(H0,H),the covariance function also satisfies the homogenous condition in Theorem 1,where the covariance function isΓ(x,y)=1/2{|x|2H+|y|2H-|x-y|2H},x,y ∈ Rd.Besides,when the generalized Gaussian noise(?)/(?)tW(t,x)in(2)is white in time and the spatial variable is fractional Brownian sheet with d=1,ε(H0)in Theorem 1 can be evaluated.Corollary 1 When H0=1/2 and the spatial variable is an one-dimension frac-tional Brownian motion with Hurst parameter H,In Chapter 4,based on Chapter 3,when the space covariance function r(x,y)of the generalized Gaussian noise(?)/(?)tW(t,x)in(1),satisfies the following asymptotic homogenous assumption where Γh(x,x)is the space covariance function satisfies the homogenous conditions which in Chapter 3.By truncation,we further obtained the precise moment asymp-totics for the solution of the equation as t→∞ or m→∞.The following is the main result of this chapter.Theorem 2 Assume that 0<H0,H<1 and 1-2H0<H<1,the space covariance function Γ(x,y)of the generalized Gaussian noise(?)/(?)tW(t,x)in(1),satisfies the asymptotic homogenous assumption(9),and for all x ∈ Rd exists some constants C0>0 such thatΓ(x,x)≤C0|x|2H,Then,we have that where ε(H0)can be respectively expressed as follows:when 1/2<H0<1,when H0=1/2.when 0<H0<1/2,where Γh(x,x)is the space covariance function satisfies the homogenous conditions which in Chapter 3,CH0=H0(2H0-1)and Hd is Cameron-Martin Space.