Precise Almost Sure Asymptotics For Two Classes Of Continuous Parabolic Anderson Models

In this paper,we consider the following two classes of continuous parabolic An-derson models.First,we study parabolic Anderson model with time-independent Gaussian field(?)where the parameter ??R\{0} and the centered generalized Gaussian field V on Rd is given by a centered Gaussian family (?)with covariance(?)Here,the k(x,y)is a positive definite kernel on Rd×Rd.For the covariance k(x,y)of the Gaussian field V,we respectively consider the following two cases:(?)The k(x,y)is stationary,which means that there exists the generalized function? such that ?(x-y)=k(x,y).Here,the ? is point-wise defined in Rd\{0} and bounded outside every neighborhood of 0,and satisfies(?)(?)The k(x,y)satisfies(?)where log+x:=(logx)V O,the g(x,y)is a bounded function on Rd×Rd and the parameter T>0 is called correlation length.In model(1),assume that the initial value u0(x)belongs to the weighted Besov space Bkv,wx q,?,and satisfies(?)We obtain the following two results.(1)Precise quenched long-time asymptotics:In the cases(?)and(?),let u(t,x)be the pathwise mild solution of model(1).Then,for all x?Rd,it holds that(?)(2)Precise spatial asymptotics:In the cases(?)and(?),let u(t,x)be the pathwise mild solution of model(1).Then,for all t>0,it holds that(?)In the above two limits,the function ?(x)on R+satisfying ?(x)>e and the equation(?) when x is enough large.Second,we also study parabolic Anderson model with time-dependent Gaussian field V(t,x)(?)where the parameter ??R\{0} and the centered generalized Gaussian field V(t,x)on R+x Rd is given by a centered Gaussian family ?(V,?);??S(R_×Rd)} with covariance(?)where T is Fourier transformation about spatial variable.We assume that the time-space covariance of the Gaussian field V(t,x)respectively satisfy the following two conditions:(?)(H2)the function ?0 is positive,and there exists some ?0 satisfying(?)such that for all (?),it holds that (?)In model(2),we still assume that the initial value u0(x)satisfies(?)Define the variation(?)where the set of functions(?)Then,we obtain the following result.Precise spatial asymptotics:For all t>0 and ??R/{0},the solution u?(t,x)of model(2)satisfies that(?)...