Temporal asymptotics for a fractional parabolic Anderson model

Abstract

In this talk, some of recent developments on the parabolic Anderson model, which is described by a class of linear stochastic partial differential equations (SPDEs) with multiplicative Gaussian noise, will be reviewed. Then we consider fractional parabolic equation of the form ∂u ∂t =−(−Δ)α/2u+u ˙ W(t,x) ∂u∂t=−(−Δ)α/2u+uW˙(t,x) , where −(−Δ)α/2 −(−Δ)α/2 with α∈(0,2] α∈(0,2] is a fractional Laplacian and ˙W is a Gaussian noise colored in space and time. The precise moment Lyapunov exponents of the Stratonovich solution and the Skorohod solution are obtained. The result is based on the work joint with X. Chen, Y. Hu and X. Song.