Parabolic Anderson model on Heisenberg groups: The Itô setting

Abstract

In this note we focus our attention on a stochastic heat equation defined on the Heisenberg group Hn of order n. This equation is written as ∂tu=12Δu+uW˙α, where Δ is the hypoelliptic Laplacian on Hn and {W˙α;α>0} is a family of Gaussian space-time noises which are white in time and have a covariance structure generated by (−Δ)−α in space. Our aim is threefold: (i) Give a proper description of the noise Wα; (ii) Prove that one can solve the stochastic heat equation in the Itô sense as soon as α>n2; (iii) Give some basic moment estimates for the solution u(t,x).