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 ${\mathbf{H}}^n$ of order $n$. This equation is written as ${\partial }_{t}u=\frac{1}{2}\Delta u+u{\dot{W}}_{\alpha }$, where $\Delta$ is the hypoelliptic Laplacian on ${\mathbf{H}}^n$ and $\{{\dot{W}}_{\alpha };\alpha >0\}$ is a family of Gaussian space-time noises which are white in time and have a covariance structure generated by $(-\Delta {)}^{-\alpha }$ in space. Our aim is threefold: (i) Give a proper description of the noise $W_{\alpha }$; (ii) Prove that one can solve the stochastic heat equation in the Itô sense as soon as $\alpha >\frac{n}{2}$; (iii) Give some basic moment estimates for the solution $u(t,x)$.