Parabolic Anderson model on Heisenberg groups: the It\^o setting

TL;DR

This paper studies a stochastic heat equation on the Heisenberg group driven by Gaussian noise, establishing conditions for well-posedness in the Itô sense and providing moment estimates for solutions.

Contribution

It introduces a rigorous framework for solving the stochastic heat equation on Heisenberg groups with specific Gaussian noise and derives solution existence and moment bounds.

Findings

Solution exists when noise parameter lpha > n/2

Provides a detailed description of the Gaussian noise structure

Establishes basic moment estimates for the solution

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 It\^{o} sense as soon as $\alpha>\frac{n}{2}$; (iii) Give some basic moment estimates for the solution $u(t,x)$.