Weighted Besov spaces on Heisenberg groups and applications to the Parabolic Anderson model

TL;DR

This paper develops weighted Besov spaces on Heisenberg groups and applies them to establish solvability conditions for a stochastic PDE, the parabolic Anderson model, driven by smoother-than-white noise.

Contribution

It introduces a new definition of weighted Besov spaces on Heisenberg groups and demonstrates their use in solving the parabolic Anderson model with specific noise conditions.

Findings

Optimal conditions for noise covariance ensuring solvability.

New definition of weighted Besov spaces on Heisenberg groups.

Application of projective Fourier transform approach.

Abstract

This article aims at a proper definition and resolution of the parabolic Anderson model on Heisenberg groups $\mathbf{H}_{n}$. This stochastic PDE is understood in a pathwise (Stratonovich) sense. We consider a noise which is smoother than white noise in time, with a spatial covariance function generated by negative powers $(-\Delta)^{-\alpha}$ of the sub-Laplacian on $\mathbf{H}_{n}$. We give optimal conditions on the covariance function so that the stochastic PDE is solvable. A large portion of the article is dedicated to a detailed definition of weighted Besov spaces on $\mathbf{H}_{n}$. This definition, related paraproducts and heat flow smoothing properties, forms a necessary step in the resolution of our main equation. It also appears to be new and of independent interest. It relies on a recent approach, called projective, to Fourier transforms on $\mathbf{H}_{n}$.