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

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 $(-\mathrm{\Delta }{)}^{-\alpha }$ 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 ${\mathbf{H}}_n$.