Weighted Besov spaces on Heisenberg groups and applications to the Parabolic Anderson model
Fabrice Baudoin, Li Chen, Che-Hung Huang, Cheng Ouyang, Samy Tindel, Jing Wang
TL;DR
The paper develops a pathwise analysis of the parabolic Anderson model on the Heisenberg group ${\mathbf{H}^n}$ by constructing a robust weighted Besov-space framework via a projective Fourier transform. It establishes solvability under precise temporal-spatial covariance constraints for the driving Gaussian noise, and introduces a comprehensive paraproduct calculus tailored to the sub-Riemannian geometry. The main technical advances include the Gevrey-class regularity, Bernstein-type estimates, and sharp heat-flow smoothing in weighted Besov spaces, enabling well-posedness results for SPDEs in this non-Euclidean setting. The results illuminate how sub-Riemannian geometry modulates stochastic exponents and provide a versatile toolkit for SPDEs driven by spatially rough, temporally colored noises on Heisenberg groups, with potential extensions to other geometric settings.
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 $(-Δ)^{-α}$ 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 $\mathbf{H}_{n}$.