Singular SPDEs on Homogeneous Lie Groups

TL;DR

This paper extends the theory of regularity structures to handle singular SPDEs on homogeneous Lie groups, including hypoelliptic operators, and applies it to solve parabolic Anderson equations on Carnot groups.

Contribution

It develops a framework for analyzing singular SPDEs with hypoelliptic operators on homogeneous Lie groups, broadening the scope beyond elliptic cases.

Findings

Successfully extended regularity structures to hypoelliptic operators.

Solved parabolic Anderson equations on Carnot groups.

Provided examples including kinetic Fokker-Planck operator.

Abstract

The aim of this article is to extend the scope of the theory of regularity structures in order to deal with a large class of singular SPDEs of the form $$\partial_t u = \mathfrak{L} u+ F(u, \xi)\ ,$$ where the differential operator $\mathfrak{L}$ fails to be elliptic. This is achieved by interpreting the base space $\mathbb{R}^{d}$ as a non-trivial homogeneous Lie group $\mathbb{G}$ such that the differential operator $\partial_t -\mathfrak{L}$ becomes a translation invariant hypoelliptic operator on $\mathbb{G}$. Prime examples are the kinetic Fokker-Planck operator $\partial_t -\Delta_v - v\cdot \nabla_x$ and heat-type operators associated to sub-Laplacians. As an application of the developed framework, we solve a class of parabolic Anderson type equations $$\partial_t u = \sum_{i} X^2_i u + u (\xi-c)$$ on the compact quotient of an arbitrary Carnot group.