Liouville results for fully nonlinear equations modeled on H\"ormander vector fields: II. Carnot groups and Grushin geometries

TL;DR

This paper establishes Liouville theorems for fully nonlinear degenerate elliptic equations on various sub-Riemannian structures, including Carnot groups and Grushin geometries, extending previous results and exploring applications to diffusion processes.

Contribution

It extends Liouville results to a broad class of sub-Riemannian geometries with explicit conditions, including Carnot and Grushin structures, and discusses applications to ergodicity of degenerate diffusions.

Findings

Liouville properties hold under specific bounds at infinity.

Explicit conditions on lower order terms are provided.

Applications to ergodic behavior of diffusion processes are outlined.

Abstract

The paper treats second order fully nonlinear degenerate elliptic equations having a family of subunit vector fields satisfying a full-rank bracket condition. It studies Liouville properties for viscosity sub- and supersolutions in the whole space, namely, that under a suitable bound at infinity from above and, respectively, from below, they must be constants. In a previous paper we proved an abstract result and discussed operators on the Heisenberg group. Here we consider various families of vector fields: the generators of a Carnot group, with more precise results for those of step 2, in particular H-type groups and free Carnot groups, the Grushin and the Heisenberg-Greiner vector fields. All these cases are relevant in sub-Riemannian geometry and have in common the existence of a homogeneous norm that we use for building Lyapunov-like functions for each operator. We give explicit sufficient conditions on the size and sign of the first and zero-th order terms in the equations and discuss their optimality. We also outline some applications of such results to the problem of ergodicity of multidimensional degenerate diffusion processes in the whole space.