A Poincar\'e cone condition in the Poincar\'e group

TL;DR

This paper extends the understanding of the Green function behavior for hypoelliptic diffusions, establishing a Wiener criterion and Poincaré cone condition for a relativistic diffusion in the Poincaré group, generalizing previous sub-elliptic results.

Contribution

It adapts Gradinaru and Ben Arous's proof to hypoelliptic diffusions, deriving new criteria for thinness in the Poincaré group setting.

Findings

Established a Wiener criterion for relativistic diffusions.

Proved a Poincaré cone condition for thinness.

Extended sub-elliptic diffusion results to hypoelliptic case.

Abstract

Gradinaru and Ben Arous described the behavior of the Green function of a sub-elliptic diffusion. We adapt their proof in the general setting of an hypoelliptic diffusion and deduce a Wiener criterion and a Poincar\'e cone condition of thinness for a relativistic diffusion in the Poincar\'e group introduced by Dudley.