Large deviations principle for sub-Riemannian random walks

TL;DR

This paper establishes a large deviations principle for random walks on stratified Lie groups, focusing on increments in generating directions and considering the sub-Riemannian geometric structure.

Contribution

It introduces a large deviations framework tailored to sub-Riemannian spaces, extending classical results to stratified Lie groups with specific increment constraints.

Findings

Proves a large deviation principle for stratified Lie group random walks.

Derives a rate function aligned with sub-Riemannian geometry.

Analyzes the impact of increment distribution constraints on deviations.

Abstract

We study large deviations for random walks on stratified (Carnot) Lie groups. For such groups, there is a natural collection of vectors which generates their Lie algebra, and we consider random walks with increments in only these directions. Under certain constraints on the distribution of the increments, we prove a large deviation principle for these random walks with a natural rate function adapted to the sub-Riemannian geometry of these spaces.