A coupling strategy for Brownian motions at fixed time on Carnot groups using Legendre expansion

TL;DR

This paper introduces a novel coupling method for sub-Riemannian Brownian motions on Carnot groups using Legendre expansion, leading to sharp decay estimates and functional inequalities.

Contribution

It presents a new coupling construction based on Legendre expansion for Brownian motions on Carnot groups, providing sharp decay estimates and functional inequalities.

Findings

Sharp total variation decay estimates between Brownian motions

Derivation of log-Harnack inequality and Bismut type formula

Establishment of reverse Poincaré inequalities

Abstract

We propose a new simple construction of a coupling at a fixed time of two sub-Riemannian Brownian motions on the Heisenberg group and on the free step 2 Carnot groups. The construction is based on a Legendre expansion of the standard Brownian motion and of the L{\'e}vy area. We deduce sharp estimates for the decay in total variation distance between the laws of the Brownian motions. Using a change of probability method, we also obtain the log-Harnack inequality, a Bismut type integration by part formula and reverse Poincar\'e inequalities for the associated semi-group.