Polynomial rate of convergence to the Yaglom limit for Brownian motion with drift

TL;DR

This paper establishes that the convergence of the law of a Brownian motion with drift, conditioned to avoid hitting zero, towards its Yaglom limit occurs at a polynomial rate of 1/t in Wasserstein distance.

Contribution

It proves a polynomial convergence rate for the conditioned process towards the Yaglom limit, applicable to a broad class of initial measures.

Findings

Wasserstein distance decays as 1/t for the conditioned process.

Convergence rate applies to initial measures with compact support and Dirac measures.

Similar polynomial rate observed for the Bessel-3 process conditioned on non-absorption.

Abstract

This paper deals with the rate of convergence in 1-Wasserstein distance of the marginal law of a Brownian motion with drift conditioned not to have reached 0 towards the Yaglom limit of the process. In particular it is shown that, for a wide class of initial measures including Dirac measures and probability measures with compact support, the Wasserstein distance decays asymptotically as 1/t. Likewise, this speed of convergence is recovered for the convergence of marginal laws conditioned not to be absorbed up to a horizon time towards the Bessel-$3$ process, when the horizon time tends to infinity.