Quantitative control of Wasserstein distance between Brownian motion and the Goldstein--Kac telegraph process

TL;DR

This paper establishes explicit bounds on the Wasserstein distance between the telegraph process and Brownian motion, providing non-asymptotic estimates for their moments using advanced coupling techniques.

Contribution

It introduces a non-asymptotic, process-level control of the Wasserstein distance between the telegraph process and Brownian motion, with new coupling-based proof methods.

Findings

Explicit Wasserstein distance bounds between processes

Non-asymptotic estimates for time average moments

Application of advanced coupling techniques

Abstract

In this manuscript, we provide a non-asymptotic process level control between the telegraph process and the Brownian motion with suitable diffusivity constant via a Wasserstein distance with quadratic average cost. In addition, we derive non-asymptotic estimates for the corresponding time average $p$-th moments. The proof relies on coupling techniques such as coin-flip coupling, synchronous coupling and the Koml\'os--Major--Tusn\'ady coupling.