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

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 coinflip coupling, synchronous coupling and the Komlós-Major-Tusnády coupling.