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 Komlós–Major–Tusnády coupling.
Dans cet article, nous fournissons un contrôle au niveau de processus et non asymptotique entre le processus télégraphique et le mouvement brownien avec une constante de diffusivité appropriée par rapport à la distance de Wasserstein et avec un coût moyen quadratique. De plus, nous dérivons des estimations non asymptotiques pour les p-ièmes moments moyens correspondants. La preuve repose sur des techniques de couplage telles que le couplage pile ou face, le couplage synchrone et le couplage Komlós–Major–Tusnády.
Funding Statement
The research has been supported by the Academy of Finland, via the Matter and Materials Profi4 university profiling action, an Academy project (project No. 339228) and the Finnish centre of excellence in Randomness and STructures (project No. 346306). JL would also like to thank the Isaac Newton Institute for Mathematical Sciences, Cambridge, for support and hospitality during the programme Frontiers in kinetic theory: connecting microscopic to macroscopic (KineCon 2022) where partial work on this paper was undertaken. This work was supported by EPSRC grant no EP/R014604/1.
Acknowledgements
The authors are grateful to the reviewer for the thorough examination of the manuscript, which has lead to a significant improvement.
Citation
Gerardo Barrera. Jani Lukkarinen. "Quantitative control of Wasserstein distance between Brownian motion and the Goldstein–Kac telegraph process." Ann. Inst. H. Poincaré Probab. Statist. 59 (2) 933 - 982, May 2023. https://doi.org/10.1214/22-AIHP1288
Information
