Brownian motion with asymptotically normal reflection in unbounded domains: from transience to stability

Brešar, Miha; Mijatović, Aleksandar; Wade, Andrew

Abstract

We quantify the asymptotic behaviour of multidimensional drifltess diffusions in domains unbounded in a single direction with asymptotically normal reflections from the boundary. We identify the critical growth/contraction rates of the domain that separate stability, null recurrence and transience. In the stable case, we prove existence and uniqueness of the invariant distribution and establish the polynomial rate of decay of its tail. We also establish matching polynomial upper and lower bounds on the rate of convergence to stationarity in total variation. All exponents are explicit in the model parameters that determine the asymptotics of the growth rate of the domain, the interior covariance and the reflection vector field.

Citation

Brešar, M., Mijatović, A., & Wade, A. (2025). Brownian motion with asymptotically normal reflection in unbounded domains: from transience to stability. Annals of Probability, 53(1), 175-222. https://doi.org/10.1214/24-AOP1703

• Journal Article Type: Article
• Acceptance Date: Mar 29, 2024
• Online Publication Date: Jan 19, 2025
• Publication Date: 2025-01
• Deposit Date: Mar 14, 2023
• Publicly Available Date: Jan 19, 2025
• Journal: The Annals of Probability
• Print ISSN: 0091-1798
• Publisher: Institute of Mathematical Statistics
• Peer Reviewed: Peer Reviewed
• Volume: 53
• Issue: 1
• Pages: 175-222
• DOI: https://doi.org/10.1214/24-AOP1703
• Public URL: https://durham-repository.worktribe.com/output/1177079
• Publisher URL: https://imstat.org/journals-and-publications/annals-of-probability/annals-of-probability-future-papers/
• Related Public URLs: https://doi.org/10.48550/arXiv.2303.06916