Reflecting Brownian motion in generalized parabolic domains: explosion and superdiffusivity

Menshikov, Mikhail V.; Mijatović, Aleksandar; Wade, Andrew R.

Abstract

For a multidimensional driftless diffusion in an unbounded, smooth, sub-linear generalized parabolic domain, with oblique reflection from the boundary, we give natural conditions under which either explosion occurs, if the domain narrows sufficiently fast at infinity, or else there is superdiffusive transience, which we quantify with a strong law of large numbers. For example, in the case of a planar domain, explosion occurs if and only if the area of the domain is finite. We develop and apply novel semimartingale criteria for studying explosions and establishing strong laws, which are of independent interest.

Citation

Menshikov, M. V., Mijatović, A., & Wade, A. R. (2023). Reflecting Brownian motion in generalized parabolic domains: explosion and superdiffusivity. Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, 59(4), 1813-1843. https://doi.org/10.1214/22-AIHP1309

• Journal Article Type: Article
• Acceptance Date: Aug 26, 2022
• Publication Date: 2023-11
• Deposit Date: Mar 3, 2022
• Publicly Available Date: Nov 30, 2023
• Journal: Annales de l'Institut Henri Poincaré
• Print ISSN: 0246-0203
• Publisher: Institute Henri Poincaré
• Peer Reviewed: Peer Reviewed
• Volume: 59
• Issue: 4
• Pages: 1813-1843
• DOI: https://doi.org/10.1214/22-AIHP1309
• Public URL: https://durham-repository.worktribe.com/output/1212811