Reflecting Brownian motion in generalized parabolic domains: explosion and superdiffusivity

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

Annales de l'Institut Henri Poincaré, 59, no. 4, November 2023, 1813–1843. DOI: 10.1214/22-AIHP1309

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.