Brownian couplings, convexity, and shy-ness

TL;DR

This paper extends the non-existence of shy couplings for reflected Brownian motions to all bounded convex domains without line segments on the boundary, using potential-theoretic methods.

Contribution

It generalizes previous results by removing smoothness and dimensional restrictions on the domain boundaries.

Findings

No shy couplings exist in bounded convex domains without boundary line segments.

Potential-theoretic methods effectively prove the non-existence of shy couplings.

Results apply to both planar and higher-dimensional convex domains.

Abstract

Benjamini, Burdzy and Chen (2007) introduced the notion of a shy coupling: a coupling of a Markov process such that, for suitable starting points, there is a positive chance of the two component processes of the coupling staying a positive distance away from each other for all time. Among other results, they showed no shy couplings could exist for reflected Brownian motions in C^2 bounded convex planar domains whose boundaries contain no line segments. Here we use potential-theoretic methods to extend this Benjamini et al. result (a) to all bounded convex domains (whether planar and smooth or not) whose boundaries contain no line segments, (b) to all bounded convex planar domains regardless of further conditions on the boundary.