A collection of results relating the geometry of plane domains and the exit time of planar Brownian motion, II

TL;DR

This paper extends previous work on the geometry of plane domains and Brownian motion exit times, providing new results on moments for specific domain types and constructing domains with particular exit time behaviors.

Contribution

It introduces new results on the moments of Brownian motion exit times for domains with modified boundaries and constructs domains with behaviors addressing open questions.

Findings

Derived moments of exit times for domains with dashed boundary lines.

Analyzed domains from periodic tilings of the plane.

Constructed wedge-like domains with specific exit time properties.

Abstract

This paper is the sequel to another with the same name (Buttigieg et al., Comput. Methods Funct. Theory, 2023), and is concerned with results of the same type. We deduce a result on the moments of the exit time of Brownian motion from domains whose boundary curve is replaced by a dashed line, and from domains arising from a periodic tiling of the plane. We also give a construction of a type of domain which is similar to a wedge domain, but the behaviour of whose exit time moments answer several questions that had been speculated upon.