On Brownian motion on the plane with membranes on rays with a common endpoint

TL;DR

This paper studies a Brownian motion on the plane with semipermeable membranes on rays emanating from a common point, providing conditions for the process to hit the origin and determining the probability of such an event.

Contribution

It establishes necessary and sufficient conditions for the Brownian motion to reach the origin and characterizes the probability of hitting the origin as zero or one.

Findings

Conditions for the process to reach the origin are derived.

The probability of hitting the origin is either zero or one.

The study advances understanding of Brownian motion with membranes on rays.

Abstract

We consider a Brownian motion on the plane with semipermeable membranes on n rays that have a common endpoint in the origin. We obtain the necessary and sufficient conditions for the process to reach the origin and we show that the probability of hitting the origin is equal to zero or one.