Brownian Motion in wedges, last passage time and the second arc-sine law

TL;DR

This paper analyzes the last passage time of planar Brownian motion in wedges formed by semi-infinite lines, deriving explicit probability laws involving arcsine functions, and relates it to reaction-diffusion problems.

Contribution

It provides explicit formulas for the last passage time distribution in wedge-shaped domains, extending Levy's classical result and connecting to reaction-diffusion systems.

Findings

Probability law expressed as sum of arcsine functions for symmetric configurations

Special case of Levy's result recovered for n=2

Links to reaction-diffusion problems with three particles

Abstract

We consider a planar Brownian motion starting from $O$ at time $t=0$ and stopped at $t=1$ and a set $F= \{OI_i ; i=1,2,..., n\}$ of $n$ semi-infinite straight lines emanating from $O$. Denoting by $g$ the last time when $F$ is reached by the Brownian motion, we compute the probability law of $g$. In particular, we show that, for a symmetric $F$ and even $n$ values, this law can be expressed as a sum of $\arcsin $ or $(\arcsin)^2 $ functions. The original result of Levy is recovered as the special case $n=2$. A relation with the problem of reaction-diffusion of a set of three particles in one dimension is discussed.