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

Abstract

We consider a planar Brownian motion starting from $O$ at time $t=0$ and stopped at $t=1$ and a set $F=\{O{I}_i;i=1,2,...,n\}$ 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.