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

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

2003-02-13

arxiv.org/abs/cond-mat/0302269v1

Physics

Condensed Matter

Disordered Systems and Neural Networks

Scientific paper

10.1088/0305-4470/36/17/101

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.

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