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

Alain Comtet 1, 2, Jean Desbois 1

Journal of Physics A 36 (2003) L255-L262

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.

• 1. Laboratoire de Physique Théorique et Modèles Statistiques (LPTMS),
CNRS : UMR8626 – Université Paris XI - Paris Sud
• 2. IHP,
Institut Henri Poincaré