Non-intersecting Brownian motions leaving from and going to several points

Abstract

Consider n non-intersecting Brownian motions on $\mathbb{R}$, depending on time $t \in [0,1]$, with $m_i$ particles forced to leave from $a_i$ at time $t=0$, $1\leq i\leq q$, and $n_j$ particles forced to end up at $b_j$ at time $t=1$, $1\leq j\leq p$. For arbitrary $p$ and $q$, it is not known if the distribution of the positions of the non-intersecting Brownian particles at a given time $0<t<1$, is the same as the joint distribution of the eigenvalues of a matrix ensemble. This paper proves the existence, for general $p$ and $q$, of a partial differential equation (PDE) satisfied by the log of the probability to find all the particles in a disjoint union of intervals $E=\cup_{i=1}^{r}[c_{2i-1},c_{2i}]\subset\mathbb{R}$ at a given time $0<t<1$. The variables are the coordinates of the starting and ending points of the particles, and the boundary points of the set $E$. The proof of the existence of such a PDE, using Virasoro constraints and the multicomponent KP hierarchy, is based on the method of elimination of the unwanted partials; that this is possible is a miracle. Unfortunately we were unable to find its explicit expression. The case $p=q=2$ will be discussed in the last section.