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Consider n non-intersecting Brownian motions on $\mathbb{R}$, depending ontime $t \in [0,1]$, 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 thedistribution of the positions of the non-intersecting Brownian particles at agiven time $0<t<1$, is the same as the joint distribution of the eigenvalues ofa matrix ensemble. This paper proves the existence, for general $p$ and $q$, ofa partial differential equation (PDE) satisfied by the log of the probabilityto 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 theparticles, and the boundary points of the set $E$. The proof of the existenceof such a PDE, using Virasoro constraints and the multicomponent KP hierarchy,is based on the method of elimination of the unwanted partials; that this ispossible is a miracle. Unfortunately we were unable to find its explicitexpression. The case $p=q=2$ will be discussed in the last section.

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