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

Consider n non-intersecting Brownian motions on R , depending on time t ∈ [ 0 , 1 ] , with m i particles forced to leave from a i at time t = 0 , 1 ≤ i ≤ q , and n j particles forced to end up at b j at time t = 1 , 1 ≤ j ≤ 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 = ∪ i = 1 r [ c 2 i − 1 , c 2 i ] ⊂ 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.