NON-INTERSECTING BROWNIAN MOTIONS LEAVING FROM AND GOING TO SEVERAL POINTS

Abstract. Consider n non-intersecting Brownian motions on R, depending on time t ∈ [0, 1], with mi particles forced to leave from ai at time t = 0, 1 ≤ i ≤ q, and nj particles forced to end up at bj 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 = ∪ri=1[c2i−1, c2i] ⊂ 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. Content