A PDE for non-intersecting Brownian motions and applications

摘要

Consider N=n_1+n_2+...+n_p non-intersecting Brownian motions on the real line, starting from the origin at t=0, with ni particles forced to reach p distinct target points β_i at time t=1, with β_1<β_2<β<β_p. This can be viewed as a diffusion process in a sector of R~N. This work shows that the transition probability, that is the probability for the particles to pass through windows ?_k at times t_k, satisfies, in a new set of variables, a non-linear PDE which can be expressed as a near-Wronskian; that is a determinant of a matrix of size p+1, with each row being a derivative of the previous, except for the last column. It is an interesting open question to understand those equations from a more probabilistic point of view. As an application of these equations, let the number of particles forced to the extreme points β_1 and β_p tend to infinity; keep the number of particles forced to intermediate points fixed (inliers), but let the target points themselves go to infinity according to a proper scale. A new critical process appears at the point of bifurcation, where the bulk of the particles forced to -√n depart from those going to √n. These statistical fluctuations near that point of bifurcation are specified by a kernel, which is a rational perturbation of the Pearcey kernel. This work also shows that such equations are an essential tool in obtaining certain asymptotic results. Finally, the paper contains a conjecture.