A PDE for Nonintersecting Brownian Motions and Applications

TL;DR

This paper derives a new non-linear PDE for non-intersecting Brownian motions with multiple targets, revealing a bifurcation phenomenon and a new critical process characterized by a perturbed Pearcey kernel.

Contribution

It introduces a novel PDE expressed as a near-Wronskian for transition probabilities of non-intersecting Brownian motions with multiple target points.

Findings

Derived a non-linear PDE for transition probabilities in non-intersecting Brownian motions.

Identified a bifurcation point where particle behavior changes, described by a perturbed Pearcey kernel.

Proposed a conjecture relating the PDE and probabilistic structures of the process.

Abstract

Consider non-intersecting Brownian motions on the real line, starting from the origin at t=0, with a number of particles forced to reach p distinct target points at time t=1. This work shows that the transition probability, that is the probability for the particles to pass through windows E_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 target points (the first and the last one) tend to infinity; keep the number of particles forced to intermediate target 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 the first target point depart from those going to the last target point. These statistical fluctuations near that point of bifurcation are specified by a kernel, which is a rational perturbation of the Pearcey kernel. Finally, the paper contains a conjecture.