Abstract

In a celebrated paper, Dyson shows that the spectrum of a n × n random Hermitian matrix, diffusing according to an Ornstein-Uhlenbeck process, evolves as n non-colliding Brownian motions held together by a drift term. The universal edge and bulk scalings for Hermitian ran-dom matrices, applied to the Dyson process, lead to the Airy and Sine processes. In particular, the Airy process is a continuous stationary process, describing the motion of the outermost particle of the Dyson Brownian motion, when the number of particles gets large, with space and time appropriately rescaled. In this paper, we answer a question posed by Kurt Johansson, to find a PDE for the joint distribution of the Airy Process at two differ-ent times. Similarly we find a PDE satisfied by the joint distribution of the Sine process. This hinges on finding a PDE for the joint distri-bution of the Dyson process, which itself is based on the joint proba-bility of the eigenvalues for coupled Gaussian Hermitian matrices. The