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In a celebrated paper, Dyson shows that the spectrum of an n\times n randomHermitian matrix, diffusing according to an Ornstein-Uhlenbeck process, evolvesas n noncolliding Brownian motions held together by a drift term. The universaledge and bulk scalings for Hermitian random matrices, applied to the Dysonprocess, lead to the Airy and Sine processes. In particular, the Airy processis a continuous stationary process, describing the motion of the outermostparticle of the Dyson Brownian motion, when the number of particles gets large,with space and time appropriately rescaled. In this paper, we answer a questionposed by Kurt Johansson, to find a PDE for the joint distribution of the Airyprocess at two different times. Similarly we find a PDE satisfied by the jointdistribution of the Sine process. This hinges on finding a PDE for the jointdistribution of the Dyson process, which itself is based on the jointprobability of the eigenvalues for coupled Gaussian Hermitian matrices. The PDEfor the Dyson process is then subjected to an asymptotic analysis, consistentwith the edge and bulk rescalings. The PDEs enable one to compute theasymptotic behavior of the joint distribution and the correlation for theseprocesses at different times t_1 and t_2, when t_2-t_1\to \infty, asillustrated in this paper for the Airy process. This paper also contains arigorous proof that the extended Hermite kernel, governing the jointprobabilities for the Dyson process, converges to the extended Airy and Sinekernels after the appropriate rescalings.

PDEs for the joint distributions of the Dyson, Airy and Sine processes