  In a celebrated paper, Dyson shows that the spectrum of an n\times n random
Hermitian matrix, diffusing according to an Ornstein-Uhlenbeck process, evolves
as n noncolliding Brownian motions held together by a drift term. The universal
edge and bulk scalings for Hermitian random 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 different times. Similarly we find a PDE satisfied by the joint
distribution of the Sine process. This hinges on finding a PDE for the joint
distribution of the Dyson process, which itself is based on the joint
probability of the eigenvalues for coupled Gaussian Hermitian matrices. The PDE
for the Dyson process is then subjected to an asymptotic analysis, consistent
with the edge and bulk rescalings. The PDEs enable one to compute the
asymptotic behavior of the joint distribution and the correlation for these
processes at different times t_1 and t_2, when t_2-t_1\to \infty, as
illustrated in this paper for the Airy process. This paper also contains a
rigorous proof that the extended Hermite kernel, governing the joint
probabilities for the Dyson process, converges to the extended Airy and Sine
kernels after the appropriate rescalings.
