We call "Dyson process" any process on ensembles of matrices in which the entries
undergo diffusion. We are interested in the distribution of the eigenvalues (or singular
values) of such matrices. In the original Dyson process it was the ensemble of n by n
Hermitian matrices, and the eigenvalues describe n curves. Given sets X_1,...,X_m the
probability that for each k no curve passes through X_k at time \tau_k is given by the
Fredholm determinant of a certain matrix kernel, the extended Hermite kernel. For this
reason we call this Dyson process the Hermite process. Similarly, when the entries of a
complex matrix undergo diffusion we call the evolution of its singular values the Laguerre
process, for which there is a corresponding extended Laguerre kernel. Scaling the Hermite
process at the edge leads to the Airy process and in the bulk to the sine process; scaling
the Laguerre process at the edge leads to the Bessel process. Generalizing and
strengthening earlier work, we assume that each X_k is a finite union of intervals and find
for the Airy process a system of partial differential equations, with the end-points of the
intervals of the X_k as independent variables, whose solution determines the probability
that for each k no curve passes through X_k at time \tau_k. Then we find the analogous
systems for the Hermite process (which is more complicated) and also for the sine process.
Finally we find an analogous system of PDEs for the Bessel process, which is the most
difficult.