  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.
