  The extended Airy kernel describes the space-time correlation functions for
the Airy process, which is the limiting process for a polynuclear growth model.
The Airy functions themselves are given by integrals in which the exponents
have a cubic singularity, arising from the coalescence of two saddle points in
an asymptotic analysis. Pearcey functions are given by integrals in which the
exponents have a quartic singularity, arising from the coalescence of three
saddle points. A corresponding Pearcey kernel appears in a random matrix model
and a Brownian motion model for a fixed time. This paper derives an extended
Pearcey kernel by scaling the Brownian motion model at several times, and a
system of partial differential equations whose solution determines associated
distribution functions. We expect there to be a limiting nonstationary process
consisting of infinitely many paths, which we call the Pearcey process, whose
space-time correlation functions are expressible in terms of this extended
kernel.
