  We study the general solution of the Fokker-Planck equation in d dimensions
with arbitrary space and time dependent diffusion matrix and drift term. We
show how to construct the solution, for arbitrary initial distributions, as an
asymptotic expansion for small time. This generalizes the well-known asymptotic
expansion of the heat-kernel for the Laplace operator on a general Riemannian
manifold. We explicitly work out the general solution to leading and
next-to-leading order in this small-time expansion, as well as to
next-to-next-to-leading order for vanishing drift. We illustrate our results on
a several examples.
