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We study the general solution of the Fokker-Planck equation in d dimensionswith arbitrary space and time dependent diffusion matrix and drift term. Weshow how to construct the solution, for arbitrary initial distributions, as anasymptotic expansion for small time. This generalizes the well-known asymptoticexpansion of the heat-kernel for the Laplace operator on a general Riemannianmanifold. We explicitly work out the general solution to leading andnext-to-leading order in this small-time expansion, as well as tonext-to-next-to-leading order for vanishing drift. We illustrate our results ona several examples.

Small-time expansion of the Fokker-Planck kernel for space and time dependent diffusion and drift coefficients