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

## TL;DR

This paper derives a small-time asymptotic expansion for the Fokker-Planck kernel with space- and time-dependent coefficients, generalizing heat kernel expansions on Riemannian manifolds, and provides explicit solutions up to second order.

## Contribution

It introduces a method to construct small-time solutions for the Fokker-Planck equation with arbitrary coefficients, extending classical heat kernel asymptotics to more general stochastic processes.

## Key findings

- Explicit leading and next-to-leading order solutions derived.
- Second-order solutions obtained for zero drift case.
- Applications demonstrated through multiple examples.

## Abstract

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.