The diffusion equation for non-Markovian Gaussian stochastic processes

TL;DR

This paper derives an exact, generalized diffusion equation for Gaussian stochastic processes without assuming Markovianity or stationarity, extending the Fokker-Planck framework to non-Markovian dynamics.

Contribution

It introduces a systematic hierarchy of equations using Wick's theorem to describe non-Markovian Gaussian diffusion processes.

Findings

Derives a closed non-Markovian diffusion equation

Generalizes the Fokker-Planck equation for non-Markovian processes

Shows Gaussianity is preserved only at infinite order

Abstract

We derive the exact evolution equation for the probability density function of particle displacements generated by arbitrary Gaussian velocity processes, when neither Markovianity and nor stationarity are assumed. Starting from the characteristic function of the density of the position, we construct a systematic hierarchy of equations based on Wick's theorem, in which the dynamics is governed by sums of geometrically connected Wick contractions. This approach yields a closed non-Markovian diffusion equation that generalizes the Fokker-Planck description and preserves Gaussianity only in the infinite-order limit.