On solutions of a class of non-Markovian Fokker-Planck equations

TL;DR

This paper presents a method to solve non-Markovian Fokker-Planck equations by expressing solutions through Markovian counterparts, classifies memory kernels, and discusses conditions for probability density validity.

Contribution

It introduces an integral decomposition approach to solve non-Markovian Fokker-Planck equations and classifies memory kernels based on solution properties.

Findings

Solutions can be expressed via Markovian equations with the same operator.

Memory kernels are classified into safe and dangerous types.

Non-Markovian equations are valid only under certain conditions.

Abstract

We show that a formal solution of a rather general non-Markovian Fokker-Planck equation can be represented in a form of an integral decomposition and thus can be expressed through the solution of the Markovian equation with the same Fokker-Planck operator. This allows us to classify memory kernels into safe ones, for which the solution is always a probability density, and dangerous ones, when this is not guaranteed. The first situation describes random processes subordinated to a Wiener process, while the second one typically corresponds to random processes showing a strong ballistic component. In this case the non-Markovian Fokker-Planck equation is only valid in a restricted range of parameters, initial and boundary conditions.