Relationship between a Non-Markovian Process and Fokker-Planck Equation

TL;DR

This paper establishes the equivalence between a Non-Markovian evolution with memory effects and a Fokker-Planck equation, revealing how feedback influences drift and diffusion, supported by analytical and numerical analysis.

Contribution

It demonstrates the equivalence of a Non-Markovian process with a Fokker-Planck equation, highlighting the role of feedback and noise in the dynamics.

Findings

Memory effects induce non-trivial drift terms.

Deterministic dynamics depend on noise strength.

Numerical results support analytical findings.

Abstract

We demonstrate the equivalence of a Non--Markovian evolution equation with a linear memory--coupling and a Fokker--Planck equation (FPE). In case the feedback term offers a direct and permanent coupling of the current probability density to an initial distribution, the corresponding FPE offers a non-trivial drift term depending itself on the diffusion parameter. As the consequence the deterministic part of the underlying Langevin equation is likewise determined by the noise strength of the stochastic part. This memory induced stochastic behavior is discussed for different initial distributions. The analytical calculations are supported by numerical results.