Generalized Fokker-Planck equation and its solution for linear non-Markovian Gaussian systems

TL;DR

This paper develops a method to derive and solve the generalized Fokker-Planck equation for Gaussian non-Markovian systems, enabling the analysis of complex stochastic processes without directly solving the PDE.

Contribution

A novel approach to derive the GFPE for Gaussian non-Markovian processes that allows PDF construction without solving the equation explicitly.

Findings

Derived GFPE and PDFs for generalized Brownian motion and harmonic potential.

Confirmed systems can be described by Einstein-Smoluchowski equation at high viscosity and long times.

Calculated work PDF and verified fluctuation theorem in a moving harmonic potential.

Abstract

In this paper we suggest a consistent approach to derivation of generalized Fokker-Planck equation (GFPE) for Gaussian non-Markovian processes with stationary increments. This approach allows us to construct the probability density function (PDF) without a need to solve the GFPE. We employ our method to obtain the GFPE and PDFs for free generalized Brownian motion and the one in harmonic potential for the case of power-law correlation function of the noise. We prove the fact that the considered systems may be described with Einstein-Smoluchowski equation at high viscosity levels and long times. We also compare the results with those obtained by other authors. At last, we calculate PDF of thermodynamical work in the stochastic system which consists of a particle embedded in a harmonic potential moving with constant velocity, and check the work fluctuation theorem for such a system.