Finite Dimensional Fokker-Planck Equations for Continuous Time Random Walks

TL;DR

This paper derives finite-dimensional fractional Fokker-Planck equations for the distributions of continuous time random walk limits at multiple times, enabling a complete characterization of these non-Markovian processes.

Contribution

It introduces finite-dimensional FFPEs for CTRWLs, providing a new way to define and analyze these non-Markovian stochastic processes.

Findings

Derived FFPEs for joint distributions at multiple times

Extended the characterization of CTRWLs beyond single-time distributions

Enabled defining CTRWLs via their finite-dimensional FFPEs

Abstract

Continuous Time Random Walk(CTRW) is a model where particle's jumps in space are coupled with waiting times before each jump. A Continuous Time Random Walk Limit(CTRWL) is obtained by a limit procedure on a CTRW and can be used to model anomalous diffusion. The distribution $p\left(dx,t\right)$ of a CTRWL $X_{t}$ satisfies a Fractional Fokker-Planck Equation(FFPE). Since CTRWLs are usually not Markovian, their one dimensional FFPE is not enough to completely define them. In this paper we find the FFPEs of the distribution of $X_{t}$ at multiple times , i.e. the distribution of the random vector $\left(X_{t_{1}},...,X_{t_{n}}\right)$ for $t_{1}<...<t_{n}$ for a large class of CTRWLs. This allows us to define CTRWLs by their finite dimensional FFPEs.