Finite Dimensional Fokker-Planck Equations for Continuous Time Random Walks

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.