Finite dimensional Fokker–Planck equations for continuous time random walk limits

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(dx,t) of a CTRWL Xt satisfies a Fractional Fokker–Planck Equation (FFPE). Since CTRWLs are usually not Markovian, their one dimensional FFPE is not enough to completely determine them. In this paper we find the FFPEs of the distribution of Xt at multiple times, i.e. the distribution of the random vector (Xt1,…,Xtn) for t1<⋯<tn for a large class of CTRWLs. This allows us to define CTRWLs by their finite dimensional FFPEs.