Variable Order Fractional Fokker-Planck Equations derived from Continuous Time Random Walks

摘要

Continuous Time Random Walk models (CTRW) of anomalous diffusion are studied,where the anomalous exponent $\beta(x) \in (0,1)$ varies in space. This type ofsituation occurs e.g. in biophysics, where the density of the intracellularmatrix varies throughout a cell. Scaling limits of CTRWs are known to haveprobability distributions which solve fractional Fokker-Planck type equations(FFPE). This correspondence between stochastic processes and FFPE solutions hasmany useful extensions e.g. to nonlinear particle interactions and reactions,but has not yet been established for FFPEs of the "variable order" type withnon-constant $\beta(x)$. In this article, variable order FFPEs (VOFFPE) are derived from scalinglimits of CTRWs. The key mathematical tool is the 1-1 correspondence of a CTRWscaling limit to a bivariate Langevin process, which tracks the cumulative sumof jumps in one component and the cumulative sum of waiting times in the other.The spatially varying anomalous exponent is modelled by spatially varying$\beta(x)$-stable L\'evy noise in the waiting time component. The VOFFPEdisplays a unique temporal scaling behaviour: at a time scale $T_0$, the factor$T_0^{-\beta(x)}$ enters in both the drift and diffusivity coefficients of theFokker-Planck operator. A consequence of the mathematical derivation of a VOFFPE from CTRW limits inthis article is that a solution of a VOFFPE can be approximated via Monte Carlosimulations. We check the consistency of VOFFPEs at two different time scales$T_0$ by calculating probability densities at a sequence of times.