Variable order fractional Fokker–Planck equations derived from Continuous Time Random Walks

Continuous Time Random Walk models (CTRW) of anomalous diffusion are studied, where the anomalous exponent β(x)∈(0,1) varies in space. This type of situation occurs e.g. in biophysics, where the density of the intracellular matrix varies throughout a cell. Scaling limits of CTRWs are known to have probability distributions which solve fractional Fokker–Planck type equations (FFPE). This correspondence between stochastic processes and FFPE solutions has many useful extensions e.g. to nonlinear particle interactions and reactions, but has not yet been sufficiently developed for FFPEs of the “variable order” type with non-constant β(x).