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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 sufficiently developed for FFPEs of the"variable order"type with non-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 spatially heterogeneous temporal scaling behaviour, with generalizeddiffusivity and drift coefficients whose units are length$^2$/time$^{\beta(x)}$resp. length/time$^{\beta(x)}$. A global change of the time scale results in aspatially varying change in diffusivity and drift.  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. Based on such simulations, we are able to confirm that the VOFFPEis consistent under a change of the global time scale.

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