We develop new theoretical framework to derive fractional diffusion equations from Heterogeneous Continuous Time Random Walk (HCTRW) model.

  We are studying the continuous limits of Heterogeneous Continuous Time Random Walk model, when a random walk is making jumps on a graph within different time-length. 

 We apply the concept of a generalized master equation to study heterogeneous continuous-time random walks on networks.

 Depending on the interpretations of the waiting time distributions the generalized master equation gives different forms of continuous equations.

Soon on arxiv: project from L.Tupikina (CRI) and D.Grebenkov (Ecole Polytechnique)

Code is available soon on github. 

We are asking a very simple question:

• what is the corresponding macroscopic diffusion equation for the microscopic behaviour of the random walk on heterogeneous media

Our new theoretical framework on Continuous Time Random Walk (HCTRW) [*] model can help to derive generalized master equation and fractional diffusion equations.

[*] L.Tupikina, D.Grebenkov “Temporal and structural heterogeneities in networks” arxiv, accept.Applied network Journ. (2019)

[**] D.Grebenkov, L. Tupikina “Heterogeneous continuous time random walk”, link to Physical Rev. E Journal  E 97, 012148 (2018)

Code is available on github.