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Stochastic solution of space-time fractional diffusion equations. (English) Zbl 1244.60080
Summary: Classical and anomalous diffusion equations employ integer derivatives, fractional derivatives, and other pseudodifferential operators in space. We show that replacing the integer time derivative by a fractional derivative subordinates the original stochastic solution to an inverse stable subordinator process whose probability distributions are Mittag-Leffler type. This leads to explicit solutions for space-time fractional diffusion equations with multiscaling space-fractional derivatives, and additional insight into the meaning of these equations.

MSC:
60J65 Brownian motion
35K57 Reaction-diffusion equations
35R11 Fractional partial differential equations
35R60 PDEs with randomness, stochastic partial differential equations
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