Space-time duality for fractional diffusion

Abstract

Zolotarev proved a duality result that relates stable densities with different indices. In this paper, we show how Zolotarev duality leads to some interesting results on fractional diffusion. Fractional diffusion equations employ fractional derivatives in place of the usual integer order derivatives. They govern scaling limits of random walk models, with power law jumps leading to fractional derivatives in space, and power law waiting times between the jumps leading to fractional derivatives in time. The limit process is a stable Lévy motion that models the jumps, subordinated to an inverse stable process that models the waiting times. Using duality, we relate the density of a spectrally negative stable process with index $1<\alpha <2$ to the density of the hitting time of a stable subordinator with index $1/\alpha$, and thereby unify some recent results in the literature. These results also provide a concrete interpretation of Zolotarev duality in terms of the fractional diffusion model.

16 pages

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• Space-Time Duality and High-Order Fractional Diffusion
• Randomly Stopped Nonlinear Fractional Birth Processes
• Explicit solutions to fractional diffusion equations via Generalized Gamma Convolution
• Bessel processes and hyperbolic Brownian motions stopped at different random times
• On self-similar Bernstein functions and corresponding generalized fractional derivatives
• Sibuya probability distributions and numerical evaluation of fractional-order operators
• Alternative probabilistic representations of Barenblatt-type solutions