Space-time duality for fractional diffusion

TL;DR

This paper explores how Zolotarev duality connects different stable densities and applies this to fractional diffusion equations, revealing new insights into the behavior of processes with power law jumps and waiting times.

Contribution

It demonstrates how Zolotarev duality can be used to relate densities in fractional diffusion models, unifying recent results and providing a concrete interpretation.

Findings

Relates stable densities with different indices via duality

Unifies recent results on fractional diffusion processes

Provides a new interpretation of Zolotarev duality in this context

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 L\'evy 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.