Probability Seminar - Jim Kelly

Institution: Michigan State University
Title: Space-Time Duality for Continuous Time Random Walks and Nonlocal Diffusion
Date: April 5, 2018
Location: C405 Wells Hall
Time: 3:00 PM - 3:50 PM

Nonlocal diffusion, Continuous Time Random Walks (CTRWs) and space-time fractional and tempered fractional PDEs are three frameworks for modeling anomalous diffusion. Waiting times with heavy-tailed distributions give rise to time-fractional PDEs, while particle jumps with large jumps yield space-fractional PDEs. Recently, space-time duality, motivated by Zolotarev's duality law for stable densities, established a link between these time-fractional and space-fractional models. We have developed a methodology based on a complex-plane analysis and Bernstein functions that generalizes space-time duality to nonlocal operators. Specifically, we show that a temporal nonlocal governing equation for an inverse subordinator is equivalent to a spatial nonlocal governing equation complemented by a nonlocal boundary condition. This method allows us to extend space-time duality for fractional diffusion to a larger range of nonlocal diffusions, including tempered fractional and distributed-order fractional PDEs. From the nonlocal diffusion perspective, space-time duality allows temporal memory to be transformed into spatial non-locality. From the CTRW perspective, space-time duality allows an ensemble of waiting times to be transformed into an equivalent ensemble of particle jumps. As an application, we show that multi-dimensional Brownian motion subordinated to an independent inverse stable subordinator in each dimension is governed by a vector space-fractional PDE. Writing a time-nonlocal governing equation in more than one dimension seems impossible.