Stochastic representation of fractional Bessel-Riesz motion

Abstract

This paper derives the stochastic solution of a Cauchy problem for the distribution of a fractional diffusion process. The governing equation involves the Bessel-Riesz derivative (in space) to model heavy tails of the distribution, and the Caputo-Djrbashian derivative (in time) to depicts the memory of the diffusion process. The solution is obtained as Brownian motion with time change in terms of the Bessel-Riesz subordinator on the inverse stable subordinator. This stochastic solution, named fractional Bessel-Riesz motion, provides a method to simulate a large class of stochastic motions with memory and heavy tails.