Fractional diffusion equations and processes with randomly varying time

Abstract

In this paper the solutions $u_{\nu }=u_{\nu }(x,t)$ to fractional diffusion equations of order $0<\nu \le 2$ are analyzed and interpreted as densities of the composition of various types of stochastic processes. For the fractional equations of order $\nu =\frac{1}{2^n}$, $n\ge 1,$ we show that the solutions $u_{1/2^n}$ correspond to the distribution of the $n$-times iterated Brownian motion. For these processes the distributions of the maximum and of the sojourn time are explicitly given. The case of fractional equations of order $\nu =\frac{2}{3^n}$, $n\ge 1,$ is also investigated and related to Brownian motion and processes with densities expressed in terms of Airy functions. In the general case we show that $u_{\nu }$ coincides with the distribution of Brownian motion with random time or of different processes with a Brownian time. The interplay between the solutions $u_{\nu }$ and stable distributions is also explored. Interesting cases involving the bilateral exponential distribution are obtained in the limit.

Published in at http://dx.doi.org/10.1214/08-AOP401 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

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• Fractional Calculus: Integral and Differential Equations of Fractional Order
• Brownian subordinators and fractional Cauchy problems
• Brownian-Time Processes: The PDE Connection and the Half-Derivative Generator
• Iterated Brownian motion in an open set
• Brownian-Time Processes: The PDE Connection II and the Corresponding Feynman-Kac Formula
• Reaction-diffusion systems and nonlinear waves

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