Fractional diffusion equations and processes with randomly varying time

TL;DR

This paper explores solutions to fractional diffusion equations of various orders, linking them to stochastic processes like iterated Brownian motion and processes with random or Brownian time, revealing new probabilistic interpretations and explicit distributions.

Contribution

It provides novel interpretations of fractional diffusion solutions as distributions of complex stochastic processes, including iterated Brownian motion and processes with random time, expanding understanding of fractional PDEs.

Findings

Solutions for orders 1/2^n correspond to iterated Brownian motion distributions.

Explicit distributions for maximum and sojourn times of these processes are derived.

Connections between fractional equations, stable distributions, and processes with random or Brownian time are established.

Abstract

In this paper the solutions $u_{\nu}=u_{\nu}(x,t)$ to fractional diffusion equations of order $0<\nu \leq 2$ are analyzed and interpreted as densities of the composition of various types of stochastic processes. For the fractional equations of order $\nu =\frac{1}{2^n}$, $n\geq 1,$ 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 $\nu =\frac{2}{3^n}$, $n\geq 1,$ 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.