Characterizations and simulations of a class of stochastic processes to model anomalous diffusion

TL;DR

This paper introduces and characterizes a flexible class of stochastic processes called generalized grey Brownian motion (ggBm) for modeling various types of anomalous diffusion, providing explicit constructions, characterizations, and simulation methods.

Contribution

It defines ggBm in an unspecified probability space, characterizes it via probability densities and the M-Wright function, and develops a random walk simulation approach.

Findings

ggBm includes fractional Brownian motion and time-fractional diffusion processes.

Explicit finite-dimensional probability densities for ggBm are derived.

A new random walk model for simulating ggBm trajectories is proposed.

Abstract

In this paper we study a parametric class of stochastic processes to model both fast and slow anomalous diffusion. This class, called generalized grey Brownian motion (ggBm), is made up off self-similar with stationary increments processes (H-sssi) and depends on two real parameters alpha in (0,2) and beta in (0,1]. It includes fractional Brownian motion when alpha in (0,2) and beta=1, and time-fractional diffusion stochastic processes when alpha=beta in (0,1). The latters have marginal probability density function governed by time-fractional diffusion equations of order beta. The ggBm is defined through the explicit construction of the underline probability space. However, in this paper we show that it is possible to define it in an unspecified probability space. For this purpose, we write down explicitly all the finite dimensional probability density functions. Moreover, we provide different ggBm characterizations. The role of the M-Wright function, which is related to the fundamental solution of the time-fractional diffusion equation, emerges as a natural generalization of the Gaussian distribution. Furthermore, we show that ggBm can be represented in terms of the product of a random variable, which is related to the M-Wright function, and an independent fractional Brownian motion. This representation highlights the $H$-{\bf sssi} nature of the ggBm and provides a way to study and simulate the trajectories. For this purpose, we developed a random walk model based on a finite difference approximation of a partial integro-differenital equation of fractional type.