A stochastic solution with Gaussian stationary increments of the symmetric space-time fractional diffusion equation

TL;DR

This paper introduces a new stochastic solution for the symmetric space-time fractional diffusion equation using Gaussian stationary increments, derived through an innovative integral representation and product of independent variables, offering an alternative to existing methods.

Contribution

The paper presents a novel integral representation for the fundamental solution and constructs a stochastic process with Gaussian stationary increments for the symmetric space-time fractional diffusion equation.

Findings

The stochastic solution is self-similar with stationary increments.

Numerical simulations with fractional Brownian motion match the fundamental solution.

The method provides an alternative to traditional approaches like CTRW and subordination.

Abstract

The stochastic solution with Gaussian stationary increments is establihsed for the symmetric space-time fractional diffusion equation when $0 < \beta < \alpha \le 2$, where $0 < \beta \le 1$ and $0 < \alpha \le 2$ are the fractional derivation orders in time and space, respectively. This solution is provided by imposing the identity between two probability density functions resulting (i) from a new integral representation formula of the fundamental solution of the symmetric space-time fractional diffusion equation and (ii) from the product of two independent random variables. This is an alternative method with respect to previous approaches such as the scaling limit of the continuos time random walk, the parametric subordination and the subordinated Langevin equation. A new integral representation formula for the fundamental solution of the space-time fractional diffusion equation is firstly derived. It is then shown that, in the symmetric case, a stochastic solution can be obtained by a Gaussian process with stationary increments and with a random wideness scale variable distributed according to an arrangement of two extremal L\'evy stable densities. This stochastic solution is self-similar with stationary increments and uniquely defined in a statistical sense by the mean and the covariance structure. Numerical simulations are carried out by choosing as Gaussian process the fractional Brownian motion. Sample paths and probability densities functions are shown to be in agreement with the fundamental solution of the symmetric space-time fractional diffusion equation.