Stochastic solutions of generalized time-fractional evolution equations

TL;DR

This paper explores stochastic solutions to a broad class of generalized time-fractional evolution equations, linking their parameters to underlying stochastic processes and providing explicit solutions for specific subclasses.

Contribution

It establishes a general relation between equation parameters and stochastic process distributions, and explicitly determines parameters for Saigo-Maeda operators.

Findings

Derived relations between parameters and stochastic processes

Explicit parameter determination for Saigo-Maeda operators

Constructed self-similar solutions via linear fractional Lévy motion

Abstract

We consider a general class of integro-differential evolution equations which includes the governing equation of the generalized grey Brownian motion and the time- and space-fractional heat equation. We present a general relation between the parameters of the equation and the distribution of the underlying stochastic processes, as well as discuss different classes of processes providing stochastic solutions of these equations. For a subclass of evolution equations, containing Saigo-Maeda generalized time-fractional operators, we determine the parameters of the corresponding processes explicitly. Moreover, we explain how self-similar stochastic solutions with stationary increments can be obtained via linear fractional L\'evy motion for suitable pseudo-differential operators in space.