Stochastic solutions of generalized time-fractional evolution equations

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 Lévy motion for suitable pseudo-differential operators in space.

Stochastic analysis for vector-valued generalized grey Brownian motion

“…In statistical physics, fractional time derivatives reflects random waiting times between particle jumps [41]. Detailed discussion of such processes is also found in [4]. For a detailed study of the special class of heavy tailed processes, see [34,35,21].…”

“…For a detailed study of the special class of heavy tailed processes, see [34,35,21]. An approach using subordination can be found in [4].…”