Space–time fractional diffusion with stochastic resetting

Abstract

In this article, we study the space–time fractional diffusion equation (STFDE) which is a generalization of the classical diffusion equation, in the presence of stochastic resetting. The STFDE is formulated by replacing the standard time and space derivatives with the Caputo and Riesz fractional derivatives, respectively, to capture anomalous diffusion behaviors. We derive analytical solutions using Laplace and Fourier transforms, and express them in terms of Fox H-functions. We obtain a closed-form expression for the stationary distribution and prove the finiteness of the mean first passage time. Additionally, we examine how stochastic resetting influences the infinite divisibility of the standard diffusion process, showing that this property is lost once resetting is introduced. The reset mechanism interrupts the Lévy process at random times, effectively altering the jump structure and destroying the self-decomposability required for infinite divisibility.