Recurrence of the random process governed with the fractional Laplacian and the Caputo time derivative

Abstract

We are addressing a parabolic equation with fractional derivatives in time and space that governs the scaling limit of continuous-time random walks with anomalous diffusion. For these equations, the fundamental solution represents the probability density of finding a particle released at the origin at time 0 at a given position and time. Using some estimates of the asymptotic behaviour of the fundamental solution, we evaluate the probability of the process returning infinite times to the origin in a heuristic way. Our calculations suggest that the process is always recurrent.

Keywords

• fractional diffusion
• continuous time random walks
• fundamental solution
• decay estimates
• caputo derivative
• fractional laplacian