Generalized diffusion process with nonlocal interactions: Continuous time random walk model and stochastic resetting

TL;DR

This paper introduces a space fractional diffusion equation with nonlocal temporal and spatial features, models it via continuous time random walks, and studies the impact of stochastic resetting leading to a nonequilibrium stationary state.

Contribution

It develops a generalized diffusion model incorporating nonlocal interactions in space and time, and analyzes the effects of stochastic resetting on its long-term behavior.

Findings

The system reaches a nonequilibrium stationary state under stochastic resetting.

The subordination approach effectively analyzes the probability density function.

Different memory kernels influence the dynamics distinctly.

Abstract

A space fractional diffusion-like equation is introduced, which embodies the nonlocality in time, represented by the memory kernel and the non-locality in space. A specific example of the nonlocal term is considered in combination with three different forms of the memory kernel. To analyse the probability density function, we utilize the subordination approach. Subsequently, the corresponding continuous time random walk model is presented. Furthermore, we investigate the effects of the stochastic resetting on the dynamics of the process and we showed that in the long time limit the system approaches a nonequilibrium stationary state.