Long time scaling behaviour for diffusion with resetting and memory

TL;DR

This paper studies a diffusion process with memory-dependent resetting, revealing various long-time behaviors of the particle's variance, including normal diffusion and ultraslow growth, depending on the memory kernel.

Contribution

It provides an analytical framework linking memory kernel forms to asymptotic diffusion behaviors in resetting processes.

Findings

Variance can grow linearly, logarithmically, or even slower depending on the memory kernel.

Different memory kernels lead to distinct long-time diffusion regimes.

The model unifies various anomalous diffusion behaviors under a common framework.

Abstract

We consider a continuous-space and continuous-time diffusion process under resetting with memory. A particle resets to a position chosen from its trajectory in the past according to a memory kernel. Depending on the form of the memory kernel, we show analytically how different asymptotic behaviours of the variance of the particle position emerge at long times. These range from standard diffusive ($\sigma^2 \sim t$) all the way to anomalous ultraslow growth $\sigma^2 \sim \ln \ln t$.