Conditioned backward and forward times of diffusion with stochastic resetting: a renewal theory approach

TL;DR

This paper uses renewal theory to analyze diffusion processes with stochastic resetting, introducing conditioned backward and forward times, and explores how reset time distributions affect these times, especially for power-law resets.

Contribution

It formally derives renewal theory results for diffusion with resets and introduces conditioned backward and forward times, providing new insights into their distributions under various reset time PDFs.

Findings

Conditioned backward and forward time PDFs are derived and validated with simulations.

Power-law reset time PDFs cause notable changes in time distributions at half-integer alpha values.

Long-time diffusion scaling interacts with reset time PDFs to influence process properties.

Abstract

Stochastic resetting can be naturally understood as a renewal process governing the evolution of an underlying stochastic process. In this work, we formally derive well-known results of diffusion with resets from a renewal theory perspective. Parallel to the concepts from renewal theory, we introduce the conditioned backward and forward times for stochastic processes with resetting to be the times since the last and until the next reset, given that the current state of the system is known. We focus on studying diffusion under Markovian and non-Markovian resetting. For these cases, we find the conditioned backward and forward time PDFs, comparing them with numerical simulations of the process. In particular, we find that for power-law reset time PDFs with asymptotic form $\varphi(t)\sim t^{-1-\alpha}$, significant changes in the properties of the conditioned backward and forward times happen at half-integer values of $\alpha$. This is due to the composition between the long-time scaling of diffusion $P(x,t)\sim 1/\sqrt{t}$ and the reset time PDF.