Diffusion under time-dependent resetting

TL;DR

This paper investigates a Brownian particle with a time-dependent resetting rate, deriving conditions for steady states, analyzing distribution behaviors, and optimizing first passage times with threshold resetting strategies.

Contribution

It introduces a framework for analyzing diffusion with time-modulated resetting rates and identifies optimal resetting strategies for minimizing first passage times.

Findings

Derived a condition for steady-state existence under time-dependent resetting.

Obtained explicit steady-state distributions for specific resetting functions.

Identified threshold resetting as a locally optimal strategy for minimizing mean first passage time.

Abstract

We study a Brownian particle diffusing under a time-modulated stochastic resetting mechanism to a fixed position. The rate of resetting r(t) is a function of the time t since the last reset event. We derive a sufficient condition on r(t) for a steady-state probability distribution of the position of the particle to exist. We derive the form of the steady-state distributions under some particular choices of r(t) and also consider the late time relaxation behavior of the probability distribution. Finally we consider first passage time properties for the Brownian particle to reach the origin and derive a formula for the mean first passage time. We study optimal properties of the mean first passage time and show that a threshold function is at least locally optimal for the problem of minimizing the mean first passage time.