Stochastic resetting and first arrival subjected to Gaussian noise and Poisson white noise

TL;DR

This paper analyzes the dynamics of an overdamped Brownian particle under Poissonian stochastic resetting in a nonthermal bath with Gaussian and Poisson white noise, deriving exact steady-state distributions and optimizing first-arrival times.

Contribution

It provides an exact analytical expression for the steady-state distribution under combined noises and investigates optimal resetting rates to minimize first-arrival times in a nonequilibrium environment.

Findings

Steady-state distribution is double exponential, unlike thermal bath cases.

Existence of an optimal resetting rate that minimizes mean first-arrival time.

The formalism generalizes diffusion-limited reactions with resetting in nonequilibrium baths.

Abstract

We study the dynamics of an overdamped Brownian particle subjected to Poissonian stochastic resetting in a nonthermal bath, characterized by a Poisson white noise and a Gaussian noise. Applying the renewal theory we find an exact analytical expression for the spatial distribution at the steady state. Unlike the single exponential distribution as observed in the case of a purely thermal bath, the distribution is double exponential. Relaxation of the transient spatial distributions to the stationary one, for the limiting cases of Poissonian rate, is investigated carefully. In addition, we study the first-arrival properties of the system in the presence of a delta-function sink with strength $\kappa$, where $\kappa=0$ and $\kappa=\infty$ correspond to fully nonreactive and fully reactive sinks, respectively. We explore the effect of two competitive mechanisms: the diffusive spread in the presence of two noises and the increase in probability density around the initial position due to stochastic resetting. We show that there exists an optimal resetting rate, which minimizes the mean first-arrival time (MFAT) to the sink for a given value of the sink strength. We also explore the effect of the strength of the Poissonian noise on MFAT, in addition to sink strength. Our formalism generalizes the diffusion-limited reaction under resetting in a nonequilibrium bath and provides an efficient search strategy for a reactant to find a target site, relevant in a range of biophysical processes.