Effects of refractory period on stochastic resetting

TL;DR

This paper analyzes how refractory periods affect stochastic resetting processes, deriving exact distributions and exploring slow relaxation phenomena, with implications for search and completion times.

Contribution

It provides an exact analytical framework for stochastic processes with refractory periods, including correlated resetting and refractory times, and computes key distributions and mean first passage times.

Findings

Exact stationary distribution with delta peak at reset point

Slow relaxation behavior with power-law refractory periods

Explicit formulas for mean first passage and joint distributions

Abstract

We consider a stochastic process undergoing resetting after which a random refractory period is imposed. In this period the process is quiescent and remains at the resetting position. Using a first-renewal approach, we compute exactly the stationary position distribution and analyse the emergence of a delta peak at the resetting position. In the case of a power-law distribution for the refractory period we find slow relaxation. We generalise our results to the case when the resetting period and the refractory period are correlated, by computing the Laplace transform of the survival probability of the process and the mean first passage time, i.e., the mean time to completion of a task. We also compute exactly the joint distribution of the active and absorption time to a fixed target.