First passage in discrete-time absorbing Markov chains under stochastic resetting

TL;DR

This paper derives exact formulas for the mean first passage time and splitting probabilities in discrete-time absorbing Markov chains under stochastic resetting, revealing conditions for optimal resetting strategies.

Contribution

It introduces a renewal approach to exactly analyze first passage properties of discrete-time Markov chains with resetting, including new formulas involving a deformed fundamental matrix.

Findings

Exact expressions for unconditional MFPT and splitting probabilities under resetting

A sufficient condition for MFPT optimization via resetting

Application to random walks and voter models demonstrating the theory

Abstract

First passage of stochastic processes under resetting has recently been an active research topic in the field of statistical physics. However, most of previous studies mainly focused on the systems with continuous time and space. In this paper, we study the effect of stochastic resetting on first passage properties of discrete-time absorbing Markov chains, described by a transition matrix $\brm{Q}$ between transient states and a transition matrix $\brm{R}$ from transient states to absorbing states. Using a renewal approach, we exactly derive the unconditional mean first passage time (MFPT) to either of absorbing states, the splitting probability the and conditional MFPT to each absorbing state. All the quantities can be expressed in terms of a deformed fundamental matrix $\brm{Z_{\gamma}}=\left[\brm{I}-(1-\gamma) \brm{Q} \right]^{-1}$ and $\brm{R}$, where $\brm{I}$ is the identity matrix, and $\gamma$ is the resetting probability at each time step. We further show a sufficient condition under which the unconditional MPFT can be optimized by stochastic resetting. Finally, we apply our results to two concrete examples: symmetric random walks on one-dimensional lattices with absorbing boundaries and voter model on complete graphs.