Master equations for continuous-time random walks with stochastic resetting

TL;DR

This paper develops a comprehensive framework for continuous-time random walks with stochastic resetting, including non-Markovian dynamics and Levy flights, deriving conditions for meaningful displacement and master equations for various resetting laws.

Contribution

It introduces a unified approach to CTRWs with stochastic resetting, deriving conditions for displacement validity and master equations for monotone resetting laws.

Findings

Resetting transforms CTRWs into processes with the same jump distribution but different counting processes.

Established the zero-law condition for meaningful displacement in reset CTRWs.

Derived master equations for CTRWs with completely monotone resetting laws.

Abstract

We study a general continuous-time random walk (CTRW), by including non-Markovian cases and L\'evy flights, under complete stochastic resetting to the initial position with an arbitrary law, which can be power-lawed as well as Poissonian. We provide three linked results. First, we show that the random walk under stochastic resetting is a CTRW with the same jump-size distribution of the non-reset original CTRW but different counting process. Later, we derive the condition for a CTRW with stochastic resetting to be a meaningful displacement process at large elapsed times, i.e., the probability to jump to any site is higher than the probability to be reset to the initial position, and we call this condition the zero-law for stochastic resetting. This law joins with the other two laws for reset random walks concerning the existence and the non-existence of a non-equilibrium stationary state. Finally, we derive master equations for CTRWs when the resetting law is a completely monotone function.