On the Kemeny time for continuous-time reversible and irreversible Markov processes with applications to stochastic resetting and to conditioning towards forever-survival

TL;DR

This paper explores the Kemeny time for continuous-time Markov processes, providing explicit calculations and properties for both reversible and irreversible cases, with applications to stochastic resetting and conditioning on survival.

Contribution

It offers a comprehensive analysis of the Kemeny time, including explicit formulas and spectral properties for various Markov processes, extending understanding to irreversible dynamics and special conditioned processes.

Findings

Kemeny time equals the average mean-first-passage-time over the steady state.

Explicit formulas for Kemeny time in systems with jumps and diffusion.

Distinct properties of Kemeny times in reversible and irreversible processes.

Abstract

For continuous-time ergodic Markov processes, the Kemeny time $\tau_*$ is the characteristic time needed to converge towards the steady state $P_*(x)$ : in real-space, the Kemeny time $\tau_*$ corresponds to the average of the Mean-First-Passage-Time $\tau(x,x_0) $ over the final configuration $x$ drawn with the steady state $P_*(x)$, which turns out to be independent of the initial configuration $x_0$; in the spectral domain, the Kemeny time $\tau_*$ corresponds to the sum of the inverses of all the non-vanishing eigenvalues $\lambda_n \ne 0 $ of the opposite generator. We describe many illustrative examples involving jumps and/or diffusion in one dimension, where the Kemeny time can be explicitly computed as a function of the system-size, via its real-space definition and/or via its spectral definition : we consider both reversible processes satisfying detailed-balance where the eigenvalues are real, and irreversible processes characterized by non-vanishing steady currents where the eigenvalues can be complex. In particular, we study the specific properties of the Kemeny times for Markov processes with stochastic resetting, and for absorbing Markov processes conditioned to survive forever.