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

Abstract

AbstractFor continuous-time ergodic Markov processes, the Kemeny timeτ∗is the characteristic time needed to converge towards the steady stateP∗(x): in real-space, the Kemeny timeτ∗corresponds to the average of the mean-first-passage-timeτ(x,x0)over the final configurationxdrawn with the steady stateP∗(x), which turns out to be independent of the initial configurationx0; in the spectral domain, the Kemeny timeτ∗corresponds to the sum of the inverses of all the non-vanishing eigenvaluesλn≠0of 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.