Milestoning Markov-jump dynamics: Stationary properties, thermodynamic consistency, kinetic hysteresis, and fluctuation symmetries

Abstract

We derive an exact coarse graining of generic Markov-jump processes into observable semi-Markov dynamics. Exact results for waiting-time distributions for jumps between observable states are derived and proved that these decompose into conditionally independent dwell and transition times. Dwell times are proved to be a local property of mesostates - they depend on the initial but not final state. Conversely, transition-path times depend on both states, trigger kinetic hysteresis, and, under suitable conditions on the hidden sub-network, are shown to obey a reflection symmetry. We characterize the stationary properties of the milestoned dynamics, prove its thermodynamic consistency, and demonstrate robustness to milestone positioning. Surprisingly, even in the limit of a time-scale separation rendering the observed dynamics approximately Markovian, the effect of kinetic hysteresis on the dissipation persists. A minimal example shows how the results lay the foundation for inferring affinities of hidden dissipative cycles from observations of transition-path times.