Interlacing Relaxation and First-Passage Phenomena in Reversible Discrete and Continuous Space Markovian Dynamics

TL;DR

This paper reveals a spectral duality between relaxation and first passage times in reversible Markov processes, enabling analytical derivation of first passage distributions from relaxation spectra.

Contribution

It introduces a spectral interlacing duality in reversible Markovian dynamics, allowing analytical computation of first passage times from relaxation eigenspectra.

Findings

Spectral interlacing between relaxation and first passage time scales.

Analytical formulas for first passage time distributions in specific models.

Application to protein folding and Ornstein-Uhlenbeck processes.

Abstract

We uncover a duality between relaxation and first passage processes in ergodic reversible Markovian dynamics in both discrete and continuous state-space. The duality exists in the form of a spectral interlacing -- the respective time scales of relaxation and first passage are shown to interlace. Our canonical theory allows for the first time to determine the full first passage time distribution analytically from the simpler relaxation eigenspectrum. The duality is derived and proven rigorously for both discrete state Markov processes in arbitrary dimension and effectively one-dimensional diffusion processes, whereas we also discuss extensions to more complex scenarios. We apply our theory to a simple discrete-state protein folding model and to the Ornstein-Uhlenbeck process, for which we obtain the exact first passage time distribution analytically in terms of a Newton series of determinants of almost triangular matrices.