Interlacing relaxation and first-passage phenomena in reversible discrete and continuous space Markovian dynamics

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.