  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.
