Mean Extinction Times in Multi-Metastable Systems: A Discrete Coarse-Grained Approach

The paper develops a coarse-grained framework for computing mean extinction times in multi-metastable systems modeled as one-step continuous-time Markov chains with an absorbing state. At the microscopic level, backward equations on finite corridors are solved to obtain closed-form series for committors, mean first-passage times, and intrawell (basin) waiting times. A renewal–reward construction then yields effective interwell transition rates written as a success probability divided by a mean cycle duration, providing an interpretable effective rate constant. These rates define a reduced Markov chain on the wells together with extinction; mean extinction times follow from a linear system, and the associated fundamental matrix quantifies pre-extinction residence times in each coarse state. This framework makes explicit how multiple escape pathways and intrawell dwell times contribute to extinction statistics in finite systems. The method is illustrated on a double-well landscape with an extinction state, using a reversible potential-to-rates mapping for the numerical example. Comparisons of alternative intrawell models and validation against exact one-step computations demonstrate accuracy at finite system sizes, including regimes where diffusion approximations are unreliable. The resulting formulas require only local rate data, remain numerically stable under strong bias, and extend directly to multiple wells and flexible boundary conditions.