A mathematical formalization of the parallel replica dynamics

The purpose of this article is to lay the mathematical foundations of a well numerical approach in computational statistical physics and molecular, namely the parallel replica dynamics introduced by A. F. Voter. The of the approach is to efficiently generate a coarse-grained evolution (in of state-to-state dynamics) of a given stochastic process. The approach consists in concurrently considering several realizations of the process, and tracking among the realizations that which, the, undergoes an important transition. Using specific properties of the generated, a computational speed-up is obtained. In the best cases, speed-up approaches the number of realizations considered. By drawing with the theory of Markov processes and, in particular, exploiting notion of quasi-stationary distribution, we provide a mathematical setting for assessing theoretically the performance of the approach, and improving it.