The parallel replica method for simulating long trajectories of Markov chains (Q2927899)

The authors consider the problem of simulating, in an efficient way, time-homogeneous Markov chains with subsets of states in which the Markov chain stays for a long time. Those states are the so-called metastable states. The authors present an algorithm which is an adaptation of the parallel replica (ParRep) dynamics generally used to simulate metastable processes. Their version is used in the discrete time setting, but it can be applied to any Markov chain. In order to describe the proposed algorithm, some basic definitions are given as well as the setting of some notation. After the statement of the preliminary information, the discrete time ParRep algorithm is described. This is performed in three steps: the decorrelation step, the dephasing step, and finally, the parallel step. A mathematical analysis of some properties of the discrete ParRep algorithm is also performed. Among them is the analysis of the properties of the quasi-stationary distributions, including conditions for their existence and uniqueness. Examples illustrating the use of the algorithm are given towards the end of the work as well as a discussion of the algorithm's performance.