REGENERATION-ENRICHED MARKOV PROCESSES WITH APPLICATION TO MONTE CARLO

We study a class of Markov processes that combine local dynamics, arising from a fixed Markov process, with regenerations arising at a state-dependent rate. We give conditions under which such processes possess a given target distribution as their invariant measures, thus making them amenable for use within Monte Carlo methodologies. Since the regeneration mechanism can compensate the choice of local dynamics, while retaining the same invariant distribution, great flexibility can be achieved in selecting local dynamics, and the mathematical analysis is simplified. We give straightforward conditions for the process to possess a central limit theorem, and additional conditions for uniform ergodicity and for a coupling from the past construction to hold, enabling exact sampling from the invariant distribution. We further consider and analyse a natural approximation of the process which may arise in the practical simulation of some classes of continuous-time dynamics.

Automated summary

This study focuses on a class of Markov processes that combine local dynamics with state-dependent regenerations, providing conditions for these processes to have a target distribution as their invariant measures. The regeneration mechanism allows for flexibility in choosing local dynamics while maintaining the same invariant distribution, simplifying mathematical analysis. Conditions for central limit theorem, uniform ergodicity, and exact sampling from the invariant distribution are provided, along with analysis of a natural approximation of the process in practical simulations of continuous-time dynamics.