  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.
