Efficient Markovian couplings: examples and counterexamples

Burdzy, Krzysztof ; Kendall, Wilfrid S.

Ann. Appl. Probab., Tome 10 (2000) no. 2, p. 362-409 / Harvested from Project Euclid

Résumé

In this paper we study the notion of an efficient coupling of Markov processes. Informally, an efficient coupling is one which couples at the maximum possible exponential rate, as given by the spectral gap. This notion is of interest not only for its own sake, but also of growing importance arising from the recent advent of methods of “perfect simulation”: it helps to establish the “price of perfection” for such methods. In general, one can always achieve efficient coupling if the coupling is allowed to “cheat”(if each component’s behavior is affected by the future behavior of the other component), but the situation is more interesting if the coupling is required to be co-adapted. We present an informal heuristic for the existence of an efficient coupling, and justify the heuristic by proving rigorous results and examples in the contexts of finite reversible Markov chains and of reflecting Brownian motion in planar domains.

Détail

Publié le : 2000-05-14
Classification:  Diffusion,  Chen-optimal coupling,  co-adapted coupling,  couplling exponent,  efficient coupling,  efficient coupling heuristic,  exact simulation,  Markov chain,  mirror coupling,  monotonicity,  perfect simulation,  price of perfection,  reflecting Brownian motion,  spectral gap,  synchronous coupling,  60J27,  60H30,  65U05