The Annals of Applied Probability

Efficient Markovian couplings: examples and counterexamples

Krzysztof Burdzy and Wilfrid S. Kendall

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Abstract

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.

Article information

Source
Ann. Appl. Probab. Volume 10, Number 2 (2000), 362-409.

Dates
First available in Project Euclid: 22 April 2002

Permanent link to this document
http://projecteuclid.org/euclid.aoap/1019487348

Digital Object Identifier
doi:10.1214/aoap/1019487348

Mathematical Reviews number (MathSciNet)
MR1768241

Zentralblatt MATH identifier
0957.60083

Subjects
Primary: 60J27: Continuous-time Markov processes on discrete state spaces 60H30: Applications of stochastic analysis (to PDE, etc.) 65U05

Keywords
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

Citation

Burdzy, Krzysztof; Kendall, Wilfrid S. Efficient Markovian couplings: examples and counterexamples. Ann. Appl. Probab. 10 (2000), no. 2, 362--409. doi:10.1214/aoap/1019487348. http://projecteuclid.org/euclid.aoap/1019487348.


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