Probability Seminar: Edward Crane - Coupling Markov chains with a common image chain
| Dates: | 20 May 2026 |
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| Times: | 15:00 - 16:00 |
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| What is it: | Seminar |
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| Organiser: | Department of Mathematics |
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Edward Crane (University of Bristol) will speak at the Probability seminar.
Title: Coupling Markov chains with a common image chain
Abstract: (Joint work with Erin Russell and Alexander Holroyd, mostly in arXiv:2604.12853)
You are given two homogeneous Markov chains X and Y with countable state spaces A and B and specified initial distributions, and you are given a subset U of A x B. How can you decide whether there exists a homogeneous Markov chain taking values in U that is a coupling of X and Y?
An interesting special case is when U is a `block diagonal' set of the form {(a,b): f(a) = g(b)}, for maps f and g from A and B to a third state space C. We construct a Markov chain coupling taking values in this set in the case where the image processes f(X) and g(Y) are equal in law to a homogeneous Markov chain on C. (In highbrow language this shows that the Ore property holds for the category of countable homogeneous Markov chains with weak lumpings as morphisms.)
I will present a simple example which shows that for the Markov chain coupling to exist it is not sufficient for f(X) and g(Y) to have the same law as processes if the common image process is not required to be a homogeneous Markov chain.
I will also explain our original motivation for this problem, which came from comparing a directed graph version of the Ráth-Tóth mean field forest fire model with the original undirected graph model.
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