ISSN 0032-9460, Problems of Information Transmission, 2007, Vol. 43, No. 4, pp. 316–330.
Pleiades Publishing, Inc., 2007.
Original Russian Text
M.L. Blank, S.A. Pirogov, 2007, published in Problemy Peredachi Informatsii, 2007, Vol. 43, No. 4, pp. 51–67.
On Quasi-successful Couplings of Markov Processes
M. L. Blank and S. A. Pirogov
Kharkevich Institute for Information Transmission Problems, RAS, Moscow
Received February 14, 2007; in ﬁnal form, August 10, 2007
Abstract—The notion of a successful coupling of Markov processes, based on the idea that
both components of a coupled system “intersect” in ﬁnite time with probability 1, is extended
to cover situations where the coupling is not necessarily Markovian and its components only
converge (in a certain sense) to each other with time. Under these assumptions the unique
ergodicity of the original Markov process is proved. The price for this generalization is the
weak convergence to the unique invariant measure instead of the strong convergence. Applying
these ideas to inﬁnite interacting particle systems, we consider even more involved situations
where the unique ergodicity can be proved only for a restriction of the original system to a
certain class of initial distributions (e.g., translation-invariant). Questions about the existence
of invariant measures with a given particle density are also discussed.
Let (X, B,) be a locally compact measurable space with a Borel σ-algebra B and a metric .
acting on this phase space is deﬁned by a family of transition probabilities
(x, A)tojumpfromapointx ∈ X toameasurablesetA ∈Bduring time t ≥ 0andbyan
initial distribution of the random variable ξ
representing the initial state of the Markov chain.
The Markov chain ξ
generates two semigroups of operators: P
(x, dy), acting on
bounded measurable functions, and P
ϕ), acting on measures.
One of the ﬁrst questions in the analysis of Markov chains is the existense/uniqueness of their
invariant (stationary) measures
(solutions to the equation P
μ = μ, ∀t) and convergence to them
for various initial distributions. We shall describe a novel approach to study these questions,
applicable to a reasonably broad class of Markov chains.
To a large extent, various coupling results about Markov chains are based on the so-called
coupling inequality applied as follows: let ξ
be two Markov chains with the same transition
probabilities and initial conditions ξ
= x and
= y deﬁned on a common probability space
(Ω, F, P
), where the distribution P
depends on x and y measurably (i.e., for any B ∈F
the value P
(B) is a measurable function of x and y). Denote by τ
a random variable, called
intersection time, equal to the ﬁrst time instant such that (ξ
)=0,∀t ≥ τ
is the moment of intersection of realizations of two Markov processes that started from the
points x, y ∈ X. Using this notation, the coupling inequality can be written as follows:
∈ A) − P
Supported in part by the Russian Foundation for Basic Research, project no. 05-01-00449, CRDF, project
RUM1-2693-MO-05, and the French Ministry of Education.
We consider only probability measures, i.e., measures μ for which μ(X)=1.