Coupling in the theory of interacting systems - Greven, Andreas (Acta Applicandae Mathematicae, 1994)
In this paper we consider the construction of couplings for Markovian evolutions on a state space of the form E, with Mathematical expression (measurable) and ω a countable group (ℤ for example). The evolutions we focus on are mainly systems of linearly interacting diffusions, with E compact. We explain and state properties of such couplings and show how they are used to obtain information on the behaviour of the evolution in finite time and as time tends to infinity. An important property of a coupling is to be a successful coupling. The latter concept is introduced here in the context of interacting systems, which is different from the classical concept for Markov chains or processes with state space ℝ. The analysis of the question when a coupling is successful depends heavily on the structure of the interaction term and is investigated in detail. We formulate some open problems and conjectures. The paper puts in perspective the coupling statements appearing in the proofs of various results and is largely based on the works of Cox and Greven, Fleischmann and Greven, Dawson and Greven, Greven, and Cox, Greven and Shiga. © 1994 Kluwer Academic Publisher
Keywords: Coupling; interacting diffusions; Mathematics Subject Classifications (1991): 60K35