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Random fields and diffusion processes. (English) Zbl 0661.60063

Calcul des probabilités, Éc. d’Été, Saint-Flour/Fr. 1985-87, Lect. Notes Math. 1362, 101-203 (1988).
[For the entire collection see Zbl 0649.00016.]
This article is a nice introduction to some connections between random fields and diffusion processes, in particular, between Markov fields and Markov processes. Chapter 1 (An introduction to random fields) contains 1) basic facts about the structure of random fields specified by a collection of local conditional probabilities; 2) Dobrushin’s contraction technique, which provides a uniqueness theorem of Gibbs measures and derives its regularity properties; 3) Shannon-McMillan’s theorem for the relative entropy, and 4) large deviations for the empirical field of a Gibbs measure.
Chapter II (Infinite dimensional diffusions) deals with some connections between Gibbs measures and infinite-dimensional diffusion processes, which induce Markov fields. From this point of view, 1) large deviations of an infinite dimensional Brownian motion are illustrated; 2) invariant measures of interacting diffusion processes are described as Gibbs measures, using time reversal; and 3) its local specification is derived.
Reviewer: M.Nisio

MSC:

60G60 Random fields
60J99 Markov processes
60F10 Large deviations

Citations:

Zbl 0649.00016