Markovianity in space and time. (English) Zbl 1131.60044

van Lieshout, M. N. M.

Denteneer, Dee (ed.) et al., Dynamics and stochastics. Festschrift in honor of M. S. Keane. Selected papers based on the presentations at the conference ‘Dynamical systems, probability theory, and statistical mechanics’, Eindhoven, The Netherlands, January 3–7, 2005, on the occasion of the 65th birthday of Mike S. Keane. Beachwood, OH: IMS, Institute of Mathematical Statistics (ISBN 0-940600-64-1/pbk). Institute of Mathematical Statistics Lecture Notes - Monograph Series 48, 154-168 (2006).

The concept of Markovianity plays an important role in space as well as in time, especially in image analysis and statistical physics. Although its precise definition depends on the context, common ingredients are conditional independence and factorization formulae that allow to break up complex, or high dimensional, probabilities into manageable, lower dimensional components. In this paper, after a brief review, the author introduces a new concept of Markovianity that aims to combine spatial and temporal conditional independence. The paper is divided into the following four sections: I. From Markov chain to Markov point process, and beyond; II. Definitions and notation; III. Hammersley-Clifford factorization; IV. Dynamic representation. See also the following monograph of the author: [M. N. M. van Lieshout, Markov point processes and their applications, London: ICP, Imperial College Press (2000; Zbl 0968.60005)].
For the entire collection see [Zbl 1113.60008].

Reviewer: Viktor Oganyan (Erevan)

MSC:

60G55	Point processes (e.g., Poisson, Cox, Hawkes processes)
60D05	Geometric probability and stochastic geometry
62M30	Inference from spatial processes

Keywords:

Hammersley-Clifford factorization; marked point process; neighbor relation; pairwise interaction; random sequential adsorption; sequential spatial process

Citations:

Zbl 0968.60005