Abstract
Markov properties and strong Markov properties for random fields are defined and discussed. Special attention is given to those defined by I. V. Evstigneev. The strong Markov nature of Markov random fields with respect to random domains such as [0, L], where L is a multidimensional extension of a stopping time, is explored. A special case of this extension is shown to generalize a result of Merzbach and Nualart for point processes. As an additional example, Evstigneev's Markov and strong Markov properties are considered for independent increment jump processes.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
REFERENCES
Cairoli, R., and Walsh, J. B. (1978). —Regions d'arrêt, localisations et prolongements de martingales. Z. Wahrsch. Verw. Gebiete. 44, 279–306.
Carnal, E., and Walsh, J. B. (1991). Markov properties for certain random fields. In E. Mayer-Wolf, E. Merzbach, and A. Schwartz (eds.), Stochastic Analysis, Academic, San Diego, California, pp. 99–110.
Christenson, C. O., and Voxman, W. L. (1977). Aspects of Topology, Marcel Dekker, Inc., New York, p. 145.
Dalang, R. C., and Walsh, J. B. (1991). The sharp Markov property of the Brownian sheet and related processes. Acta Math.
Dalang, R. C., and Walsh, J. B. (1992). The sharp Markov property of Levy sheets. Ann. Prob. 20(2), 591–626.
Daley, D. J., and Vere-Jones, D. (1988). An Introduction to the Theory of Point Processes, Springer-Verlag, Berlin-Heidelberg-New York, p. 35.
Dudley, R. M. (1982). Empirical and Poisson processes on classes of sets or functions too large for central limit theorems. Z. Wahrsch. Verve. Gebiete. 61, 355–368.
Dugundji, J. (1966). Topology, Allyn and Bacon, Inc., Boston, p. 358.
Evstigneev, I. V. (1977). Markov times for random fields. Theory Prob. Appl. 22, 563–569.
Evstigneev, I. V. (1982). Extremal problems and the strong Markov property of random fields. Russian Math. Surveys. 37(5), 183–184.
Evstigneev, I. V. (1988). Stochastic extremal problems and the strong Markov property of random fields. Russian Math. Surveys 43(2), 1–49.
Kallianpur, G., and Mandrekar, V. (1974). The Markov property for generalized Gaussian random fields. Ann. Inst. Fourier. 24(2), 143–167.
Kinateder, K. K. J. (1991). Measurability and nonanticipating conditions of set-valued random functions and related multidimensional strong Markov properties. Preprint.
Kunch, H. (1979). Gaussian Markov random fields. J. Fac. Sci. Univ. Tokyo. Sect. IA Math 26.
Lévy, P. (1948). Processus Stochastiques et Mouvement Brownien, Gauthier-Villars, Paris.
McKean, H. P. (1963). Brownian motion with a several dimensional time. Theory Prob. Appl. 8, 335–354.
Mandrekar, V. (1983). Markov Properties for Random Fields. Probabilistic Analysis and Related Topics, Academic Press, pp. 161–193.
Merzbach, E., and Nualart, D. (1990). Markov properties for point processes on the plane. Ann. Prob. 18, 342–358.
Molchan, G. M. (1971). Characterization of Gaussian fields with Markov property. Soviet Math. Dokl. 12, 563–567.
Nelson, E. (1973). Constructions of quantum fields from Markov fields. J. Funct. Anal. 12, 97–112.
Pitt, L. D. (1971). Markov property for Gaussian processes with a multi-dimensional time. J. Rat. Mech. Anal. 43, 368–391.
Rozanov, Y. A. (1982). Markov Random Fields, Springer-Verlag, Berlin/Heidelberg/New York.
Simon, B. (1974). The P(ϕ) 2 Model of Euclidean (Quantum) Field Theory, Princeton Series in Physics, Princeton University Press, Princeton, New Jersey.
Wong, E., and Zakai, M. (1976). Weak martingales and stochastic integrals in the plane. Ann. Prob. 4, 570–587.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Kinateder, K.K.J. Strong Markov Properties for Markov Random Fields. Journal of Theoretical Probability 13, 1101–1114 (2000). https://doi.org/10.1023/A:1007822209798
Issue Date:
DOI: https://doi.org/10.1023/A:1007822209798