Summary
A natural definition of the Markov property for multi-parameter random processes (random fields) is the following. Let {X t,t∈ℝN} be a multiparameter process. For any set D in ℝN let σ D denote the σ-field generated by {X t , t∈D}. The field {X t,t∈D} is said to be Markov (or Markov of degree 1 [6], or sharp Markov) if, for any bounded open set D with smooth boundary, σ D and σ D c are conditionally independent given σ δD . It has been known for some time that to find interesting examples of Markov processes under this definition; it is necessary to consider generalized random functions. In this paper we show that a natural framework for the Markov property of multiparameter processes is a class of generalized random differential forms (i.e., random currents). Our principal objective is to relate the Markovian nature of an isotropic gaussian current to its spectral properties.
References
Agmon, S.: Lectures on elliptic boundary value problems. Princeton, N.J.: D. Van Nostrand Co. 1965
Albeverio, S., Høegh-Krohn, R., Holden, H.: Markov processes on infinite dimensional spaces, Markov fields, and Markov consurfaces. Proceedings, Bremen Conf., Arnold, L. and Kotelenez, P. (eds.). Dordrecht: Reidel 1985
Dobrushin, R.L., Surgailis, D.: On the innovation problem for Gaussian Markov Random Fields. Z. Wahrscheinlichkeitstheor. Verw. Geb., 49, 275–291 (1979)
Flanders, H.: Differential Forms. New York: Academic Press 1963
Ito, K.: Isotropic random current. Proceedings, Third Berkeley Symp. on Math. Stat. and Probab. pp. 125–32, 1956
Mandrekar, V.: Markov properties for random fields. Probab. Anal. Rel. Topics. Bharucha-Reid, A.T. (ed.), vol. 3, pp. 161–93. New York London: Academic Press 1983
McKean, H., Jr.: Brownian motion with a several-dimensional time. Theory Probab. Appl. 8, 335–54 (1963)
Pitt, L.D.: A Markov property for Gaussian processes with a multidimensional parameter. Arch. Rational Mech. Anal. 43, 367–91 (1971)
deRham, G.: Differentiable Manifolds. Berlin Heidelberg New York: Springer 1982
Reed, M., Simon, B.: Methods of modern mathematical physics II: Fourier analysis, self-adjointness. New York London: Academic Press 1975
Rozanov, Y.A.: Markov Random Fields. New York London: Academic Press 1982
Westenholtz, C. von: Differential forms in mathematical physics. Amsterdam: North Holland 1981
Wong, E.: Homogeneous Gauss-Markov random fields. Ann. Math. Stat. 40, 1625–34 (1969)
Wong, E., Zakai, M.: Markov processes on the plane. Stochastics 15, 311–333 (1985)
Wong, E., Zakai, M.: Multiparameter martingale differential forms. Probab. Th. Rel. Fields 74, 429–453 (1987)
Wong, E., Zakai, M.: Spectral representation of isotropic random currents. To appear
Yaglom, A.M.: Some classes of random fields in n-dimensional space related to stationary random processes. Theory Prob. Appl. 2, 273–320 (1957)
Author information
Authors and Affiliations
Additional information
Work supported by the Army Research Office, Grant No. DAAG 29-85-K-0233
Work done while at the University of California at Berkeley
Rights and permissions
About this article
Cite this article
Wong, E., Zakai, M. Isotropic Gauss-Markov currents. Probab. Th. Rel. Fields 82, 137–154 (1989). https://doi.org/10.1007/BF00340015
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF00340015
Keywords
- Stochastic Process
- Probability Theory
- Statistical Theory
- Markov Process
- Random Process