×

On the sharp Markov property for Gaussian random fields and spectral synthesis in spaces of Bessel potentials. (English) Zbl 1040.60042

Authors’ summary: Let \(\Phi=\{\varphi(x):x\in\mathbb{R}^2\}\) be a Gaussian random field on the plane. For \(A\subset\mathbb{R}^2\), we investigate the relationship between the \(\sigma\)-field \(\mathcal{F}(\Phi,A)=\sigma\{\varphi(x): x\in A\}\) and the infinitesimal or germ \(\sigma\)-field \(\bigcap_{\varepsilon>0}\mathcal{F}(\Phi,A_\varepsilon)\), where \(A_\varepsilon\) is an \(\varepsilon\)-neighborhood of \(A\). General analytic conditions are developed giving necessary and sufficient conditions for the equality of these two \(\sigma\)-fields. These conditions are potential theoretic in nature and are formulated in terms of the reproducing kernel Hilbert space associated with \(\Phi\). The Bessel fields \(\Phi_\beta\) satisfying the pseudo-partial differential equation \((I-\Delta)^{\beta/2}=\dot{W}(x)\), \(\beta>1\), for which the reproducing kernel Hilbert spaces are identified as spaces of Bessel potentials \(\mathcal{L}^{\beta,2}\), are studied in detail and the conditions for equality are conditions for spectral synthesis in \(\mathcal{L}^{\beta,2}\). The case \(\beta=2\) is of special interest, and we deduce sharp conditions for the sharp Markov property to hold here, complementing the work of R. C. Dalang and J. B. Walsh [Acta Math. 168, 153–218 (1992; Zbl 0759.60056)] on the Brownian sheet.

MSC:

60G60 Random fields
31B15 Potentials and capacities, extremal length and related notions in higher dimensions
31B25 Boundary behavior of harmonic functions in higher dimensions
60G15 Gaussian processes
60H15 Stochastic partial differential equations (aspects of stochastic analysis)

Citations:

Zbl 0759.60056
PDF BibTeX XML Cite
Full Text: DOI Euclid

References:

[1] ADAMS, D. R. and HEDBERG, L. I. (1996). Function Spaces and Potential Theory. A Series of Comprehensive Studies in Mathematics 314. Springer, Berlin. · Zbl 0834.46021
[2] AGMON, S. (1965). Lectures on Elliptic Boundary Value Problems. Van Nostrand, Princeton, NJ. · Zbl 0142.37401
[3] BENHENNI, K. (1998). Approximating integrals of stochastic processes: Extensions. J. Appl. Probab. 35 843-855. · Zbl 0924.62097
[4] BEURLING, A. (1948). On the spectral sy nthesis of bounded functions. Acta Math. 81 225-238.
[5] DALANG, R. C. and WALSH, J. B. (1992). The sharp Markov property of the Brownian sheet and related processes. Acta Math. 168 153-218. · Zbl 0759.60056
[6] Dy M, H. and MCKEAN, H. P. (1976). Gaussian Processes, Function Theory, and the Inverse Spectral Problem. Academic, New York. · Zbl 0327.60029
[7] FALCONER, K. J. (1985). The Geometry of Fractal Sets. Cambridge Univ. Press. · Zbl 0587.28004
[8] HEDBERG, L. I. (1980). Spectral sy nthesis and stability in Sobolev spaces. Euclidean Harmonic Analy sis. Lecture Notes in Math. 779 73-103. Springer, Berlin. · Zbl 0469.31003
[9] HEDBERG, L. I. (1981). Spectral sy nthesis in Sobolev spaces, and uniqueness of solutions of the Dirichlet problem. Acta Math. 147 237-264. · Zbl 0504.35018
[10] KATZNELSON, Y. (1968). An Introduction to Harmonic Analy sis. Dover, New York. · Zbl 0169.17902
[11] KONDRAT’EV, V. A. and EIDEL’MAN, S. D. (1979). On conditions on the boundary surface in the theory of elliptic boundary value problems. Soviet Math. Dokl. 20 561-563. · Zbl 0422.35034
[12] LANDKOF, N. S. (1972). Foundation of Modern Potential Theory. Springer, Berlin. · Zbl 0253.31001
[13] MCKEAN, H., JR. (1963). Brownian motion with a several dimensional time. Theory Probab. Appl. 8 335-354. · Zbl 0124.08702
[14] PITERBARG, L. I. (1983). On the structure of the infinitesimal -algebra of Gaussian processes and fields. Theory Probab. Appl. 31 484-492. · Zbl 0614.60033
[15] PITT, L. D. (1971). A Markov property for Gaussian processes with multidimensional parameter. Arch. Rational Mech. Anal. 43 367-391. · Zbl 0277.60025
[16] PITT, L. D. (1973). Some problems in the spectral theory of stationary processes on Rd. Indiana Univ. Math. J. 23 343-365. · Zbl 0285.60022
[17] PITT, L. D. (1975). Stationary Gaussian Markov fields on Rd with a deterministic component. J. Multivariate Anal. 5 20-32. · Zbl 0317.60016
[18] PITT, L. D. and ROBEVA, R. S. (1994). On the sharp Markov property for the Whittle field in 2-dimensions. In Stochastic Analy sis on Infinite Dimensional Spaces (H. Kunita and H.-H. Kuo, eds.) 242-254. Longman, Harlow, UK. · Zbl 0814.60047
[19] PITT, L. D., ROBEVA, R. S. and WANG, D. Y. (1995). An error analysis for the numerical calculation of certain random integrals I. Ann. Appl. Probab. 5 171-197. · Zbl 0821.62058
[20] ROBEVA, R. S. (1997). On the sharp Markov property for Gaussian random fields and the problem of spectral sy nthesis in certain function spaces. Ph.D. dissertation, Univ. Virginia.
[21] ROZANOV, YU. A. (1982). Markov Random Fields. Springer, Berlin. · Zbl 0498.60057
[22] SAKS, S. (1937). Theory of the Integral, 2nd ed. Hafner, New York. · Zbl 0017.30004
[23] STEIN, E. M. (1970). Singular Integrals and Differentiability Properties of Functions. Princeton Univ. Press. · Zbl 0207.13501
[24] STOKE, B. (1984). Differentiability properties of Bessel potentials and Besov functions. Ark. Mat. 22 269-286. · Zbl 0562.31011
[25] STRICHARTZ, R. S. (1967). Multipliers on fractional Sobolev spaces. J. Math. Mech. 16 1031-1060. · Zbl 0145.38301
[26] TORCHINSKI, A. (1986). Real-Variable Methods in Harmonic Analy sis. Academic, New York.
[27] TRIEBEL, H. (1984). Theory of Function Spaces. Birkhäuser, Berlin. · Zbl 0556.46018
[28] TSUJI, M. (1975). Potential Theory in Modern Function Theory. Chelsea, New York. · Zbl 0322.30001
[29] WHITTLE, P. (1954). On stationary processes in the plane. Biometrica 41 434-449. JSTOR: · Zbl 0058.35601
[30] Zy GMUND, A. (1959). Trigonometric Series, 2nd ed. Cambridge Univ. Press.
[31] CHARLOTTESVILLE, VIRGINIA 22904-4137 E-MAIL: ldp@virginia.edu WEB: www.math.virginia.edu/faculty/pitt DEPARTMENT OF MATHEMATICAL SCIENCES SWEET BRIAR COLLEGE URL:
[32] SWEET BRIAR, VIRGINIA 24595 E-MAIL: robeva@sbc.edu WEB: www.mathsci.sbc.edu/robeva.htm URL:
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.