Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Support properties of Gaussian processes over Schwartz space


Author: M. Ann Piech
Journal: Proc. Amer. Math. Soc. 53 (1975), 460-462
MSC: Primary 60G15; Secondary 28A40, 81.60
DOI: https://doi.org/10.1090/S0002-9939-1975-0391242-8
MathSciNet review: 0391242
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We utilize the concept of an abstract Wiener space to prove a converse to a theorem of Minlos, thereby obtaining necessary and sufficient conditions for a Hilbert subspace of $ \mathcal{S}'({{\text{R}}^d})$ to support a given Gaussian process over $ \mathcal{S}({{\text{R}}^d})$.


References [Enhancements On Off] (What's this?)

  • [1] John T. Cannon, Continuous sample paths in quantum field theory, Comm. Math. Phys. 35 (1974), 215–233. MR 0351338
  • [2] G. Velo and A. Wightman (eds.), Constructive quantum field theory, Springer-Verlag, Berlin-New York, 1973. The 1973 “Ettore Majorana” International School of Mathematical Physics, Erice (Sicily), 26 July–5 August 1973; Lecture Notes in Physics, Vol. 25. MR 0395513
  • [3] R. M. Dudley, Jacob Feldman, and L. Le Cam, On seminorms and probabilities, and abstract Wiener spaces, Ann. of Math. (2) 93 (1971), 390–408. MR 0279272, https://doi.org/10.2307/1970780
  • [4] I. M. Gel′fand and G. E. Shilov, Generalized functions. Vol. 1, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1964 [1977]. Properties and operations; Translated from the Russian by Eugene Saletan. MR 0435831
    I. M. Gel′fand and G. E. Shilov, Generalized functions. Vol. 2, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1968 [1977]. Spaces of fundamental and generalized functions; Translated from the Russian by Morris D. Friedman, Amiel Feinstein and Christian P. Peltzer. MR 0435832
    I. M. Gel′fand and G. E. Shilov, Generalized functions. Vol. 3, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1967 [1977]. Theory of differential equations; Translated from the Russian by Meinhard E. Mayer. MR 0435833
    I. M. Gel′fand and N. Ya. Vilenkin, Generalized functions. Vol. 4, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1964 [1977]. Applications of harmonic analysis; Translated from the Russian by Amiel Feinstein. MR 0435834
    I. M. Gel′fand, M. I. Graev, and N. Ya. Vilenkin, Generalized functions. Vol. 5, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1966 [1977]. Integral geometry and representation theory; Translated from the Russian by Eugene Saletan. MR 0435835
  • [5] Leonard Gross, Abstract Wiener spaces, Proc. Fifth Berkeley Sympos. Math. Statist. and Probability (Berkeley, Calif., 1965/66) Univ. California Press, Berkeley, Calif., 1967, pp. 31–42. MR 0212152
  • [6] Edward Nelson, Construction of quantum fields from Markoff fields, J. Functional Analysis 12 (1973), 97–112. MR 0343815
  • [7] G. Velo and A. Wightman (eds.), Constructive quantum field theory, Springer-Verlag, Berlin-New York, 1973. The 1973 “Ettore Majorana” International School of Mathematical Physics, Erice (Sicily), 26 July–5 August 1973; Lecture Notes in Physics, Vol. 25. MR 0395513
  • [8] M. Reed and L. Rosen, Support properties of the free measure for Boson fields, Comm. Math. Phys. 36 (1974), 123–132. MR 0351340

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 60G15, 28A40, 81.60

Retrieve articles in all journals with MSC: 60G15, 28A40, 81.60


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1975-0391242-8
Keywords: Gaussian process, Schwartz space, free Boson field, abstract Wiener space
Article copyright: © Copyright 1975 American Mathematical Society