Hostname: page-component-78c5997874-mlc7c Total loading time: 0 Render date: 2024-11-10T20:05:07.709Z Has data issue: false hasContentIssue false
  • English
  • Français

Gaussian integrals on Wiener spaces

Part of: Markov processes Stochastic processes Set functions and measures on spaces with additional structure

Published online by Cambridge University Press:  14 July 2016

Adam D. Helfer  and
Zhongxin Zhao
Adam D. Helfer
Affiliation:
University of Missouri
Zhongxin Zhao
Affiliation:
University of Missouri
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider integrals on Wiener space of the forms E(exp K(x)) and E(exp K(x) |L(x) = l) where K is a quadratic form and L a system of linear forms. We give explicit formulas for these integrals in terms of the operators K and L, in the case that these arise from quasilinear functions in the sense of Zhao (1981). As examples, we recover Lévy's area formula in the plane, and derive new formulas for the probability density of the radius of gyration tensor for Brownian paths.

Keywords

ABSTRACT WIENER SPACESGAUSSIAN INTEGRALSSHAPE OF BROWNIAN PATHSSTOCHASTIC AREA

MSC classification

Primary: 60G15: Gaussian processes
Secondary: 60G35: Signal detection and filtering 60J65: Brownian motion 28C20: Set functions and measures and integrals in infinite-dimensional spaces (Wiener measure, Gaussian measure, etc.)
Type
Research Papers
Information
Journal of Applied Probability , Volume 29 , Issue 1 , March 1992 , pp. 46 - 55
Copyright
Copyright © Applied Probability Trust 1992 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

∗

Postal address for both authors: Department of Mathematics, University of Missouri, Columbia, MO 65211, USA.

References

Biane, Ph. and Yor, M. (1987) Variations sur une formule de Paul Lévy. Ann. Inst. H. Poincaré 23, 359–377.Google Scholar
Bismut, J. (1984) The Atiyah-Singer theorems I, II. J. Funct. Anal. 57, 56–99 and 329–348.Google Scholar
Bismut, J. (1986) Probability and geometry. In Probability and Analysis, ed. Letta, G. and Pratelli, M., Lecture Notes in Mathematics 1206, Springer-Verlag, Berlin.Google Scholar
Chaleyat-Maurel, M. (1981) Densités des diffusions invariantes sur certains groupes nilpotents. Soc. Math. France Astérisque 84-85, 203–214.Google Scholar
Chan, T. (1990) Indefinite quadratic functionals of Gaussian processes and least-action paths. Preprint.Google Scholar
Chiang, Tzu-Shuh, Chow, Yunshyong and Lee, Yuh-Jia (1987) A formula for . Proc. Amer. Math. Soc. 100, 721–724.Google Scholar
Chiang, Tzu-Shuh, Chow, Yunshyong and Lee, Yuh-Jia (1990) An abstract Wiener space approach to certain functional integrals. Preprint.Google Scholar
Donati-Martin, C. and Yor, M. (1991) Fubini's theorem for double Wiener integrals and the variance of the Brownian path. Ann. Inst. H. Poincaré 27.Google Scholar
Duplantier, B. (1989) Areas of planar Brownian curves. J. Phys. A: Math. Gen. 22, 3033–3048.Google Scholar
Eichinger, B. E. (1985) Shape distributions for Gaussian molecules. Macromolecules 18, 211–216.Google Scholar
Fixman, M. (1961) Radius of gyration of polymer chains. J. Chem. Phys. 36, 306–318.Google Scholar
Gaveau, B. (1977) Principe de moindre action, propagation de la chaleur et estimées sous elliptiques sur certains groupes nilpotents. Acta Math. 139, 95–153.Google Scholar
Gradshteyn, I. S. and Ryzhik, I. M. (1980) Tables of Integrals, Series and Products. Academic Press, New York.Google Scholar
Gross, L. (1967) Abstract Wiener spaces. Proc. 5th Berkeley Symp. Math. Statist. Prob. 2(1), 31–42.Google Scholar
Helmes, K. and Schwane, A. (1983) Lévy stochastic area formula in higher dimensions. J. Funct. Anal. 54, 177–192.Google Scholar
Knight, F. (1981) Essentials of Brownian motion and diffusion. Math. Surveys 18, American Mathematical Society, Providence, RI.Google Scholar
Kreé, P. (1986) A remark on Paul Lévy's stochastic area formula. In Aspects of Mathematics and its Applications, ed. Barroso, J. A., Elsevier, Amsterdam.Google Scholar
Lévy, P. (1951) Wiener's random function, and other Laplacian random functions. Proc. 2nd Berkeley Symp. Math. Statist. Prob. pp. 171–187.Google Scholar
McAonghusa, P. and Pulé, J. P. (1989) An extension of Lévy's stochastic area formula. Stochastics 26, 247–255.Google Scholar
Pitman, J. and Yor, M. (1982) A decomposition for Bessel bridges. Z. Wahrscheinlichkeitsth. 59, 425–457.Google Scholar
Rudnick, J. and Gaspari, G. (1987) The shapes of random walks. Science 237, 384–389.Google Scholar
Xia, Dao-Xing (1972) Measure Theory and Integration on Infinite-Dimensional Spaces. Academic Press, New York.Google Scholar
Yor, M. (1989) On stochastic areas and averages of planar Brownian curves. J. Phys. A: Math. Gen. 22, 3049–3057.Google Scholar
Yor, M. (1991) Une explication du théorème de Ciesielski-Taylor. Ann. Inst. H. Poincaré 27.Google Scholar
Zhao, Zhongxin (1981) Quasilinear transformations in abstract Wiener spaces. Sci. Sinica 24, 1–12.Google Scholar