The Annals of Probability

Random fields and the geometry of Wiener space

Jonathan E. Taylor and Sreekar Vadlamani

Full-text: Open access

Abstract

In this work we consider infinite dimensional extensions of some finite dimensional Gaussian geometric functionals called the Gaussian Minkowski functionals. These functionals appear as coefficients in the probability content of a tube around a convex set $D\subset\mathbb{R}^{k}$ under the standard Gaussian law $N(0,I_{k\times k})$. Using these infinite dimensional extensions, we consider geometric properties of some smooth random fields in the spirit of [Random Fields and Geometry (2007) Springer] that can be expressed in terms of reasonably smooth Wiener functionals.

Article information

Source
Ann. Probab. Volume 41, Number 4 (2013), 2724-2754.

Dates
First available in Project Euclid: 3 July 2013

Permanent link to this document
https://projecteuclid.org/euclid.aop/1372859765

Digital Object Identifier
doi:10.1214/11-AOP730

Mathematical Reviews number (MathSciNet)
MR3112930

Zentralblatt MATH identifier
1284.60100

Subjects
Primary: 60G60: Random fields 60H05: Stochastic integrals 60H07: Stochastic calculus of variations and the Malliavin calculus
Secondary: 53C65: Integral geometry [See also 52A22, 60D05]; differential forms, currents, etc. [See mainly 58Axx]

Keywords
Wiener space Malliavin calculus random fields

Citation

Taylor, Jonathan E.; Vadlamani, Sreekar. Random fields and the geometry of Wiener space. Ann. Probab. 41 (2013), no. 4, 2724--2754. doi:10.1214/11-AOP730. https://projecteuclid.org/euclid.aop/1372859765


Export citation

References

  • [1] Adler, R. J. and Taylor, J. E. (2007). Random Fields and Geometry. Springer, New York.
  • [2] Airault, H. and Malliavin, P. (1988). Intégration géométrique sur l’espace de Wiener. Bull. Sci. Math. (2) 112 3–52.
  • [3] Kunita, H. (1997). Stochastic Flows and Stochastic Differential Equations. Cambridge Studies in Advanced Mathematics 24. Cambridge Univ. Press, Cambridge.
  • [4] Malliavin, P. (1997). Stochastic Analysis. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 313. Springer, Berlin.
  • [5] Nualart, D. (2006). The Malliavin Calculus and Related Topics, 2nd ed. Springer, Berlin.
  • [6] Nualart, D. and Zakai, M. (1989). The partial Malliavin calculus. In Séminaire de Probabilités, XXIII. Lecture Notes in Math. 1372 362–381. Springer, Berlin.
  • [7] Ramer, R. (1974). On nonlinear transformations of Gaussian measures. J. Funct. Anal. 15 166–187.
  • [8] Ren, J. and Röckner, M. (2000). Ray Hölder-continuity for fractional Sobolev spaces in infinite dimensions and applications. Probab. Theory Related Fields 117 201–220.
  • [9] Ren, J. and Röckner, M. (2005). A remark on sets in infinite dimensional spaces with full or zero capacity. In Stochastic Analysis: Classical and Quantum 177–186. World Scientific, Hackensack, NJ.
  • [10] Shigekawa, I. (1994). Sobolev spaces of Banach-valued functions associated with a Markov process. Probab. Theory Related Fields 99 425–441.
  • [11] Simon, B. (2005). Trace Ideals and Their Applications, 2nd ed. Mathematical Surveys and Monographs 120. Amer. Math. Soc., Providence, RI.
  • [12] Sugita, H. (1988). Positive generalized Wiener functions and potential theory over abstract Wiener spaces. Osaka J. Math. 25 665–696.
  • [13] Takeda, M. (1984). $(r,p)$-capacity on the Wiener space and properties of Brownian motion. Z. Wahrsch. Verw. Gebiete 68 149–162.
  • [14] Takemura, A. and Kuriki, S. (2002). On the equivalence of the tube and Euler characteristic methods for the distribution of the maximum of Gaussian fields over piecewise smooth domains. Ann. Appl. Probab. 12 768–796.
  • [15] Üstünel, A. S. and Zakai, M. (2000). Transformation of Measure on Wiener Space. Springer, Berlin.
  • [16] Watanabe, S. (1993). Fractional order Sobolev spaces on Wiener space. Probab. Theory Related Fields 95 175–198.
  • [17] Worsley, K. J. (1994). Local maxima and the expected Euler characteristic of excursion sets of $\chi^{2}$, $F$ and $t$ fields. Adv. in Appl. Probab. 26 13–42.
  • [18] Worsley, K. J. (1995). Boundary corrections for the expected Euler characteristic of excursion sets of random fields, with an application to astrophysics. Adv. in Appl. Probab. 27 943–959.
  • [19] Worsley, K. J., Marrett, S., Neelin, P., Vandal, A. C., Friston, K. J. and Evans, A. C. (1996). Aunified statistical approach for determining significant signals in images of cerebral activation. Humar Brain Mapping 4 58–73.