Electronic Communications in Probability

A type of Gauss' divergence formula on Wiener spaces

Yoshiki Otobe

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Abstract

We will formulate a type of Gauss' divergence formula on sets of functions which are greater than a specific value of which boundaries are not regular. Such formula was first established by L. Zambotti in 2002 with a profound study of stochastic processes. In this paper we will give a much shorter and simpler proof for his formula in a framework of the Malliavin calculus and give alternate expressions. Our approach also enables to show that such formulae hold in other Gaussian spaces.

Article information

Source
Electron. Commun. Probab., Volume 14 (2009), paper no. 44, 457-463.

Dates
Accepted: 30 October 2009
First available in Project Euclid: 6 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ecp/1465234753

Digital Object Identifier
doi:10.1214/ECP.v14-1498

Mathematical Reviews number (MathSciNet)
MR2559095

Zentralblatt MATH identifier
1189.60112

Subjects
Primary: 60H07: Stochastic calculus of variations and the Malliavin calculus
Secondary: 28C20: Set functions and measures and integrals in infinite-dimensional spaces (Wiener measure, Gaussian measure, etc.) [See also 46G12, 58C35, 58D20, 60B11]

Keywords
divergence formulae on the Wiener spaces integration by parts formulae on the Wiener spaces

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

Citation

Otobe, Yoshiki. A type of Gauss' divergence formula on Wiener spaces. Electron. Commun. Probab. 14 (2009), paper no. 44, 457--463. doi:10.1214/ECP.v14-1498. https://projecteuclid.org/euclid.ecp/1465234753


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References

  • Enchev, O.; Stroock, D. W. Rademacher's theorem for Wiener functionals. Ann. Probab. 21 (1993), no. 1, 25–33.
  • Florit, Carme; Nualart, David. A local criterion for smoothness of densities and application to the supremum of the Brownian sheet. Statist. Probab. Lett. 22 (1995), no. 1, 25–31.
  • Fukushima, Masatoshi. $BV$ functions and distorted Ornstein Uhlenbeck processes over the abstract Wiener space. J. Funct. Anal. 174 (2000), no. 1, 227–249.
  • Fukushima, Masatoshi; Hino, Masanori. On the space of BV functions and a related stochastic calculus in infinite dimensions. J. Funct. Anal. 183 (2001), no. 1, 245–268.
  • Funaki, Tadahisa; Ishitani, Kensuke. Integration by parts formulae for Wiener measures on a path space between two curves. Probab. Theory Related Fields 137 (2007), no. 3-4, 289–321.
  • Goodman, Victor. A divergence theorem for Hilbert space. Trans. Amer. Math. Soc. 164 (1972), 411–426.
  • Hariya, Yuu. Integration by parts formulae for Wiener measures restricted to subsets in $Bbb Rsp d$. J. Funct. Anal. 239 (2006), no. 2, 594–610.
  • Hino, Masanori. On Dirichlet spaces over convex sets in infinite dimensions. Finite and infinite dimensional analysis in honor of Leonard Gross (New Orleans, LA, 2001), 143–156, Contemp. Math., 317, Amer. Math. Soc., Providence, RI, 2003.
  • Hino, Masanori; Uchida, Hiroto. Reflecting Ornstein-Uhlenbeck processes on pinned path spaces. Proceedings of RIMS Workshop on Stochastic Analysis and Applications, 111–128, RIMS Kôkyûroku Bessatsu, B6, Res. Inst. Math. Sci. (RIMS), Kyoto, 2008.
  • Lanjri Zadi, Noureddine; Nualart, David. Smoothness of the law of the supremum of the fractional Brownian motion. Electron. Comm. Probab. 8 (2003), 102–111 (electronic).
  • Nualart, David. The Malliavin calculus and related topics. Second edition. Probability and its Applications (New York). Springer-Verlag, Berlin, 2006. xiv+382 pp. ISBN: 978-3-540-28328-7; 3-540-28328-5
  • Nualart, D.; Pardoux, É. White noise driven quasilinear SPDEs with reflection. Probab. Theory Related Fields 93 (1992), no. 1, 77–89.
  • Shigekawa, Ichiro. Vanishing theorem of the Hodge-Kodaira operator for differential forms on a convex domain of the Wiener space. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 6 (2003), suppl., 53–63.
  • Sugita, Hiroshi. Positive generalized Wiener functions and potential theory over abstract Wiener spaces. Osaka J. Math. 25 (1988), no. 3, 665–696.
  • Watanabe, S. Lectures on stochastic differential equations and Malliavin calculus. Notes by M. Gopalan Nair and B. Rajeev. Tata Institute of Fundamental Research Lectures on Mathematics and Physics, 73. Published for the Tata Institute of Fundamental Research, Bombay; by Springer-Verlag, Berlin, 1984. iii+111 pp. ISBN: 3-540-12897-2
  • Zambotti, Lorenzo. Integration by parts formulae on convex sets of paths and applications to SPDEs with reflection. Probab. Theory Related Fields 123 (2002), no. 4, 579–600.