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Chaos expansions of double intersection local time of Brownian motion in and renormalization

Author

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  • Imkeller, Peter
  • Perez-Abreu, Victor
  • Vives, Josep

Abstract

Double intersection local times [alpha](x,.) of Brownian motion which measure the size of the set of time pairs (s, t), s [not equal to] t, for which Wt and Ws + x coincide can be developed into series of multiple Wiener-Ito integrals. These series representations reveal on the one hand the degree of smoothness of [alpha](x,.) in terms of eventually negative order Sobolev spaces with respect to the canonical Dirichlet structure on Wiener space. On the other hand, they offer an easy access to renormalization of [alpha](x,.) as x --> 0. The results, valid for any dimension d, describe a pattern in which the well known cases d = 2, 3 are naturally embedded.

Suggested Citation

  • Imkeller, Peter & Perez-Abreu, Victor & Vives, Josep, 1995. "Chaos expansions of double intersection local time of Brownian motion in and renormalization," Stochastic Processes and their Applications, Elsevier, vol. 56(1), pages 1-34, March.
    • Handle: RePEc:eee:spapps:v:56:y:1995:i:1:p:1-34
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    Cited by:

    1. Shi, Qun & Yu, Xianye, 2017. "Fractional smoothness of derivative of self-intersection local times," Statistics & Probability Letters, Elsevier, vol. 129(C), pages 241-251.
    2. Marie F. Kratz & José R. León, 2001. "Central Limit Theorems for Level Functionals of Stationary Gaussian Processes and Fields," Journal of Theoretical Probability, Springer, vol. 14(3), pages 639-672, July.
    3. Franco Flandoli & Peter Imkeller & Ciprian A. Tudor, 2014. "2D-Stochastic Currents over the Wiener Sheet," Journal of Theoretical Probability, Springer, vol. 27(2), pages 552-575, June.
    4. Albeverio, Sergio & Hu, Yaozhong & Zhou, Xian Yin, 1997. "A remark on non-smoothness of the self-intersection local time of planar Brownian motion," Statistics & Probability Letters, Elsevier, vol. 32(1), pages 57-65, February.
    5. Yan, Litan & Shen, Guangjun, 2010. "On the collision local time of sub-fractional Brownian motions," Statistics & Probability Letters, Elsevier, vol. 80(5-6), pages 296-308, March.
    6. Naitzat, Gregory & Adler, Robert J., 2017. "A central limit theorem for the Euler integral of a Gaussian random field," Stochastic Processes and their Applications, Elsevier, vol. 127(6), pages 2036-2067.
    7. H. Uemura, 2004. "Tanaka Formula for Multidimensional Brownian Motions," Journal of Theoretical Probability, Springer, vol. 17(2), pages 347-366, April.
    8. Bojdecki, Tomasz & Gorostiza, Luis G., 1995. "Self-intersection local time for Gaussian '(d)-processes: Existence, path continuity and examples," Stochastic Processes and their Applications, Elsevier, vol. 60(2), pages 191-226, December.
    9. Uemura, H., 2008. "Generalized positive continuous additive functionals of multidimensional Brownian motion and their associated Revuz measures," Stochastic Processes and their Applications, Elsevier, vol. 118(10), pages 1870-1891, October.
    10. Rudenko, Alexey, 2012. "Some properties of the Itô–Wiener expansion of the solution of a stochastic differential equation and local times," Stochastic Processes and their Applications, Elsevier, vol. 122(6), pages 2454-2479.

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