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Generalized positive continuous additive functionals of multidimensional Brownian motion and their associated Revuz measures

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  • Uemura, H.

Abstract

We extend the notion of positive continuous additive functionals of multidimensional Brownian motions to generalized Wiener functionals in the setting of Malliavin calculus. We call such a functional a generalized PCAF. The associated Revuz measure and a characteristic of a generalized PCAF are also extended adequately. By making use of these tools a local time representation of generalized PCAFs is discussed. It is known that a Radon measure corresponds to a generalized Wiener functional through the occupation time formula. We also study a condition for this functional to be a generalized PCAF and the relation between the associated Revuz measure of the generalized PCAF corresponding to Radon measure and this Radon measure. Finally we discuss a criterion to determine the exact Meyer-Watanabe's Sobolev space to which this corresponding functional belongs.

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  • 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.
    • Handle: RePEc:eee:spapps:v:118:y:2008:i:10:p:1870-1891
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    References listed on IDEAS

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    1. 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.
    2. Bass, Richard, 1984. "Joint continuity and representations of additive functionals of d-dimensional Brownian motion," Stochastic Processes and their Applications, Elsevier, vol. 17(2), pages 211-227, July.
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    Cited by:

    1. Shigeyoshi Ogawa & Hideaki Uemura, 2014. "On a Stochastic Fourier Coefficient: Case of Noncausal Functions," Journal of Theoretical Probability, Springer, vol. 27(2), pages 370-382, June.

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