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Surface Integrals in Hilbert Spaces for General Measures and Applications

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Book cover Stochastic Geometric Mechanics (CIB-SGM 2015)

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Abstract

We present a construction of surface integrals with respect to some non Gaussian probability measures in Hilbert spaces.

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Notes

  1. 1.

    For differentiable functions \(\varphi :H\mapsto \mathbb R\) we denote by \(D_k\varphi (x)\) the derivative of \(\varphi \) in the direction of \(e_k\) at x, and by \(D\varphi (x)\) the gradient at x. By \(C_b(H)\) (resp. \(UC_b(H)\)) we mean the space of all real continuous (resp. uniformly continuous) and bounded mappings \(\varphi :H\rightarrow \mathbb R\) endowed with the sup norm \(\Vert \cdot \Vert _{\infty }\). Moreover, \(C^1_b(H)\) (resp. \(UC_b^1 (H)\)) is the subspace of \(C_b(H)\) (resp. \(UC_b(H)\)) of all continuously differentiable functions, with bounded (resp. uniformly continuous and bounded) derivative.

  2. 2.

    When \(m=1\) the space \(W_{Q^{1/2}}^{1,2}(H,\nu _m)\) coincides with the Malliavin space \(D^{1,2}(H,\mu _1)\).

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Author information

Authors and Affiliations

  1. Scuola Normale Superiore di Pisa, Pisa, Italy

    Giuseppe Da Prato

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  1. Giuseppe Da Prato
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Correspondence to Giuseppe Da Prato .

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Editors and Affiliations

  1. Institute of Applied Mathematics and HCM, University of Bonn, Bonn, Germany

    Sergio Albeverio

  2. Instituto Superior Técnico, University of Lisbon, Lisbon, Portugal

    Ana Bela Cruzeiro

  3. Department of Mathematics, Imperial College London, London, United Kingdom

    Darryl Holm

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Da Prato, G. (2017). Surface Integrals in Hilbert Spaces for General Measures and Applications. In: Albeverio, S., Cruzeiro, A., Holm, D. (eds) Stochastic Geometric Mechanics . CIB-SGM 2015. Springer Proceedings in Mathematics & Statistics, vol 202. Springer, Cham. https://doi.org/10.1007/978-3-319-63453-1_3

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