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.
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.
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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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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