Malliavin calculus for non-Gaussian differentiable measures and surface measures in Hilbert spaces. (English) Zbl 1392.28012
The authors defined Sobolev spaces with respect to a non degenerate Borel probability measure \(\nu\). They also proved properties of the Sobolev spaces that are useful to construct the surface measures.
In Theorem 3.4, which is one of the main results of this paper, the authors construct surface measures in a Hilbert space under Hypothesis 1.1 and 1.3. Their approach comes from the general geometric measure theory and relies on the theory of the functions with bounded variation.
Examples concerned with Gaussian measure, weighted Gaussian measures, an infinite product of non-Gaussian measure, and some invariant measures of stochastic PDEs are given in Sections 6–8. Also integration by parts formulae on sublevel sets of good functions that involve surface integrals are proved.
In Theorem 3.4, which is one of the main results of this paper, the authors construct surface measures in a Hilbert space under Hypothesis 1.1 and 1.3. Their approach comes from the general geometric measure theory and relies on the theory of the functions with bounded variation.
Examples concerned with Gaussian measure, weighted Gaussian measures, an infinite product of non-Gaussian measure, and some invariant measures of stochastic PDEs are given in Sections 6–8. Also integration by parts formulae on sublevel sets of good functions that involve surface integrals are proved.
Reviewer: Byoung Soo Kim (Seoul)
MSC:
| 28C20 | Set functions and measures and integrals in infinite-dimensional spaces (Wiener measure, Gaussian measure, etc.) |
| 60H15 | Stochastic partial differential equations (aspects of stochastic analysis) |
| 35R15 | PDEs on infinite-dimensional (e.g., function) spaces (= PDEs in infinitely many variables) |
Keywords:
infinite dimensional analysis; probability measures in Hilbert spaces; surface integrals in Hilbert spaces; invariant measures; stochastic PDEsReferences:
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