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2019 Hörmander’s theorem for semilinear SPDEs
Andris Gerasimovičs, Martin Hairer
Electron. J. Probab. 24: 1-56 (2019). DOI: 10.1214/19-EJP387


We consider a broad class of semilinear SPDEs with multiplicative noise driven by a finite-dimensional Wiener process. We show that, provided that an infinite-dimensional analogue of Hörmander’s bracket condition holds, the Malliavin matrix of the solution is an operator with dense range. In particular, we show that the laws of finite-dimensional projections of such solutions admit smooth densities with respect to Lebesgue measure. The main idea is to develop a robust pathwise solution theory for such SPDEs using rough paths theory, which then allows us to use a pathwise version of Norris’s lemma to work directly on the Malliavin matrix, instead of the “reduced Malliavin matrix” which is not available in this context. On our way of proving this result, we develop some new tools for the theory of rough paths like a rough Fubini theorem and a deterministic mild Itô formula for rough PDEs.


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Andris Gerasimovičs. Martin Hairer. "Hörmander’s theorem for semilinear SPDEs." Electron. J. Probab. 24 1 - 56, 2019.


Received: 15 November 2018; Accepted: 5 November 2019; Published: 2019
First available in Project Euclid: 13 November 2019

zbMATH: 07142926
MathSciNet: MR4040992
Digital Object Identifier: 10.1214/19-EJP387

Primary: 60H07, 60H15
Secondary: 60H05


Vol.24 • 2019
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