Abstract
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 Hormander’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 Ito formula for rough PDEs.
Highlights
We consider a broad class of semilinear stochastic partial differential equations (SPDEs) with multiplicative noise driven by a finite-dimensional Wiener process
The goal of this paper is to generalise the series of articles [HM06, BM07, HM11] where the authors developed Malliavin calculus for semilinear stochastic partial differential equations (SPDEs) with additive degenerate noise and showed non-degeneracy of the Malliavin matrix under Hörmander’s bracket condition
We do show that in the Brownian case the solutions constructed here coincide with those obtained from Itô calculus, which connects our result with existing objects and allows us to exploit information known for the solutions to Itô SPDEs like Malliavin differentiability, a priori bounds and global existence
Summary
The goal of this paper is to generalise the series of articles [HM06, BM07, HM11] where the authors developed Malliavin calculus for semilinear stochastic partial differential equations (SPDEs) with additive degenerate noise and showed non-degeneracy of the Malliavin matrix under Hörmander’s bracket condition. They introduce operator-valued rough paths and use a slightly different kind of local (in time) expansion of the controlled processes, taking into account the solution to the linearised equation This means that we no longer compare Yt to Ys at small scales, but instead to eL(t−s)Ys. More formally, we replace (1.5) by an expansion of the type. We do show that in the Brownian case the solutions constructed here coincide with those obtained from Itô calculus, which connects our result with existing objects and allows us to exploit information known for the solutions to Itô SPDEs like Malliavin differentiability, a priori bounds and global existence Such information might be much harder to obtain for more general Gaussian rough paths. We show in Theorem 8.8 how this immediately yields smooth densities for finitedimensional marginals of the solution
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