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The stochastic wave equation in high dimensions: Malliavin differentiability and absolute continuity. (English) Zbl 1288.60079

Authors’ abstract: We consider the class of nonlinear stochastic partial differential equations studied in [D. Conus and R. C. Dalang, Electron. J. Probab. 13, 629–670 (2008; Zbl 1187.60049)]. Equivalent formulations using integration with respect to a cylindrical Brownian motion and also the Skorohod integral are established. It is proved that the random field solution to these equations at any fixed point \((t,x) \in [0,T] \times \mathbb{R}^d\) is differentiable in the Malliavin sense. For this, an extension of the integration theory in Conus and Dalang [loc. cit.] to Hilbert space valued integrands is developed, and commutation formulae of the Malliavin derivative and stochastic and pathwise integrals are proved. In the particular case of equations with additive noise, we establish the existence of density for the law of the solution at \((t,x) \in ]0,T] \times \mathbb{R}^d\). The results apply to the stochastic wave equation in spatial dimension \(d\geq 4 \).”

MSC:

60H15 Stochastic partial differential equations (aspects of stochastic analysis)
60H07 Stochastic calculus of variations and the Malliavin calculus
60H20 Stochastic integral equations
60H05 Stochastic integrals

Citations:

Zbl 1187.60049
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