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The stochastic wave equation in high dimensions: Malliavin differentiability and absolute continuity


 
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1. Title Title of document The stochastic wave equation in high dimensions: Malliavin differentiability and absolute continuity
 
2. Creator Author's name, affiliation, country Marta Sanz-Solé; University of Barcelona; Spain
 
2. Creator Author's name, affiliation, country André Süss; University of Barcelona; Spain
 
3. Subject Discipline(s)
 
3. Subject Keyword(s) stochastic integration; stochastic partial differential equations; stochastic wave equation; Malliavin calculus; densities.
 
3. Subject Subject classification Primary: 60H15, 60H07; Secondary: 60H20, 60H05.
 
4. Description Abstract We consider the class of non-linear stochastic partial differential equations studied in [Conus-Dalang, 2008]. 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-Dalang, 2008] 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 $d\ge 4$.
 
5. Publisher Organizing agency, location
 
6. Contributor Sponsor(s) MICINN-FEDER MTM 2009-07203 from the \textit{Direcci\'on General de Investigaci\'on, Ministerio de Econom\'{\i}a y Competitividad, Spain
 
7. Date (YYYY-MM-DD) 2013-06-21
 
8. Type Status & genre Peer-reviewed Article
 
8. Type Type
 
9. Format File format PDF
 
10. Identifier Uniform Resource Identifier http://ejp.ejpecp.org/article/view/2341
 
10. Identifier Digital Object Identifier 10.1214/EJP.v18-2341
 
11. Source Journal/conference title; vol., no. (year) Electronic Journal of Probability; Vol 18
 
12. Language English=en en
 
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