On the density of systems of non-linear spatially homogeneous SPDEs

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• dc.contributor.author Nualart, Eulàlia
• dc.date.accessioned 2021-02-22T08:57:11Z
• dc.date.available 2021-02-22T08:57:11Z
• dc.date.issued 2012
• dc.description.abstract In this paper, we consider a system of k second-order nonlinear stochastic partial differential equations with spatial dimension , driven by a q-dimensional Gaussian noise, which is white in time and with some spatially homogeneous covariance. The case of a single equation and a one-dimensional noise has largely been studied in the literature. The first aim of this paper is to give a survey of some of the existing results. We will start with the existence, uniqueness and Hölder's continuity of the solution. For this, the extension of Walsh's stochastic integral to cover some measure-valued integrands will be recalled. We will then recall the results concerning the existence and smoothness of the density, as well as its strict positivity, which are obtained using techniques of Malliavin calculus. The second aim of this paper is to show how these results extend to our system of stochastic partial differential equations (SPDEs). In particular, we give sufficient conditions in order to have existence and smoothness of the density on the set where the columns of the diffusion matrix span . We then prove that the density is strictly positive in a point if the connected component of the set where the columns of the diffusion matrix span which contains this point has a non-void intersection with the support of the law of the solution. We will finally check how all these results apply to the case of the stochastic heat equation in any space dimension and the stochastic wave equation in dimension.en
• dc.format.mimetype application/pdf
• dc.identifier.citation Nualart E. On the density of systems of non-linear spatially homogeneous SPDEs. Stochastics. 2012 Feb 28;85(1):48-70. DOI: 10.1080/17442508.2011.653567
• dc.identifier.doi http://dx.doi.org/10.1080/17442508.2011.653567
• dc.identifier.issn 1744-2508
• dc.identifier.uri http://hdl.handle.net/10230/46562
• dc.language.iso eng
• dc.publisher Taylor & Francis
• dc.relation.ispartof Stochastics. 2012 Feb 28;85(1):48-70
• dc.rights © Taylor & Francis. This is an electronic version of an article published in "Nualart E. On the density of systems of non-linear spatially homogeneous SPDEs. Stochastics. 2012;85(1): 48-70". Stochastics is available online at: www.tandfonline.com/doi/abs/10.1080/17442508.2011.653567
• dc.rights.accessRights info:eu-repo/semantics/openAccess
• dc.subject.keyword Spatially homogeneous Gaussian noiseen
• dc.subject.keyword Malliavin calculusen
• dc.subject.keyword Non-linear stochastic partial differential equationsen
• dc.subject.keyword Strict positivity of the densityen
• dc.title On the density of systems of non-linear spatially homogeneous SPDEsen
• dc.type info:eu-repo/semantics/article
• dc.type.version info:eu-repo/semantics/acceptedVersion