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Existence and Regularity of the Density for Solutions to Semilinear Dissipative Parabolic SPDEs

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Abstract

We prove existence and smoothness of the density of the solution to a nonlinear stochastic heat equation on \(L^2(\mathcal{O})\) (evaluated at fixed points in time and space), where \(\mathcal{O}\) is an open bounded domain in ℝd with smooth boundary. The equation is driven by an additive Wiener noise and the nonlinear drift term is the superposition operator associated to a real function which is assumed to be (maximal) monotone, continuously differentiable, and growing not faster than a polynomial. The proof uses tools of the Malliavin calculus combined with methods coming from the theory of maximal monotone operators.

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Authors and Affiliations

  1. Department of Mathematics, University College London, Gower Street, London, WC1E 6BT, United Kingdom

    Carlo Marinelli

  2. Libera Università di Bolzano, 39100, Bolzano, Italy

    Carlo Marinelli

  3. Department of Economics and Business, University Pompeu Fabra and Graduate School of Economics, Ramón Trias Fargas 25–27, 08005, Barcelona, Spain

    Eulalia Nualart

  4. Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193, Bellaterra, Barcelona, Spain

    Lluís Quer-Sardanyons

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  1. Carlo Marinelli
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  2. Eulalia Nualart
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  3. Lluís Quer-Sardanyons
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Correspondence to Lluís Quer-Sardanyons.

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Marinelli, C., Nualart, E. & Quer-Sardanyons, L. Existence and Regularity of the Density for Solutions to Semilinear Dissipative Parabolic SPDEs. Potential Anal 39, 287–311 (2013). https://doi.org/10.1007/s11118-012-9330-9

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  • DOI: https://doi.org/10.1007/s11118-012-9330-9

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