Existence and Regularity of the Density for Solutions to Semilinear Dissipative Parabolic SPDEs |
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Authors: | Carlo Marinelli Eulalia Nualart Lluís Quer-Sardanyons |
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Institution: | 1. Department of Mathematics, University College London, Gower Street, London, WC1E 6BT, United Kingdom 2. Libera Università di Bolzano, 39100, Bolzano, Italy 3. Department of Economics and Business, University Pompeu Fabra and Graduate School of Economics, Ramón Trias Fargas 25–27, 08005, Barcelona, Spain 4. Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193, Bellaterra, Barcelona, Spain
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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 ? d 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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