Abstract
We prove absolute continuity of the law of the solution, evaluated at fixed points in time and space, to a parabolic dissipative stochastic PDE on L2(G), where G is an open bounded domain in \(\mathbb {R}^{d}\) with smooth boundary. The equation is driven by a multiplicative Wiener noise and the nonlinear drift term is the superposition operator associated to a real function that is assumed to be monotone, locally Lipschitz continuous, and growing not faster than a polynomial. The proof, which uses arguments of the Malliavin calculus, crucially relies on the well-posedness theory in the mild sense for stochastic evolution equations in Banach spaces.
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Acknowledgments
The authors thank the anonymous referees for a careful reading of the manuscript. The first-named author is sincerely grateful to Prof. S. Albeverio for several very pleasant stays at the Interdisziplinäres Zentrum für Komplexe Systeme, Universität Bonn, where most of the work for this paper was done. The second-named author is supported by the grants MTM2015-67802P and PGC2018-097848-B-I00 (Ministerio de Economía y Competitividad).
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Marinelli, C., Quer-Sardanyons, L. Absolute Continuity of Solutions to Reaction-Diffusion Equations with Multiplicative Noise. Potential Anal 57, 243–261 (2022). https://doi.org/10.1007/s11118-021-09914-3
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DOI: https://doi.org/10.1007/s11118-021-09914-3