Absolute continuity of solutions to reaction-diffusion equations with multiplicative noise

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 \(L^2(G)\), where \(G\) is an open bounded domain in \(\mathbb{R}^d\) with smooth boundary. The equation is driven by a multiplicative Wiener noise and t...

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Bibliographic Details
Published in:arXiv.org 2019-05
Main Authors: Marinelli, Carlo, Quer-Sardanyons, Lluís
Format: Article
Language:English
Subjects:
Banach spaces
Continuity
Mathematical analysis
Operators (mathematics)
Polynomials
Reaction-diffusion equations
Smooth boundaries
Superposition (mathematics)
Well posed problems
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Summary: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 \(L^2(G)\), 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 which 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.
ISSN:2331-8422