Existence and regularity of the density for the solution to semilinear dissipative parabolic SPDEs

Details Existence and regularity of the density for the solution to semilinear dissipative parabolic SPDEs Existence and regularity of the density for the solution to semilinear dissipative parabolic SPDEs

2012-02-21

arxiv.org/abs/1202.4610v2

Mathematics

Probability

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Scientific paper

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 $\mathbb{R}^d$. 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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