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

Abstract

We prove existence and smoothness of the density of the solution to a nonlinear stochastic heat equation on L2(O) (evaluated at fixed points in time and space), where 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.