Absolute continuity of the law for solutions of stochastic differential equations with boundary noise

Abstract

We study the existence and regularity of the density for the solution [Formula: see text] (with fixed [Formula: see text] and [Formula: see text]) of the heat equation in a bounded domain [Formula: see text] driven by a stochastic inhomogeneous Neumann boundary condition with stochastic term. The stochastic perturbation is given by a fractional Brownian motion process. Under suitable regularity assumptions on the coefficients, by means of tools from the Malliavin calculus, we prove that the law of the solution has a smooth density with respect to the Lebesgue measure in [Formula: see text].