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

TL;DR

This paper proves that solutions to a heat equation with stochastic Neumann boundary conditions driven by fractional Brownian motion have smooth probability densities, using Malliavin calculus to establish their regularity.

Contribution

It establishes the existence and smoothness of the density for solutions of a stochastic heat equation with boundary noise driven by fractional Brownian motion, under regularity assumptions.

Findings

Law of the solution has a smooth density.

Uses Malliavin calculus for regularity analysis.

Applicable to stochastic PDEs with boundary noise.

Abstract

We study existence and regularity of the density for the solution $u(t,x)$ (with fixed $t > 0$ and $x \in D$) of the heat equation in a bounded domain $D \subset \mathbb R^d$ 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 $\mathbb R$.