Absolute continuity of the law for solutions of stochastic differential equations 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 $\mathbb{R}$.

25 pages

References in corpus (1)

• Long-time behavior of stochastically perturbed neuronal networks

Cited by in corpus (2)

• Differentiability in infinite dimension and the Malliavin calculus
• Well-posedness of a reaction-diffusion model with stochastic dynamical boundary conditions