Heat equation perturbed by boundary noise

Abstract

We will consider the second order parabolic equation with a very rough Dirichlet boundary condition defined as a derivative of the Wiener process evolving in space and time. We will show that such an equation has a Markovian solution taking values in a certain non-standard weighted $L^p$-space. We will also show some properties of the associated transition semigroup such as irreducibility and strong Feller property. Finally, we will present conditions for the existence and uniqueness of the invariant measure.