Linear parabolic equation with Dirichlet white noise boundary conditions

Beniamin Goldys and Szymon Peszat

Abstract

We study inhomogeneous Dirichlet boundary value problems associated to a linear parabolic equation \(\frac{du}{dt}=Au\) with strongly elliptic operator \(A\) on bounded and unbounded domains with white noise boundary data. Our main assumption is that the heat kernel of the corresponding homogeneous problem enjoys the Gaussiantype estimates taking into account the distance to the boundary. Under mild assumptions about the domain, we show that \(A\) generates a \(C_0\)-semigroup in weighted \(L^p\)-spaces where the weight is a proper power of the distance from the boundary. We also prove some smoothing properties and exponential stability of the semigroup. Finally, we reformulate the Cauchy-Dirichlet problem with white noise boundary data as an evolution equation in the weighted space and prove the existence of Markovian solutions.

Keywords: stochastic partial differential equations, white noise boundary conditions, Ornstein–Uhlenbeck process, Dirichlet boundary conditions.

AMS Subject Classification: Primary 60G15; secondary 60H15, 60J99.