  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 Gaussian type 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.
