Second order PDEs with Dirichlet white noise boundary conditions

In this paper, we study inhomogeneous Dirichlet boundary problems associated to the Poisson and heat equations on bounded and unbounded domains with smooth boundary and random boundary data. The main novelty of this work is a convenient framework for the analysis of equations excited by the white in time and/or space noise on the boundary. Our approach allows us to show the existence and uniqueness of weak solutions in the space of distributions. We also prove that the solutions can be identified as smooth functions inside the domain, and finally, the rate of their blow up at the boundary is estimated. A large class of noises including Wiener and fractional Wiener space-time white noise, homogeneous noise and Lévy noise are considered.