  In this paper we study the Poisson and heat equations on bounded and
unbounded domains with smooth boundary with random Dirichlet boundary
conditions. The main novelty of this work is a convenient framework for the
analysis of such 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. Then we 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 L\'evy noise is considered.
