The fundamental solution for a degenerate parabolic Dirichlet problem

We study a homogeneous parabolic Dirichlet problem involving a possibly degenerate Ornstein-Uhlenbeck operator in a half-space H+ of Rn. We find an explicit formula for the fundamental solution. Under the Hörmander condition of hypoellipticity, we prove a global regularity result in spaces of continuous and bounded functions. We extend our explicit formula to the infinite-dimensional setting.