Quantum field theory on manifolds with a boundary

We discuss quantum theory of fields \phi defined on (d+1)-dimensional manifold {\cal M} with a boundary {\cal B}. The free action W_{0}(\phi) which is a bilinear form in \phi defines the Gaussian measure with a covariance (Green function) {\cal G}. We discuss a relation between the quantum field theory with a fixed boundary condition \Phi and the theory defined by the Green function {\cal G}. It is shown that the latter results by an average over \Phi of the first. The QFT in de Sitter space is treated as an example. It is shown that quantum fields on the boundary are more regular than the ones on de Sitter space