White noise stochastic BVP's and Cimmino theory

The analysis of the Molodensky problem in spherical approximation can be reduced to the simple Dirichlet problem. If the boundary data are noisy (white noise), it is requested to explain what is the meaning of the solution of a B.V.P. with data of this kind. This can be correctly done in the framework of generalized random field theory and an equivalent principle can be stated such that the solution of the stochastic problem exists and is unique iff the analogous deterministic problem has a unique solution with L-2(S) boundary data. This result can be achieved by Cimmino's theory which is also reviewed here for the ease of the reader.