Elliptic equations and gaussian processes

We consider a Gaussian process $P$ on the space of distributions generated by a polynomial in the Laplace operator. We prove some support properties for $P$. As a byproduct we strenghten earlier results on the stochastic Dirichlet problem on bounded regions $\Lambda\subset R^d$. We describe in this way the conditional $P-$-distribution of the restriction to $\Lambda$ od a distribution $\varphi$, supposing that $\varphi$ is known outside $\Lambda$: a somewhat detailed description of the sinularity of $\varphi$ in $\Lambda$ is given.