On the Gaussian homogeneous fields with given conditional distributions

Let $\eta(t)$, $t\in E$, and $\zeta(t)\in E$, be two independent Gaussian fields in $r$-dimensional cancellated space $E$ and suppose that $\eta(t)$, $t\in E$, is a homogeneous field. Consider $\xi(t)=\eta(t)+\zeta(t)$, $t\in E$. Let $T\subset E$ be an arbitrary finite set and $\mathfrak B_T$ be the $\sigma$-algebra, generated by all random variables $\xi(t)$, $t\notin T$. The main question considered in this paper concerns the conditions for $\zeta(t)$, $t\in E$ , to be the field of conditional expectations of $\xi(t)$, $t\in E$, relative to $\mathfrak B=\bigcap\limits_T\mathfrak B_T$. Theorem 1, 2 solves the problem in the case when $\xi(t)$, $t\in E$, is a homogeneous field and theorem 4 when $\xi(t)$, $t\in E$, is a markovian field.