The purpose of this paper is to investigate way of dependency of Gaussian random fields X(D) indexed by a domain D in d-dimensional Euclidean space Rd
. Our main tool is variational calculus, where the boundary of a domain varies and deforms and we appeal to the white noise analysis. We therefore assume that X(D) is expressed white noise integral of the form
(0.1) X(D) = X(D, W)=∫D F(D, u)W(u)du,
where W is the Rd
-parameter white noise and the kernel F(D, u) is a square integrable function over Rd
, and where D is a bounded domain with smooth boundary.