On the comparison of the distribution of the supremum of random fields represented by stochastic integrals

In this paper we compare the distribution of the supremum of the Gaussian random fields Z(P) = .CpG(P, P.) dW(P.) and U(P) = .CpdW(P'), where CP are circles of fixed radius, dW is a white noise field and G are special deterministic response functions. The results obtained permit us to establish upper bounds for the distribution of the supremum of Z(P) by applying some well-known inequalities on U(P). The comparison of the suprema is carried out also, when CP = .2, between fields with different response functions.