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We obtain sufficient conditions of stochastic equivalence of Gaussian randomfields with special covariance function. These results generalize Doob'stransformation (condition of stochastic equivalence of a Gaussian and a Wienerprocesses) to the case of random fields (condition of stochastic equivalence ofa Gaussian process and a Brownian sheet). We look at the problem of finding thedistribution of supremum of a Brownian sheet on a set with a dimension lowerthan the dimension of the field. We consider the probability of a Browniansheet with a certain drift to attain zero level. The obtained results cansignificantly simplify the problem of finding distributions of functionals of aBrownian sheet by reducing it to the problem of finding distributions onparallelepipeds with dimension lower than the dimension of the field. Weconsider examples that verify validity of the obtained theorem by modelingcorresponding fields and comparing empirical probabilities with theoreticalones.

On the distribution of maximum of a Brownian sheet restricted to a lower-dimensional set