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

TL;DR

This paper establishes conditions under which Gaussian random fields are stochastically equivalent, extending classical results to Brownian sheets, and analyzes the distribution of their supremum on lower-dimensional sets.

Contribution

It generalizes Doob's transformation to Gaussian fields and provides a method to simplify distribution calculations of Brownian sheet functionals on lower-dimensional sets.

Findings

Derived sufficient conditions for stochastic equivalence of Gaussian fields.

Reduced complex distribution problems to lower-dimensional parallelepipeds.

Validated theoretical results through modeling and empirical probability comparisons.

Abstract

We obtain sufficient conditions of stochastic equivalence of Gaussian random fields with special covariance function. These results generalize Doob's transformation (condition of stochastic equivalence of a Gaussian and a Wiener processes) to the case of random fields (condition of stochastic equivalence of a Gaussian process and a Brownian sheet). We look at the problem of finding the distribution of supremum of a Brownian sheet on a set with a dimension lower than the dimension of the field. We consider the probability of a Brownian sheet with a certain drift to attain zero level. The obtained results can significantly simplify the problem of finding distributions of functionals of a Brownian sheet by reducing it to the problem of finding distributions on parallelepipeds with dimension lower than the dimension of the field. We consider examples that verify validity of the obtained theorem by modeling corresponding fields and comparing empirical probabilities with theoretical ones.