Estimation of first crossing time distribution for N-parameter Brownian motion processes relative to upper class boundaries

Xt is a Brownian sheet defined for t belonging to the positive orthant of RN, for which the covariance function is given by E(XsXt = [Pi]i = 1N min(si,ti). Functions [phi] with suitable growth conditions are classified as lower or upper class near the origin according as Xt does or does not exceed [radical sign][not partial differential](t) [phi]([not partial differential](t)) infinitely often as [not partial differential](t) --> 0 ([not partial differential](t) = [Pi] ti). S. Orey and W. E. Pruitt (Ann. Probab. 1 (1973), 138-163) obtained the necessary and sufficient condition in terms of the convergence of a generalized Kolmogorov-type integral. The distribution of the related first crossing time is considered and in the process an interpretation for the integrand in the Kolmogorov test is obtained.