Lower functions of empirical processes and Brownian sheets

We give a complete characterization of the lower functions for the two-parameter Brownian sheet and for the uniform empirical process via integral criteria. Our result for the Brownian sheet can be viewed as a solution, in a very special case (the general case remains an open question), of the following problem: find the escape rate of infinite-dimensional Brownian motion. Our result for the empirical process disproves (and provides the correct form of) a conjecture in [G. R. Shorack and J. A. Wellner, Wiley, New York, 1986.].