Small deviations for some multi-parameter Gaussian processes

We prove some general bounds for the probability that a multi-parameter Gau ssian process has very small values. These results, when applied to a certa in class functional Brownian sheets, yield the exact rate for their so-call ed small ball probability. We show by example how to use such results to co mpute the Hausdorff dimension of some exceptional sets determined by maxima l increments.