Probability tails of Gaussian extrema

We study the supremum of 'the' standard isonormal linear process L on a subset of a real Hilbert space H. Upper and lower bounds on the probability that supx[epsilon] LX>[lambda], [lambda] large, are found. We treat a number of examples. These include the distribution of the maximum of certain 'locally stationary' processes on 1, as well as those of the rectangle indexed, pinned Brownian sheet in k and the half-plane indexed pinned sheet in 2. We also consider Brownian motion indexed by convex sets in [0, 1]2.