Metric Entropy of Integration Operators and Small Ball Probabilities for the Brownian Sheet Original Research Article

Let Td: L2([0, 1]d)→C([0, 1]d) be the d-dimensional integration operator. We show that its Kolmogorov and entropy numbers decrease with order at least k−1(log k)d−1/2. From this we derive that the small ball probabilities of the Brownian sheet on [0, 1]d under the C([0, 1]d)-norm can be estimated from below by exp(−Cε−2 |log ε|2d−1), which improves the best known lower bounds considerably. We also get similar results with respect to certain Orlicz norms.