Small Deviations of Stable Processes via Metric Entropy

Let X=(X(t))t ¥ T be a symmetric a-stable, 0 < a < 2, process with paths in the dual Eg of a certain Banach space E. Then there exists a (bounded, linear) operator u from E into some La(S, s) generating X in a canonical way. The aim of this paper is to compare the degree of compactness of u with the small deviation (ball) behavior of f(e)=−log P(||X||E* < e) as eQ 0. In particular, we prove that a lower bound for the metric entropy of u implies a lower bound for f(e) under an additional assumption on E. As applications we obtain upper small deviation estimates for weighted a-stable Levy motions, linear fractional a-stable motions and d-dimensional a-stable sheets. Our results rest upon an integral representation of La-valued operators as well as on small deviation results for Gaussian processes due to Kuelbs and Li and to the authors.