Small Deviations of Stable Processes via Metric Entropy

Abstract

Let X = (X(t))t 2 T be a symmetric {stable, 0 < < 2, process with paths in the dual E of a certain Banach space E. Then there exists a (bounded, linear) operator u from E into some L (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 °(") =...In particular, we prove that a lower bound for the metric entropy of u implies a lower bound for °(") under an additional assumption on E. As applications we obtain lower small deviation estimates for weighted {stable Levy motions, linear fractional {stable motions and d{dimensional {stable Levy sheets. Our results rest upon an integral representation of L {valued operators as well as on small deviation results for Gaussian processes due to Kuelbs and Li and to the authors.