Browsing by Author "Li, Wenbo"
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Item The First Time of a Brownian Motion from an Unbounded Convex Domain(Department of Mathematical Sciences, 2002) Li, WenboConsider the first exit time, ˝D of a (d + 1)-dimensional Brownian motion from an unbounded open domain D = (x; y) 2 R d+1 : y > f(x); x 2 R d starting at (x0; f(x0) + 1) 2 R d+1 for some x0 2 R d , where the function f(x) on R d is convex and f(x) ! 1 as the Euclidean norm j x j ! 1 . Very general estimates for the asymptotics of log P (˝D > t) are given by using Gaussian techniques. In particular, for f(x) = exp fj x j p g , p > 0, lim t !1 tItem Limiting behaviors for Brownian motion reflected on Brownian motion(Department of Mathematical Sciences, 2002) Chen, X.; Li, WenboSuppose that g(t) and Wt are independent Brownian motions starting from g(0) = W0 = 0. Consider the Brownian motion Yt re ected on g(t), obtained from Wt by the means of the Skorohod lemma. The upper and lower limiting behaviors of Yt are presented. The upper tail estimate on exit time is computed via principal eigenvalue.Item Quadratic Functionals and Small Ball Probabilities for the m-fold Brownian Motion(Department of Mathematical Sciences, 2002) Chen, X.; Li, WenboLet the Gaussian process Xm(t) be the m-fold integrated Brownian motion for positive integer m. The Laplace transform of the quadratic functional of Xm(t) is found by using an appropriate self-adjoint integral operator. The result is then used to show the power of a general connection between small ball probabilities for Gaussian process. The connection is discovered by introducing an independent random shift. Various interplay between our results and principal eigenvalues for non-uniform elliptic generators on an unbounded domain are discussed.Item Small Deviations of Stable Processes via Metric Entropy(Department of Mathematical Sciences, 2002) Li, Wenbo; Linde, W.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.