A Remark on Zeros of Brownian Motion
نویسنده
چکیده
Let {W (t), t ≥ 0} be a standard Brownian motion. If I is a bounded interval on which W has no zero, an almost sure lower bound to inf{|W (t)|, t ∈ I} can be provided, when I is taken from a given countable family of intervals covering the positive half-line. 1 Main Result Let {W (t), t ≥ 0} be a standard Brownian motion. Let I be some bounded interval of R. Suppose W (t) 6= 0, for all t ∈ I. What can be said about the size of inf{|W (t)|, t ∈ I}? This one only depends on the location of I and of the size of I. The object of this note is to prove more precisely the following result. Theorem 1 Let θk ≥ 0 be such that TN = ∑ k≤N θk ↑ ∞ and denote IN = [TN , TN+1]. Let ηk ≥ 0 be such that
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تاریخ انتشار 2009