Isolated Zeros for Brownian Motion with Variable Drift
Isolated Zeros for Brownian Motion with Variable Drift
复制标题
具有可变漂移的布朗运动的孤立零点
DOI：
10.1214/ejp.v16-927
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发表时间：
2010
影响因子：
1.4
通讯作者：
J. Ruscher
中科院分区：
文献类型：
--
作者：
Ton'ci Antunovi'c;K. Burdzy;Y. Peres;J. Ruscher
众所周知，标准一维布朗运动 $B(t)$ 几乎肯定不存在孤立零点。我们证明，对于任何 $\alpha<1/2$，都存在 alpha-Holder 连续函数 $f$，其中过程 $B-f$ 具有正概率的孤立零点。我们还证明，对于任何连续函数 $f$，$B-f$ 的零集具有至少 $1/2$ 的豪斯多夫维数，并且概率为正，并且如果 $f$ 是 $1/2$-Holder 连续或有界变分，则 $1/2$ 是 Hausdorff 维数的上限。
It is well known that standard one-dimensional Brownian motion $B(t)$ has no isolated zeros almost surely. We show that for any $\alpha<1/2$ there are alpha-Holder continuous functions $f$ for which the process $B-f$ has isolated zeros with positive probability. We also prove that for any continuous function $f$, the zero set of $B-f$ has Hausdorff dimension at least $1/2$ with positive probability, and $1/2$ is an upper bound on the Hausdorff dimension if $f$ is $1/2$-Holder continuous or of bounded variation.