Hausdorff dimension of the boundary of bubbles of additive Brownian motion and of the Brownian sheet

IF 16.4 1区化学 Q1 CHEMISTRY, MULTIDISCIPLINARY

Accounts of Chemical Research Pub Date : 2017-02-27 DOI:10.4064/dm811-9-2021

R. Dalang, T. Mountford

引用次数: 0

Abstract

We first consider the additive Brownian motion process $(X(s_1,s_2),\ (s_1,s_2) \in \mathbb{R}^2)$ defined by $X(s_1,s_2) = Z_1(s_1) - Z_2 (s_2)$, where $Z_1$ and $Z_2 $ are two independent (two-sided) Brownian motions. We show that with probability one, the Hausdorff dimension of the boundary of any connected component of the random set $\{(s_1,s_2)\in \mathbb{R}^2: X(s_1,s_2) >0\}$ is equal to $$ \frac{1}{4}\left(1 + \sqrt{13 + 4 \sqrt{5}}\right) \simeq 1.421\, . $$ Then the same result is shown to hold when $X$ is replaced by a standard Brownian sheet indexed by the nonnegative quadrant.

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加性布朗运动气泡和布朗薄片边界的豪斯多夫维数

我们首先考虑由$X（s_1，s_2）=Z_1（s_1）-Z_2（s_2）$定义的加性布朗运动过程$（X（s_2，s_1），\in\mathbb{R}^2）$，其中$Z_1$和$Z_2$是两个独立的（双侧）布朗运动。我们在概率一的情况下证明了随机集$\{（s_1，s_2）\in\mathbb｛R｝^2：X（s_1、s_2）>0\｝$的任意连通分量的边界的Hausdorff维数等于$\frac｛1｝｛4｝\left（1+\sqrt｛13+4\sqrt{5｝｝\right）\simeq 1.421\，.$$。然后，当$X$被非负象限索引的标准布朗表取代时，同样的结果也成立。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Accounts of Chemical Research 化学-化学综合

CiteScore

31.40

自引率

1.10%

发文量

312

审稿时长

2 months

期刊介绍： Accounts of Chemical Research presents short, concise and critical articles offering easy-to-read overviews of basic research and applications in all areas of chemistry and biochemistry. These short reviews focus on research from the author’s own laboratory and are designed to teach the reader about a research project. In addition, Accounts of Chemical Research publishes commentaries that give an informed opinion on a current research problem. Special Issues online are devoted to a single topic of unusual activity and significance. Accounts of Chemical Research replaces the traditional article abstract with an article "Conspectus." These entries synopsize the research affording the reader a closer look at the content and significance of an article. Through this provision of a more detailed description of the article contents, the Conspectus enhances the article's discoverability by search engines and the exposure for the research.