加性布朗运动气泡和布朗薄片边界的豪斯多夫维数

IF 1.5 3区数学 Q1 MATHEMATICS

Dissertationes Mathematicae Pub Date : 2017-02-27 DOI:10.4064/dm811-9-2021

R. Dalang, T. Mountford

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引用次数: 0

摘要

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。

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

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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来源期刊

Dissertationes Mathematicae 数学-数学

CiteScore

2.80

自引率

0.00%

发文量

审稿时长

>12 weeks

期刊介绍： DISSERTATIONES MATHEMATICAE publishes long research papers (preferably 50-100 pages) in any area of mathematics. An important feature of papers accepted for publication should be their utility for a broad readership of specialists in the domain. In particular, the papers should be to some reasonable extent self-contained. The paper version is considered as primary. The following criteria are taken into account in the reviewing procedure: correctness, mathematical level, mathematical novelty, utility for a broad readership of specialists in the domain, language and editorial aspects. The Editors have adopted appropriate procedures to avoid ghostwriting and guest authorship.