arXivDaily arXiv每日学术速递 周一至周五更新
arXiv 1310.7002math.PRmath.CA
收藏

变漂移分数阶布朗运动的维数

Dimension of Fractional Brownian motion with variable drift

Yuval Peres, Perla Sousi

阅读 PDF
arXiv 页面 ↗HTML 原文 ↗

摘要：

设$X$是$\mathbb{R}^d$中的分数布朗运动.对于任意Borel函数$f:[0,1] \to \mathbb{R}^d$，我们用$f$表示$X+f$的图像和图的豪斯多夫维数.这是新的情况下，即使是布朗运动和连续的$f$，它是已知的，这方面几乎肯定是常数。该表达式涉及到适应的抛物线尺寸以前使用的泰勒和沃森的热方程的极性特征。当$f$图是自仿射McMullen-Bedford地毯时，我们得到了$X+f$图的维数的一个显式公式.特别地，我们证明了它可以严格大于$f$图和$X$图的豪斯多夫维数的最大值.尽管存在随机扰动，但$X+f$的图的Minkowski维数和Hausdorff维数可能不一致。

英文摘要：

Let $X$ be a fractional Brownian motion in $\mathbb{R}^d$. For any Borel function $f:[0,1] \to \mathbb{R}^d$, we express the Hausdorff dimension of the image and the graph of $X+f$ in terms of $f$. This is new even for the case of Brownian motion and continuous $f$, where it was known that this dimension is almost surely constant. The expression involves an adaptation of the parabolic dimension previously used by Taylor and Watson to characterize polarity for the heat equation. In the case when the graph of $f$ is a self-affine McMullen-Bedford carpet, we obtain an explicit formula for the dimension of the graph of $X+f$ in terms of the generating pattern. In particular, we show that it can be strictly bigger than the maximum of the Hausdorff dimension of the graph of $f$ and that of $X$. Despite the random perturbation, the Minkowski and Hausdorff dimension of the graph of $X+f$ can disagree.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

↑