论文标题
分数布朗图像的尺寸
Dimensions of fractional Brownian images
论文作者
论文摘要
本文涉及中间尺寸，即在Hausdorff和盒子尺寸之间插值的尺寸的频谱。潜在理论方法用于产生尺寸界限，以用于Hölder图和某些随机过程下的集合图像。我们将其应用于计算用能力定义的尺寸轮廓，计算索引$α$分数运动下的Borel集合的尺寸的几乎悬而未决的值。作为推论，这建立了Borel集合的配置文件的连续性，并允许我们获得明确的条件，以表明集合的Hausdorff尺寸如何影响Hölder图像的典型盒子尺寸，例如Projections。所使用的方法提出了针对相关问题的一般策略；可以从分析该集合的特定分数布朗图像中学到有关集合的维度信息。总而言之，在投影的设置中，我们在中间维度方面获得了特殊集合的Hausdorff维度的界限。
This paper concerns the intermediate dimensions, a spectrum of dimensions that interpolate between the Hausdorff and box dimensions. Potential theoretic methods are used to produce dimension bounds for images of sets under Hölder maps and certain stochastic processes. We apply this to compute the almost-sure value of the dimension of Borel sets under index-$α$ fractional Brownian motion in terms of dimension profiles defined using capacities. As a corollary, this establishes continuity of the profiles for Borel sets and allows us to obtain an explicit condition showing how the Hausdorff dimension of a set may influence the typical box dimension of Hölder images such as projections. The methods used propose a general strategy for related problems; dimensional information about a set may be learned from analysing particular fractional Brownian images of that set. To conclude, we obtain bounds on the Hausdorff dimension of exceptional sets, with respect to intermediate dimensions, in the setting of projections.
