Dimension of fractional Brownian motion with variable drift

• 成果类型: Article
• 署名作者: Peres, Yuval; Sousi, Perla
• 署名单位: Microsoft; University of Cambridge
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-015-0645-5
• 发表日期: 2016
• 页码: 771-794
• 关键词: hausdorff dimension; carpets; sets

摘要

Let X be a fractional Brownian motion in . For any Borel function , we express the Hausdorff dimension of the image and the graph of 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 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 can disagree.