Dimensions of fractional Brownian images
Abstract
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.
Citation
Burrell , S 2021 , ' Dimensions of fractional Brownian images ' , Journal of Theoretical Probability , vol. First Online . https://doi.org/10.1007/s10959-021-01131-3
Publication
Journal of Theoretical Probability
Status
Peer reviewed
ISSN
0894-9840Type
Journal article
Description
Funding: Carnegie Trust for the Universities of Scotland, grant number: PHD060287 and London Mathematical Society, grant number: ECF-1920-85.Collections
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