Dimensions of fractional Brownian images

Stuart Burrell

Research output: Contribution to journalArticlepeer-review


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.
Original languageEnglish
Number of pages22
JournalJournal of Theoretical Probability
VolumeFirst Online
Early online date1 Oct 2021
Publication statusE-pub ahead of print - 1 Oct 2021


  • Intermediate dimensions
  • Box dimension
  • Hausdorff dimension
  • Fractional Brownian motion
  • Capacity
  • Exceptional directions


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