Hausdorff and Fourier dimension of graph of continuous additive processes

Dexter Dysthe and Chun-Kit Lai

Stochastic Processes and their Applications, 2023, vol. 155, issue C, 355-392

Abstract: An additive process is a stochastic process with independent increments and that is continuous in probability. In this paper, we study the almost sure Hausdorff and Fourier dimension of the graph of continuous additive processes with zero mean. Such processes can be represented as Xt=BV(t) where B is Brownian motion and V is a continuous increasing function. We show that these dimensions depend on the local uniform Hölder indices. In particular, if V is locally uniformly bi-Lipschitz, then the Hausdorff dimension of the graph will be 3/2. We also show that the Fourier dimension almost surely is positive if V admits at least one point with positive lower Hölder regularity.

Keywords: Additive processes; Brownian motions; Fourier dimension; Hausdorff dimension (search for similar items in EconPapers)
Date: 2023
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DOI: 10.1016/j.spa.2022.10.010

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