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Abstract
In a previous article (Int. Math. Res. Not. 2014:10 (2014), 2730–2745) T. Orponen and the authors proved that the Fourier dimension of the graph of any real-valued function on R is bounded above by 1. This partially answered a question of Kahane (1993) by showing that the graph of the Wiener process Wt (Brownian motion) is almost surely not a Salem set. In this article we complement this result by showing that the Fourier dimension of the graph of Wt is almost surely 1. In the proof we introduce a method based on Itô calculus to estimate Fourier transforms by reformulating the question in the language of Itô drift-diffusion processes and combine it with the classical work of Kahane on Brownian images.
| Original language | English |
|---|---|
| Pages (from-to) | 115-132 |
| Number of pages | 18 |
| Journal | Analysis & PDE |
| Volume | 11 |
| Issue number | 1 |
| Early online date | 17 Sept 2017 |
| DOIs | |
| Publication status | Published - 2018 |
Keywords
- Brownian motion
- Wiener process
- Itô calculus
- Itô drift-diffusion process
- Fourier transform
- Fourier dimension
- Salem set
- Graph
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Dive into the research topics of 'On the Fourier analytic structure of the Brownian graph'. Together they form a unique fingerprint.Projects
- 1 Finished
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Fractal Geometry and Dimension: Fractal Geometry and dimension theory
Fraser, J. (PI)
1/09/16 → 30/06/18
Project: Fellowship
