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On the Fourier
analytic structure of the Brownian graph
Jonathan M. Fraser and Tuomas Sahlsten |
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Vol. 11 (2018), No. 1, 115–132
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Abstract |
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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 is bounded above by . This partially answered a question of Kahane (1993) by showing that the graph of the Wiener process (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 is almost surely . 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. |
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Keywords
Brownian motion, Wiener process, Itô calculus, Itô
drift-diffusion process, Fourier transform, Fourier
dimension, Salem set, graph
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Mathematical Subject Classification 2010
Primary: 42B10, 60H30
Secondary: 11K16, 60J65, 28A80
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Milestones
Received: 28 March 2016
Revised: 19 July 2017
Accepted: 5 September 2017
Published: 17 September 2017
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Authors |
| Jonathan M. Fraser | |
| School of Mathematics and
Statistics University of St Andrews St Andrews United Kingdom |
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| Tuomas Sahlsten | |
| School of Mathematics University of Manchester Manchester United Kingdom |
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