On the Fourier analytic structure of the Brownian graph
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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 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.
Fraser , J M & Sahlsten , T 2018 , ' On the Fourier analytic structure of the Brownian graph ' Analysis & PDE , vol 11 , no. 1 , pp. 115-132 . DOI: 10.2140/apde.2018.11.115
Analysis & PDE
© 2017, Publisher / the Author(s). This work has been made available online in accordance with the publisher’s policies. This is the final published version of the work, which was originally published at https://doi.org/10.2140/apde.2018.11.115
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