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 . https://doi.org/10.2140/apde.2018.11.115
Analysis & PDE
© 2018, Mathematical Sciences Publishers. 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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