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On the Fourier analytic structure of the Brownian graph

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Fraser_2017_apde_v11_n1_p03_s.pdf (622.9Kb)
Date
2018
Author
Fraser, Jonathan MacDonald
Sahlsten, Tuomas
Funder
The Leverhulme Trust
Grant ID
RF-2016-500
Keywords
Brownian motion
Wiener process
Itô calculus
Itô drift-diffusion process
Fourier transform
Fourier dimension
Salem set
Graph
QA Mathematics
T-NDAS
BDC
R2C
Metadata
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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.
Citation
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
Publication
Analysis & PDE
Status
Peer reviewed
DOI
https://doi.org/10.2140/apde.2018.11.115
ISSN
1948-206X
Type
Journal article
Rights
© 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
Collections
  • University of St Andrews Research
URI
http://hdl.handle.net/10023/11846

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