On the Fourier analytic structure of the Brownian graph

Jonathan MacDonald Fraser, Tuomas Sahlsten

Research output: Contribution to journalArticlepeer-review

1 Citation (Scopus)

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 languageEnglish
Pages (from-to)115-132
Number of pages18
JournalAnalysis & PDE
Volume11
Issue number1
Early online date17 Sept 2017
DOIs
Publication statusPublished - 2018

Keywords

  • Brownian motion
  • Wiener process
  • Itô calculus
  • Itô drift-diffusion process
  • Fourier transform
  • Fourier dimension
  • Salem set
  • Graph

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