On the Fourier analytic structure of the Brownian graph
| Autor: | Jonathan M. Fraser, Tuomas Sahlsten |
|---|---|
| Přispěvatelé: | The Leverhulme Trust, University of St Andrews. Pure Mathematics |
| Jazyk: | angličtina |
| Rok vydání: | 2018 |
| Předmět: |
Pure mathematics
28A80 Dimension (graph theory) T-NDAS Dynamical Systems (math.DS) 01 natural sciences Graph Wiener process 010104 statistics & probability symbols.namesake Mathematics::Probability Classical Analysis and ODEs (math.CA) FOS: Mathematics 42B10 60J65 Almost surely QA Mathematics 0101 mathematics Mathematics - Dynamical Systems 42B10 60H30 (primary) 60J65 28A80 (secondary) QA Brownian motion R2C Mathematics Complement (set theory) 11K16 Numerical Analysis Applied Mathematics Itô calculus Probability (math.PR) 010102 general mathematics Itô drift-diffusion process Function (mathematics) graph Fourier dimension Fourier transform Mathematics - Classical Analysis and ODEs symbols Graph (abstract data type) BDC 60H30 Analysis Mathematics - Probability Salem set |
| Zdroj: | Anal. PDE 11, no. 1 (2018), 115-132 Sahlsten, T & Fraser, J 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 |
| DOI: | 10.2140/apde.2018.11.115 |
| Popis: | In a previous article (\textit{Int. Math. Res. Not.} 2014, 2730--2745) T. Orponen and the authors proved that the Fourier dimension of the graph of any real-valued function on $\mathbb{R}$ is bounded above by $1$. This partially answered a question of Kahane ('93) by showing that the graph of the Wiener process $W_t$ (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 $W_t$ is almost surely $1$. In the proof we introduce a method based on Ito calculus to estimate Fourier transforms by reformulating the question in the language of Ito drift-diffusion processes and combine it with the classical work of Kahane on Brownian images. 17 pages, 2 figures. v3: removed the equidistribution section. To appear in Analysis & PDE |
| Databáze: | OpenAIRE |
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