A probabilistic approach to the Φ-variation of classical fractal functions with critical roughness

2020/09/11 by Xiyue Han, Han, Xiyue, Alexander Schied +3 · 1 citation
Mathematics · Physics and Astronomy · #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Mathematical Dynamics and Fractals #Probability (math.PR) #Stochastic processes and statistical mechanics #Theoretical and Computational Physics

paper · pdf · doi:10.48550/arxiv.2009.05420

openalex publication_date 2020/09/11 · openalex created_date 2024/04/11 · openalex updated_date 2026/07/28

Abstract

We consider Weierstraß and Takagi-van der Waerden functions with critical degree of roughness. In this case, the functions have vanishing p^th variation for all p>1 but are also nowhere differentiable and hence not of bounded variation either. We resolve this apparent puzzle by showing that these functions have finite, nonzero, and linear Wiener--Young Φ-variation along the sequence of b-adic partitions, where Φ(x)=x/√(-log x). For the Weierstraß functions, our proof is based on the martingale central limit theorem (CLT). For the Takagi--van der Waerden functions, we use the CLT for Markov chains if a certain parameter b is odd, and the standard CLT for b even.

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