2015/04/19 by Angel, Omer, Balka, Richárd, Máthé, András +1
#26A16 #26A45 #28A78 #54E52 #60G17 #60G22 #60J65 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Probability (math.PR)
paper · doi:10.48550/arxiv.1504.04789
For 0max\1-α,α\ and B\colon A→ ℝ is of bounded variation. Furthermore, almost surely, there exists no set A⊂ [0,1] such that dimM A>1-α and B\colon A→ ℝ is β-Hölder continuous for some β>α. The zero set and the set of record times of B witness that the above theorems give the optimal dimensions. We also prove similar restriction theorems for deterministic self-affine functions and generic α-Hölder continuous functions. Finally, let \B(t): t∈ [0,1]\ be a two-dimensional Brownian motion. We prove that, almost surely, there is a compact set D⊂ [0,1] such that dimH D≥ 1/3 and B\colon D→ ℝ2 is non-decreasing in each coordinate. It remains open whether 1/3 is best possible.