Limsup Random Fractals

By Davar Khoshnevisan, Yuval Peres, Yimin Xiao. 2000

75 citations recorded by the source corpus.

Orey and Taylor (1974) introduced sets of ``fast points'' where Brownian increments are exceptionally large, ${\rm F}(\lambda):=\{ t\in[0,1]: \limsup_{h\to 0}{ | X(t+h)-X(t)| / \sqrt{ 2h|\log h|}} \ge \lambda\}$. They proved that for $\lambda \in (0,1]$, the Hausdorff dimension of ${\rm F}(\lambda)$ is $1-\lambda^2$ a.s. We prove that for any analytic set $E \subset [0,1]$, the supremum of the $\lambda$ such that $E$ intersects ${\rm F}(\lambda)$ a.s. equals $\sqrt{\text{dim}_p E }$, where $\text{dim}_p E$ is the packing dimension of $E$. We derive this from a general result that applies to many other random fractals defined by limsup operations. This result also yields extensions of certain ``fractal functional limit laws'' due to Deheuvels and Mason (1994). In particular, we prove that for any absolutely continuous function $f$ such that $f(0)=0$ and the energy $\int_0^1 |f'|^2 \, dt $ is lower than the packing dimension of $E$, there a.s. exists some $t \in E$ so that $f$ can be uniformly approximated in $[0,1]$ by normalized Brownian increments $s \mapsto [X(t+sh)-X(t)] / \sqrt{ 2h|\log h|}$; such uniform approximation is a.s. impossible if the energy of $f$ is higher than the packing dimension of $E$.

Subfields: Condensed Matter Physics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/2ad456215e91fa58bfa33c523725fbc4bbb04353

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In Condensed Matter Physics

  1. B. Mandel
  2. Márton Elekes
  3. Z. Buczolich
  4. R. Balka
  5. Yimin Xiao

In Mathematical Physics

  1. B. Mandel
  2. Márton Elekes
  3. Z. Buczolich
  4. R. Balka
  5. Yimin Xiao
  6. Dongsheng Wu
  7. Antoine Ayache
  8. Davar Khoshnevisan

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