Lévy's modulus of continuity theorem

Lévy's modulus of continuity theorem is a theorem that gives a result about an almost sure behaviour of an estimate of the modulus of continuity for Wiener process, that is used to model what's known as Brownian motion.

Lévy's modulus of continuity theorem is named after the French mathematician Paul Lévy.

Statement of the result

Let $B:[0,1]\times \Omega \to \mathbb {R}$ be a standard Wiener process. Then, almost surely,

\lim _{h\to 0}\sup _{t,t'\leq 1;|t-t'|\leq h}{\frac {|B_{t'}-B_{t}|}{\sqrt {2h\log(1/h)}}}=1.

In other words, the sample paths of Brownian motion have modulus of continuity

\omega _{B}(\delta )=c{\sqrt {2\delta \log(1/\delta )}}

with probability one, for $c>1$ and sufficiently small $\delta >0$ .^[1]

References

^ Lévy, P. Author Profile Théorie de l’addition des variables aléatoires. 2. éd. (French) page 172 Zbl 0056.35903 (Monographies des probabilités.) Paris: Gauthier-Villars, XX, 387 p. (1954)

Paul Pierre Lévy, Théorie de l'addition des variables aléatoires. Gauthier-Villars, Paris (1937).

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