# Lévy's modulus of continuity theorem

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 {displaystyle B:[0,1]times Omega to mathbb {R} } be a standard Wiener process. Then, almost surely, {displaystyle lim _{hto 0}sup _{t,t'leq 1;|t-t'|leq h}{frac {|B_{t'}-B_{t}|}{sqrt {2hlog(1/h)}}}=1.} In other words, the sample paths of Brownian motion have modulus of continuity {displaystyle omega _{B}(delta )={sqrt {2delta log(1/delta )}}} with probability one, and for sufficiently small {displaystyle delta >0} .[1] See also Some properties of sample paths of the Wiener process 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). Categories: Probability theoremsPaul Lévy (mathematician)

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