Lévy's modulus of continuity
- Lévy's modulus of continuity
In mathematics, Lévy's modulus of continuity is a theorem that gives an almost sure estimate of the modulus of continuity for Brownian motion. It is due to the French mathematician Paul Pierre Lévy.
tatement of the result
Let be a standard Brownian motion. Then, almost surely,:
In other words, the sample paths of Brownian motion have modulus of continuity
:
with probability one, and for sufficiently small .
References
* P.P. Lévy. "Théorie de l'addition des variables aléatoires." Gauthier-Villars, Paris (1937).
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