Lévy's modulus of continuity
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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.
[edit] Statement of the result
Let ![B : [0, 1] \times \Omega \to \mathbb{R}](../../../math/0/f/b/0fbac746b43a16012f6a277e2ba22e1a.png)
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 δ > 0.
[edit] Reference
- P.P. Lévy. Théorie de l'addition des variables aléatoires. Gauthier-Villars, Paris (1937).
