Lévy's modulus of continuity theorem

In mathematics, Lévy's modulus of continuity theorem gives a result about the almost sure behaviour of an estimate of the modulus of continuity for the Wiener process, which models Brownian motion. It is due to 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,

$\lim_{h \to 0} \sup_{0 \leq t \leq 1 - h} \frac{| B_{t+ h} - B_{t} |}{\sqrt{2 h \log (1 / h)}} = 1.$

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

$\omega_{B} (\delta) = \sqrt{2 \delta \log (1 / \delta)}$

with probability one, and for sufficiently small $\delta > 0$.

See also

• Some properties of sample paths of the Wiener process

References

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