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} be a standard Brownian motion. 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 δ > 0.

[edit] References

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