Lévy's modulus of continuity theorem

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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[edit]

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[edit]

References[edit]

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