Wiener process
Modulus of continuity

Local modulus of continuity: lim sup ε → 0 + | w (ε) | 2 ε log log (1 / ε) = 1, almost surely. {\displaystyle \limsup _{\varepsilon \to 0+}{\frac {|w(\varepsilon)|}{\sqrt {2\varepsilon \log \log(1/\varepsilon)}}}=1,\qquad {\text{almost surely}}.}

Global modulus of continuity (Lévy): lim sup ε → 0 + sup 0 ≤ s < t ≤ 1, t − s ≤ ε | w (s) − w (t) | 2 ε log (1 / ε) = 1, almost surely. {\displaystyle \limsup _{\varepsilon \to 0+}\sup _{0\leq s<t\leq 1,t-s\leq \varepsilon }{\frac {|w(s)-w(t)|}{\sqrt {2\varepsilon \log(1/\varepsilon)}}}=1,\qquad {\text{almost surely}}.}
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