The Annals of Probability

Lp moduli of continuity of Gaussian processes and local times of symmetric Lévy processes

Michael B. Marcus and Jay Rosen

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Let X={X(t), tR+} be a real-valued symmetric Lévy process with continuous local times {Ltx, (t, x)∈R+×R} and characteristic function EeiλX(t)=e(λ). Let


If σ02(h) is concave, and satisfies some additional very weak regularity conditions, then for any p≥1, and all tR+,


for all a, b in the extended real line almost surely, and also in Lm, m≥1. (Here η is a normal random variable with mean zero and variance one.)

This result is obtained via the Eisenbaum Isomorphism Theorem and depends on the related result for Gaussian processes with stationary increments, {G(x), xR1}, for which E(G(x)−G(y))2=σ02(xy);


for all a, bR1, almost surely.

Article information

Ann. Probab., Volume 36, Number 2 (2008), 594-622.

First available in Project Euclid: 29 February 2008

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J55: Local time and additive functionals 60G15: Gaussian processes 60G17: Sample path properties

Gaussian processes local times Levy processes


Marcus, Michael B.; Rosen, Jay. L p moduli of continuity of Gaussian processes and local times of symmetric Lévy processes. Ann. Probab. 36 (2008), no. 2, 594--622. doi:10.1214/009117907000000277.

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