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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Abstract

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

\[\sigma_{0}^{2}(x-y)=\frac{4}{\pi}\int^{\infty}_{0}\frac{\sin^{2}({\lambda(x-y)}/{2})}{{\psi(\lambda)}}\,d\lambda.\]

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

\[\lim_{h\downarrow0}\int_{a}^{b}\biggl|{\frac{L^{x+h}_{t}-L^{x}_{t}}{\sigma_{0}(h)}}\biggr|^{p}\,dx=2^{p/2}E|\eta|^{p}\int_{a}^{b}|L^{x}_{t}|^{p/2}\,dx\]

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);

\[\lim_{h\to0}\int_{a}^{b}\biggl|\frac{G(x+h)-G(x)}{\sigma_{0}(h)}\biggr|^{p}\,dx=E|\eta|^{p}(b-a)\]

for all a, bR1, almost surely.

Article information

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

Dates
First available in Project Euclid: 29 February 2008

Permanent link to this document
https://projecteuclid.org/euclid.aop/1204306961

Digital Object Identifier
doi:10.1214/009117907000000277

Mathematical Reviews number (MathSciNet)
MR2393991

Zentralblatt MATH identifier
1260.60156

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

Keywords
Gaussian processes local times Levy processes

Citation

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. https://projecteuclid.org/euclid.aop/1204306961


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References

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  • Yor, M. (1983). Derivatives of self-intersection local times. Séminaire de Probabilités XVII. Lecture Notes in Math. 986 89–106. Springer, Berlin.