Translator Disclaimer
March 2008 Lp moduli of continuity of Gaussian processes and local times of symmetric Lévy processes
Michael B. Marcus, Jay Rosen
Ann. Probab. 36(2): 594-622 (March 2008). DOI: 10.1214/009117907000000277

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.

Citation

Download Citation

Michael B. Marcus. Jay Rosen. "Lp moduli of continuity of Gaussian processes and local times of symmetric Lévy processes." Ann. Probab. 36 (2) 594 - 622, March 2008. https://doi.org/10.1214/009117907000000277

Information

Published: March 2008
First available in Project Euclid: 29 February 2008

zbMATH: 1260.60156
MathSciNet: MR2393991
Digital Object Identifier: 10.1214/009117907000000277

Subjects:
Primary: 60G15, 60G17, 60J55

Rights: Copyright © 2008 Institute of Mathematical Statistics

JOURNAL ARTICLE
29 PAGES


SHARE
Vol.36 • No. 2 • March 2008
Back to Top