$L^p$ moduli of continuity of Gaussian processes and local times of symmetric Lévy processes

Abstract

Let $X=\{X(t),t\in R_{+}\}$ be a real-valued symmetric Lévy process with continuous local times $\{L_t^x,(t,x)\in R_{+}\times R\}$ and characteristic function $E{e}^{i\lambda X(t)}=e^{-t\psi (\lambda )}$. Let \[σ^2_0(x-y)=\frac{4}π\int^{\infty}_0\frac{\sin^2({λ(x- y)}/{2})}{ψ(λ)} dλ.\] If ${\sigma }_0^2(h)$ is concave, and satisfies some additional very weak regularity conditions, then for any $p\ge 1$, and all $t\in R_{+}$, \[\lim_{h\downar row0}\int_a^b\biggl|{\frac{L^{x+h}_t-L^x_t}{σ_0(h)}}\biggr|^p dx =2^{p/2}E|η|^p\int_a^b|L^x_t|^{p/2} dx\] for all $a,b$ in the extended real line almost surely, and also in $L^m$, $m\ge 1$. (Here $\eta$ 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, $\{G(x),x\in R^1\}$, for which $E(G(x)-G(y){)}^2={\sigma }_0^2(x-y)$; \[\lim_{h\to0}\int_a^b\biggl|\frac{G (x+h)-G(x)}{σ_0(h)}\biggr|^p dx=E|η|^p(b-a)\] for all $a,b\in R^1$, almost surely.

Published in at http://dx.doi.org/10.1214/009117907000000277 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

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