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

Michael B. Marcus, Jay Rosen

Published 2006-07-26, updated 2008-05-12Version 2

Let $X=\{X(t),t\in R_+\}$ be a real-valued symmetric L\'{e}vy process with continuous local times $\{L^x_t,(t,x)\in R_+\times R\}$ and characteristic function $Ee^{i\lambda X(t)}=e^{-t\psi(\lambda)}$. Let \[\sigma^2_0(x-y)=\frac{4}{\pi}\int^{\infty}_0\frac{\sin^2({\lambda(x- y)}/{2})}{{\psi(\lambda)}} d\lambda.\] If $\sigma^2_0(h)$ is concave, and satisfies some additional very weak regularity conditions, then for any $p\ge1$, and all $t\in R_+$, \[\lim_{h\downar row0}\int_a^b\biggl|{\frac{L^{x+h}_t-L^x_t}{\sigma_0(h)}}\biggr|^p dx =2^{p/2}E|\eta|^p\int_a^b|L^x_t|^{p/2} dx\] for all $a,b$ in the extended real line almost surely, and also in $L^m$, $m\ge1$. (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 $E(G(x)-G(y))^2=\sigma_0^2(x-y)$; \[\lim_{h\to0}\int_a^b\biggl|\frac{G (x+h)-G(x)}{\sigma_0(h)}\biggr|^p dx=E|\eta|^p(b-a)\] for all $a,b\in R^1$, almost surely.