$L^p$ 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 $\{L^x_t,(t,x)\in R_+\times R\}$ and characteristic function $Ee^{iλX(t)}=e^{-tψ(λ)}$. Let \[σ^2_0(x-y)=\frac{4}π\int^{\infty}_0\frac{\sin^2({λ(x- y)}/{2})}{ψ(λ)} dλ.\] If $σ^2_0(h)$ is concave, and satisfies some additional very weak regularity conditions, then for any $p\ge1$, and all $t\in R_+$, \[\lim_{h\downar row0}\int_a^b\biggl|{\frac{L^{x+h}_t-L^x_t}{σ_0(h)}}\biggr|^p dx =2^{p/2}E|η|^p\int_a^b|L^x_t|^{p/2} dx\] for all $a,b$ in the extended real line almost surely, and also in $L^m$, $m\ge1$. (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, $\{G(x),x\in R^1\}$, for which $E(G(x)-G(y))^2=σ_0^2(x-y)$; \[\lim_{h\to0}\int_a^b\biggl|\frac{G (x+h)-G(x)}{σ_0(h)}\biggr|^p dx=E|η|^p(b-a)\] for all $a,b\in R^1$, almost surely.