Lp moduli of continuity of Gaussian processes and local times of symmetric Lévy processes

Abstract

Let X={X(t), t∈R₊} be a real-valued symmetric Lévy process with continuous local times {L_t^x, (t, x)∈R₊×R} and characteristic function Ee^iλX(t)=e^−tψ(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 t∈R₊, $$\lim_{h\downarrow0}\int_{a}^{b}\biggl|{\frac{L^{x+h}_{t}-L^{x}_{t}}{\sigma_{0}(h)}}\biggr|^{p}\,dx=2^{p/2}E|\eta|^{p}\int_{a}^{b}|L^{x}_{t}|^{p/2}\,dx$$ for all a, b in the extended real line almost surely, and also in L^m, 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), x∈R¹}, for which E(G(x)−G(y))²=σ₀²(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∈R¹, almost surely.