Let X={X(t), t∈R+} be a real-valued symmetric Lévy process with continuous local times {Ltx, (t, x)∈R+×R} and characteristic function EeiλX(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 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), 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.