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

Let X = {X(t), t ∈ R+} be a real-valued symmetric Lévy process with continuous local times {L x t , (t, x) ∈ R+ × R} and characteristic function Ee iλX(t) = e −tψ(λ). Let σ 2 0 (x − y) = 4 π ∞ 0 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 ≥ 1, and all t ∈ R+, lim h↓0 b a L x+h t − L x t σ0(h) p dx = 2 p/2 E|η| p b a |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 ∈ R 1 }, for which E(G(x) − G(y)) 2 = σ 2 0 (x − y); lim h→0 b a G(x + h) − G(x) σ0(h) p dx = E|η| p (b − a) for all a, b ∈ R 1 , almost surely. 1. Introduction. We obtain L p moduli of continuity for a very wide class of continuous Gaussian processes and local times of symmetric Lévy processes. To introduce them, we first state our results for the local times of the Brownian motion and see how they compare with related results.