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