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\(L^p\) moduli of continuity of Gaussian processes and local times of symmetric Lévy processes. (English) Zbl 1260.60156

Summary: Let \(X= \{(t),\, t\in\mathbb{R}_+\}\) be a real-valued symmetric Lévy process with continuous local times \(\{L^X_t, (t,x)\in\mathbb{R}_+\times \mathbb{R}\}\) and characteristic function \(\text{E}e^{i\lambda X(t)}= e^{-t\psi(\lambda)}\). Let \[ \sigma^2_0(x- y)={4\over\pi} \int^\infty_0 {\sin^2(\lambda(x- y)/2)\over\psi(\lambda)}\,d\lambda. \] If \(\sigma^2_0(h)\) is concave, and satisfies some additional very weak regularity conditions, then for any \(p\geq 1\), and all \(t\in\mathbb{R}_+\), \[ \lim_{h\downarrow 0} \int^b_a\,\Biggl|{L^{x+h}_t- L^x_t\over\sigma_0(h)}\Biggr|^p\,dx= 2^{p/2} \text{E}|\eta|^p \int^b_a |L^x_t|^{p/2}\,dx \] for all \(a\), \(b\) in the extended real line almost surely, and also in \(L^m\), \(m\geq 1\). (Here \(\eta\) is a normal random variable with mean zero and variance one.)
The result is obtained via the Eisenbaum isomorphism theorem and depends on the related result for Gaussian processes with stationary increments, \(\{G(x), x\in\mathbb{R}^1\}\), for which \(\text{E}(G(x)- G(y))^2= \sigma^2_0(x- y)\); \[ \lim_{h\to 0} \int^b_a \Biggl|{G(x+ h)- G(x)\over \sigma_0(h)}\Biggr|^p\,dx= \text{E}|\eta|^p(b- a) \] for all \(a,b\in\mathbb{R}^1\), almost surely.

MSC:

60J55 Local time and additive functionals
60G15 Gaussian processes
60G17 Sample path properties
60G51 Processes with independent increments; Lévy processes

References:

[1] Kahane, J.-P. (1985). Some Random Series of Functions , 2nd ed. Cambridge Univ. Press. · Zbl 0571.60002
[2] Marcus, M. B. and Rosen, J. (2006). Markov Processes, Gaussian Processes, and Local Times . Cambridge Univ. Press. · Zbl 1129.60002
[3] Wschebor, M. (1992). Sur les accroissments du processus de Wiener. C. R. Acad. Sci. Paris Ser. I Math. 315 1293-1296. · Zbl 0770.60075
[4] Yor, M. (1983). Derivatives of self-intersection local times. Séminaire de Probabilités XVII. Lecture Notes in Math. 986 89-106. Springer, Berlin.
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