Marcus, Michael B.; Rosen, Jay \(L^p\) moduli of continuity of Gaussian processes and local times of symmetric Lévy processes. (English) Zbl 1260.60156 Ann. Probab. 36, No. 2, 594-622 (2008). 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. Cited in 1 ReviewCited in 9 Documents MSC: 60J55 Local time and additive functionals 60G15 Gaussian processes 60G17 Sample path properties 60G51 Processes with independent increments; Lévy processes Keywords:Gaussian processes; local times; Lévy processes × Cite Format Result Cite Review PDF Full Text: DOI arXiv 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. This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.