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Lévy measures for a class of Markov semigroups in one dimension

Author: Ken iti Sato
Journal: Trans. Amer. Math. Soc. 148 (1970), 211-231
MSC: Primary 60.60
MathSciNet review: 0266309
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Abstract: Given a Markov semigroup of linear operators in the space of realvalued continuous functions on the line vanishing at infinity, we prove that the Lévy measure exists if the domain of the infinitesimal generator contains $ \mathcal{D}_K(D_m D_s^+)$, the domain of William Feller's generalized second order differential operator restricted to functions with compact supports. We give estimate of singularity of the Lévy measure and representation of the infinitesimal generator. Conversely, given Lévy measure or the form of infinitesimal generator, existence of the corresponding Markov semigroup is shown under some conditions. The case of circles is also discussed.

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Keywords: Lévy measure, Markov semigroup, Markov process, infinitesimal generator, generalized second order differential operator, dispersive operator, Cauchy process, perturbation of semigroup generators
Article copyright: © Copyright 1970 American Mathematical Society

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