Lévy measures for a class of Markov semigroups in one dimension

Sato, Ken iti

doi:10.1090/S0002-9947-1970-0266309-5

Lévy measures for a class of Markov semigroups in one dimension
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by Ken iti Sato

Trans. Amer. Math. Soc. 148 (1970), 211-231

DOI: https://doi.org/10.1090/S0002-9947-1970-0266309-5

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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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