Probability Theory and Related Fields

, Volume 151, Issue 1–2, pp 95–123 | Cite as

The Lévy–Khintchine type operators with variable Lipschitz continuous coefficients generate linear or nonlinear Markov processes and semigroups

  • Vassili N. KolokoltsovEmail author


Ito’s construction of Markovian solutions to stochastic equations driven by a Lévy noise is extended to nonlinear distribution dependent integrands aiming at the effective construction of linear and nonlinear Markov semigroups and the corresponding processes with a given pseudo-differential generator. It is shown that a conditionally positive integro-differential operator (of the Lévy–Khintchine type) with variable coefficients (diffusion, drift and Lévy measure) depending Lipschitz continuously on its parameters (position and/or its distribution) generates a linear or nonlinear Markov semigroup, where the measures are metricized by the Wasserstein–Kantorovich metrics. This is a non-trivial but natural extension to general Markov processes of a long known fact for ordinary diffusions.


Stochastic equations driven by Lévy noise Nonlinear integrators Wasserstein–Kantorovich metric Pseudo-differential generators Linear and nonlinear Markov semigroups 

Mathematics Subject Classification (2000)

60J25 60H05 


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© Springer-Verlag 2010

Authors and Affiliations

  1. 1.Department of StatisticsUniversity of WarwickCoventryUK

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