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The Lévy–Khintchine type operators with variable Lipschitz continuous coefficients generate linear or nonlinear Markov processes and semigroups
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  • Published: 21 April 2010

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

  • Vassili N. Kolokoltsov1 

Probability Theory and Related Fields volume 151, pages 95–123 (2011)Cite this article

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Abstract

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.

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Authors and Affiliations

  1. Department of Statistics, University of Warwick, Coventry, CV4 7AL, UK

    Vassili N. Kolokoltsov

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  1. Vassili N. Kolokoltsov
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Correspondence to Vassili N. Kolokoltsov.

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Kolokoltsov, V.N. The Lévy–Khintchine type operators with variable Lipschitz continuous coefficients generate linear or nonlinear Markov processes and semigroups. Probab. Theory Relat. Fields 151, 95–123 (2011). https://doi.org/10.1007/s00440-010-0293-8

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  • Received: 09 October 2007

  • Revised: 31 March 2010

  • Published: 21 April 2010

  • Issue Date: October 2011

  • DOI: https://doi.org/10.1007/s00440-010-0293-8

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Keywords

  • 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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