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The Lévy-Khintchine type operators with variable Lipschitz continuous coefficients generate linear or nonlinear Markov processes and semigroups. / Kolokoltsov, Vassili N.

In: Probability Theory and Related Fields, Vol. 151, No. 1, 10.2011, p. 95-123.

Research output: Contribution to journal › Article › peer-review

Harvard

Kolokoltsov, VN 2011, 'The Lévy-Khintchine type operators with variable Lipschitz continuous coefficients generate linear or nonlinear Markov processes and semigroups', Probability Theory and Related Fields, vol. 151, no. 1, pp. 95-123. https://doi.org/10.1007/s00440-010-0293-8

APA

Kolokoltsov, V. N. (2011). The Lévy-Khintchine type operators with variable Lipschitz continuous coefficients generate linear or nonlinear Markov processes and semigroups. Probability Theory and Related Fields, 151(1), 95-123. https://doi.org/10.1007/s00440-010-0293-8

Vancouver

Kolokoltsov VN. The Lévy-Khintchine type operators with variable Lipschitz continuous coefficients generate linear or nonlinear Markov processes and semigroups. Probability Theory and Related Fields. 2011 Oct;151(1):95-123. https://doi.org/10.1007/s00440-010-0293-8

Author

Kolokoltsov, Vassili N. / The Lévy-Khintchine type operators with variable Lipschitz continuous coefficients generate linear or nonlinear Markov processes and semigroups. In: Probability Theory and Related Fields. 2011 ; Vol. 151, No. 1. pp. 95-123.

BibTeX

@article{2d4c33d546884262abc27a6adf858db4,
title = "The L{\'e}vy-Khintchine type operators with variable Lipschitz continuous coefficients generate linear or nonlinear Markov processes and semigroups",
abstract = "Ito's construction of Markovian solutions to stochastic equations driven by a L{\'e}vy 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 L{\'e}vy-Khintchine type) with variable coefficients (diffusion, drift and L{\'e}vy 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.",
keywords = "Linear and nonlinear Markov semigroups, Nonlinear integrators, Pseudo-differential generators, Stochastic equations driven by L{\'e}vy noise, Wasserstein-Kantorovich metric",
author = "Kolokoltsov, {Vassili N.}",
year = "2011",
month = oct,
doi = "10.1007/s00440-010-0293-8",
language = "English",
volume = "151",
pages = "95--123",
journal = "Probability Theory and Related Fields",
issn = "0178-8051",
publisher = "Springer Nature",
number = "1",

}

RIS

TY - JOUR

T1 - The Lévy-Khintchine type operators with variable Lipschitz continuous coefficients generate linear or nonlinear Markov processes and semigroups

AU - Kolokoltsov, Vassili N.

PY - 2011/10

Y1 - 2011/10

N2 - 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.

AB - 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.

KW - Linear and nonlinear Markov semigroups

KW - Nonlinear integrators

KW - Pseudo-differential generators

KW - Stochastic equations driven by Lévy noise

KW - Wasserstein-Kantorovich metric

UR - http://www.scopus.com/inward/record.url?scp=80052811790&partnerID=8YFLogxK

U2 - 10.1007/s00440-010-0293-8

DO - 10.1007/s00440-010-0293-8

M3 - Article

AN - SCOPUS:80052811790

VL - 151

SP - 95

EP - 123

JO - Probability Theory and Related Fields

JF - Probability Theory and Related Fields

SN - 0178-8051

IS - 1

ER -

ID: 86493481