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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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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DOI: https://doi.org/10.1007/s00440-010-0293-8
Keywords
- Stochastic equations driven by Lévy noise
- Nonlinear integrators
- Wasserstein–Kantorovich metric
- Pseudo-differential generators
- Linear and nonlinear Markov semigroups