The Lévy–Khintchine type operators with variable Lipschitz continuous coefficients generate linear or nonlinear Markov processes and semigroups
Kolokoltsov, V. N. (Vasiliĭ Nikitich)
- Publisher: Springer
Ito's construction of Markovian solutions to stochastic equations driven by a\ud Lévy noise is extended to nonlinear distribution dependent integrands aiming at\ud 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\ud variable coeffcients (diffusion, drift and Lévy measure) depending Lipschitz continuously on its parameters (position and/or its distribution) generates a linear or\ud nonlinear Markov semigroup, where the measures are metricized by the Wasserstein-Kantorovich metrics. This is a nontrivial but natural extension to general Markov\ud processes of a long known fact for ordinary diffusions.