The L\'evy-Khintchine type operators with variable Lipschitz continuous coefficients generate linear or nonlinear Markov processes and semigroups

TL;DR

This paper extends Ito's construction to nonlinear Markov processes driven by Le9vy noise with variable coefficients, establishing conditions under which these generate linear or nonlinear Markov semigroups with Lipschitz continuous parameters.

Contribution

It introduces a framework for constructing linear and nonlinear Markov semigroups from Le9vy-Khintchine type operators with Lipschitz continuous variable coefficients, generalizing known diffusion results.

Findings

Variable coefficient operators generate Markov semigroups.

Lipschitz continuity ensures well-defined processes.

Extension from diffusions to general Markov processes.

Abstract

Ito's construction of Markovian solutions to stochastic equations driven by a L\'evy 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\'evy-Khintchine type) with variable coefficients (diffusion, drift and L\'evy 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 nontrivial but natural extension to general Markov processes of a long known fact for ordinary diffusions.