The Lévy-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 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 nontrivial but natural extension to general Markov processes of a long known fact for ordinary diffusions.

References in corpus (2)

• Generalized Continuous-Time Random Walks (CTRW), Subordination by Hitting Times and Fractional Dynamics
• Integration by parts formula for locally smooth laws and applications to sensitivity computations

Cited by in corpus (4)

• Mean Field Games and Nonlinear Markov Processes
• Coupling property and gradient estimates of Lévy processes via the symbol
• Stochastic monotonicity and duality for one-dimensional Markov processes
• The law of large numbers for quantum stochastic filtering and control of many particle systems