Limit Theorems for Cylindrical Martingale Problems associated with L\'evy Generators

TL;DR

This paper establishes limit theorems for cylindrical martingale problems linked to Lévy generators and explores conditions for the Feller property, with applications to infinite-dimensional SDEs and SPDEs driven by Lévy and Wiener noise.

Contribution

It provides new limit theorems for cylindrical martingale problems and characterizes the Feller property for well-posed problems with continuous coefficients.

Findings

Derived conditions for the existence of weak solutions to infinite-dimensional SDEs with Lévy noise.

Established criteria for the Feller property of solutions to SPDEs driven by Wiener noise.

Provided continuity and growth conditions for limit theorems in infinite-dimensional stochastic equations.

Abstract

We prove limit theorems for cylindrical martingale problems associated to L\'evy generators. Furthermore, we give sufficient and necessary conditions for the Feller property of well-posed problems with continuous coefficients. We discuss two applications. First, we derive continuity and linear growth conditions for the existence of weak solutions to infinite-dimensional stochastic differential equations driven by L\'evy noise. Second, we derive continuity, local boundedness and linear growth conditions for limit theorems and the Feller property of weak solutions to stochastic partial differential equations driven by Wiener noise.