Limit Theorems for Cylindrical Martingale Problems Associated with Lévy Generators

Abstract

We prove limit theorems for cylindrical martingale problems associated to Lévy 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 Lévy 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.

The Martingale Problem Method Revisited

“…Note that these random variables are not necessarily continuous in the Skorokhod topology when q has point masses, as projections to fixed times are in general not continuous in the Skorokhod topology. Limit theorems for certain types of SPDEs and VSDEs were proved in [1,7,29]. However, for processes with fixed times of discontinuity we are not aware of any systematic study.…”