Martingale problems for large deviations of Markov processes

The martingale problems provide a powerful tool for characterizing Markov processes, especially in addressing convergence issues. For each n, let metric space E-valued process Xn be a solution of the martingale problems (i.e.is a martingale), the convergence of in some sense usually implies the weak convergence of Xn=>X, where X is some process characterized by . Our goal here is to establish similar results for another type of limit theorem - large deviations: defining , thenis a martingale. We prove that the convergence of nonlinear operators implies the large deviation principle for the Xns, where the rate function is characterized by a nonlinear transformation of . Furthermore, a 'running cost' interpretation from control theory can be given to this function. The main assumption is a regularity condition on in the sense that for each , bounded viscosity solution ofis unique. This paper considers processes in CE[0,T].