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 An martingale problems (i.e.f(Xn(t))−f(Xn(0))−∫0tAnf(Xn(s)) dsis a martingale), the convergence of Anf→Af in some sense usually implies the weak convergence of Xn⇒X, where X is some process characterized by A. Our goal here is to establish similar results for another type of limit theorem – large deviations: defining Hnf=e−fAnef, thenexpf(Xn(t))−f(Xn(0))−∫0tHnf(Xn(s)) dsis a martingale. We prove that the convergence of nonlinear operators 1nHn(nf)→Hf implies the large deviation principle for the Xns, where the rate function is characterized by a nonlinear transformation L of H. Furthermore, a ‘running cost’ interpretation from control theory can be given to this function. The main assumption is a regularity condition on H in the sense that for each f0∈D(H), bounded viscosity solution of−∂tu(t,x)+(Hu(t,·))(x)=0; u(0,x)=f0(x)is unique. This paper considers processes in CE[0,T].