Invariant Measure for the Markov Process Corresponding to a PDE System

In this paper, we consider the Markov process (X.(t), Z.(t)) corresponding to a weakly coupled elliptic PDE system with a small parameter . > 0. We first prove that (X.(t), Z.(t)) has the Feller continuity by the coupling method, and then prove that (X.(t), Z. t)) has an invariant measure µ.(·) by the Foster.Lyapunov inequality. Finally, we establish a large deviations principle for µ.(·) as the small parameter . tends to zero.