We continue the study of the asymptotic behavior of Markov processes $(X^\varepsilon(t), \nu^\varepsilon(t))$ corresponding to systems of elliptic PDE with a small parameter $\varepsilon > 0$. In the present paper we consider the case where the process $(X^\varepsilon(t), \nu^\varepsilon(t))$ can leave a given domain $D$ only due to large deviations from the degenerate process $(X^0(t), \nu^0(t))$. In this way we study the limit behavior of solutions of corresponding Dirichlet problems.
"Large Deviations for Markov Processes Corresponding to PDE Systems." Ann. Probab. 21 (2) 1015 - 1044, April, 1993. https://doi.org/10.1214/aop/1176989280