Asymptotic behaviors on the small parameter exit problems and the singularly perturbation problems

We consider the small parameter exit problems for diffusion processes and the associated singular perturbed Dirichlet problems. We investigate the asymptotic relations between the mean exit time and the principal eigenvalue. Two problems are considered under the gradient condition for the corresponding dynamical system. One is under the uniqueness of deepest valley, where we show that the product of the mean exit time and the principal eigenvalue converges to one exponentially fast. The other is related to the sharp asymptotics of the mean exit times, the eigenvalues and eigenfunctions, where we characterize the scaling limits of them by the Markov chain which appears metastable behavior of the corresponding diffusion process. To do this, we extend the methods used in our previous papers [10] and [11]