The exit problem for small random perturbations of dynamical systems with a hyperbolic fixed point

We consider the Markov diffusion process ξ^∈(t), transforming when ɛ=0 into the solution of an ordinary differential equation with a turning point ℴ of the hyperbolic type. The asymptotic behevior as ɛ→0 of the exit time, of its expectation of the probability distribution of exit points for the process ξ^∈(t) is studied. These indicate also the asymptotic behavior of solutions of the corresponding singularly perturbed elliptic boundary value problems.