Limit set of inhomogeneous Ornstein-Uhlenbeck processes, destabilization and annealing

We consider the Diffusion Process obtained by perturbing a dynamical system having a single equilibrium point x, by a fixed time-inhomogeneous Gaussian process whose intensity tends to 0 at infinity. We establish criteria for the exit time from a neighborhood of x to be a.s. finite by linking this fact with the structure of the limit set at infinity. We are also able to compute this limit set for inhomogeneous Ornstein-Uhlenbeck processes associated to linear systems. An application is given to simulated annealing.