Gaussian Perturbations of Dynamical Systems. Neighborhood of an Equilibrium Point

Perturbations of dynamical systems with a single asymptotically stable equilibrium (or limit cycle) by the white noise multiplied by a small factor are studied in Chap. 4. We calculate the action functional and introduce an important notion of quasi-potential. Properties of the quasi-potential are studied. One problem considered in this chapter (and in more general case, in Chap. 6) is the exit problem. The logarithmic asymptotics of the exit time and the most probable exit path (first exit from a domain containing the equilibrium point occurring near this path with probability close to 1 as the diffusion coefficient tends to zero) are calculated. The logarithmic asymptotics of the stationary distribution of the perturbed process is calculated in this chapter as well. The main tool for obtaining the results in Chap. 4, and also in Chap. 6, is considering cycles between successive reaching of a small neighborhood of the equilibrium point and leaving of a slightly larger neighborhood, then reaching the small neighborhood again, etc. Applications to the Dirichlet problem for elliptic equations with a small parameter are considered.