Markov Perturbations on Large Time Intervals

Perturbations of dynamical systems with multiple stable attractors are considered in Chap. 6. Random perturbations, in a long enough time, lead to transitions between the basins of attractors. Applying the results and constructions of Chaps. 3–5, we describe the most probable sequence of transitions and transition paths, calculate the logarithmic asymptotics of the transition times and the limiting behavior of the stationary distribution as the intensity of the noise tends to zero. A hierarchy of cycles and an important notion of metastable state (sublimit distribution) are introduced in Chap. 6. A special technique associated with finite graphs is developed in this chapter. This technique allows to express many results in an explicit form. The hierarchy of cycles and metastability are closely related to asymptotic problems for the eigenvalues and eigenfunctions of the generator of the perturbed system. It is explained in Chap. 6 that such an effect as stochastic resonance is a manifestation of metastability and should be considered within the framework of the large-deviation theory.