Small Random Perturbations on a Finite Time Interval

Summary.

Chapter 2 treats the results of the law-of-large-numbers type and of the central-limit-theorem type for diffusion processes with small diffusion and similar families of stochastic processes with decreasing randomness. These results allow considering the perturbed dynamics on finite time intervals and some asymptotic problems for elliptic and parabolic partial differential equations.

The rest of the book is dealing mainly with long-time influence of small perturbations. The results of this type can be, roughly, divided in two parts: problems related to the large deviation theory (Chaps. 3–6, a part of Chap. 7, and Chap. 10), and problems where some version of the averaging principle can be applied (Chaps. 7–9). A special attention is given to applications of these results to second order partial differential equations of elliptic type with a small parameter in higher derivatives.