Random Perturbations of Dynamical Systems

Asymptotical problems have always played an important role in probability theory. In classical probability theory dealing mainly with sequences of independent variables, theorems of the type of laws of large numbers, theorems of the type of the central limit theorem, and theorems on large deviations constitute a major part of all investigations. In recent years, when random processes have become the main subject of study, asymptotic investigations have continued to playa major role. We can say that in the theory of random processes such investigations play an even greater role than in classical probability theory, because it is apparently impossible to obtain simple exact formulas in problems connected with large classes of random processes. Asymptotical investigations in the theory of random processes include results of the types of both the laws of large numbers and the central limit theorem and, in the past decade, theorems on large deviations. Of course, all these problems have acquired new aspects and new interpretations in the theory of random processes.

Table of Contents:
1 Random Perturbations.- §1. Probabilities and Random Variables.- §2. Random Processes. General Properties.- §3. Wiener Process. Stochastic Integral.- §4. Markov Processes and Semigroups.- §5. Diffusion Processes and Differential Equations.- 2 Small Random Perturbations on a Finite Time Interval.- §1. Zeroth Approximation.- §2. Expansion in Powers of a Small Parameter.- §3. Elliptic and Parabolic Differential Equations with a Small Parameter at the Derivatives of Highest Order.- 3 Action Functional.- §1. Laplace’s Method in a Function Space.- §2. Exponential Estimates.- §3. Action Functional. General Properties.- §4. Action Functional for Gaussian Random Processes and Fields.- 4 Gaussian Perturbations of Dynamical Systems. Neighborhood of an Equilibrium Point.- §1. Action Functional.- §2. The Problem of Exit from a Domain.- §3. Properties of the Quasipotential. Examples.- §4. Asymptotics of the Mean Exit Time and Invariant Measure for the Neighborhood of an Equilibrium Position.- §5. Gaussian Perturbations of General Form.- 5 Perturbations Leading to Markov Processes.- §1. Legendre Transformation.- §2. Locally Infinitely Divisible Processes.- §3. Special Cases. Generalizations.- §4. Consequences. Generalization of Results of Chapter 4.- 6 Markov Perturbations on Large Time Intervals.- §1. Auxiliary Results. Equivalence Relation.- §2. Markov Chains Connected with the Process $$(X_t^\varepsilon, \,{\text{P}}_x^\varepsilon )$$.- §3. Lemmas on Markov Chains.- §4. The Problem of the Invariant Measure.- §5. The Problem of Exit from a Domain.- §6. Decomposition into Cycles. Sublimit Distributions.- §7. Eigenvalue Problems.- 7 The Averaging Principle. Fluctuations in Dynamical Systems with Averaging.- §1. The Averaging Principle in the Theory of Ordinary Differential Equations.- §2. The Averaging Principle when the Fast Motion is a Random Process.- §3. Normal Deviations from an Averaged System.- §4. Large Deviations from an Averaged System.- §5. Large Deviations Continued.- §6. The Behavior of the System on Large Time Intervals.- §7. Not Very Large Deviations.- §8. Examples.- §9. The Averaging Principle for Stochastic Differential Equations.- 8 Stability Under Random Perturbations.- §1. Formulation of the Problem.- §2. The Problem of Optimal Stabilization.- §3. Examples.- 9 Sharpenings and Generalizations.- §1. Local Theorems and Sharp Asymptotics.- §2. Large Deviations for Random Measures.- §3. Processes with Small Diffusion with Reflection at the Boundary.- References.