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AuthorFreidlin, M. I. author
TitleRandom Perturbations of Dynamical Systems [electronic resource] / by M. I. Freidlin, A. D. Wentzell
ImprintNew York, NY : Springer US, 1984
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Descript VIII, 328 p. online resource


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


1 Random Perturbations -- ยง1. Probabilities and Random Variables -- ยง2. Random Processes. General Properties -- ยง3. Wiener Process. Stochastic Integral -- ยง4. Markov Processes and Semigroups -- ยง5. Diffusion Processes and Differential Equations -- 2 Small Random Perturbations on a Finite Time Interval -- ยง1. Zeroth Approximation -- ยง2. Expansion in Powers of a Small Parameter -- ยง3. Elliptic and Parabolic Differential Equations with a Small Parameter at the Derivatives of Highest Order -- 3 Action Functional -- ยง1. Laplaceโ{128}{153}s Method in a Function Space -- ยง2. Exponential Estimates -- ยง3. Action Functional. General Properties -- ยง4. Action Functional for Gaussian Random Processes and Fields -- 4 Gaussian Perturbations of Dynamical Systems. Neighborhood of an Equilibrium Point -- ยง1. Action Functional -- ยง2. The Problem of Exit from a Domain -- ยง3. Properties of the Quasipotential. Examples -- ยง4. Asymptotics of the Mean Exit Time and Invariant Measure for the Neighborhood of an Equilibrium Position -- ยง5. Gaussian Perturbations of General Form -- 5 Perturbations Leading to Markov Processes -- ยง1. Legendre Transformation -- ยง2. Locally Infinitely Divisible Processes -- ยง3. Special Cases. Generalizations -- ยง4. Consequences. Generalization of Results of Chapter 4 -- 6 Markov Perturbations on Large Time Intervals -- ยง1. Auxiliary Results. Equivalence Relation -- ยง2. Markov Chains Connected with the Process $$(X_t̂\varepsilon, \,{\text{P}}_x̂\varepsilon )$$ -- ยง3. Lemmas on Markov Chains -- ยง4. The Problem of the Invariant Measure -- ยง5. The Problem of Exit from a Domain -- ยง6. Decomposition into Cycles. Sublimit Distributions -- ยง7. Eigenvalue Problems -- 7 The Averaging Principle. Fluctuations in Dynamical Systems with Averaging -- ยง1. The Averaging Principle in the Theory of Ordinary Differential Equations -- ยง2. The Averaging Principle when the Fast Motion is a Random Process -- ยง3. Normal Deviations from an Averaged System -- ยง4. Large Deviations from an Averaged System -- ยง5. Large Deviations Continued -- ยง6. The Behavior of the System on Large Time Intervals -- ยง7. Not Very Large Deviations -- ยง8. Examples -- ยง9. The Averaging Principle for Stochastic Differential Equations -- 8 Stability Under Random Perturbations -- ยง1. Formulation of the Problem -- ยง2. The Problem of Optimal Stabilization -- ยง3. Examples -- 9 Sharpenings and Generalizations -- ยง1. Local Theorems and Sharp Asymptotics -- ยง2. Large Deviations for Random Measures -- ยง3. Processes with Small Diffusion with Reflection at the Boundary -- References

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