Author | Freidlin, M. I. author |
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Title | Random Perturbations of Dynamical Systems [electronic resource] / by M. I. Freidlin, A. D. Wentzell |
Imprint | New York, NY : Springer US, 1984 |
Connect to | http://dx.doi.org/10.1007/978-1-4684-0176-9 |
Descript | VIII, 328 p. online resource |
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. Laplaceโs 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