Random Perturbations of Dynamical Systems, Volume 260Springer Science & Business Media, 1998 - 430 pages This volume is concerned with various kinds of limit theorems for stochastic processes defined as a result of random perturbations of dynamical systems, especially with the long-time behavior of the perturbed system. In particular, exit problems, metastable states, optimal stabilization, and asymptotics of stationary distributions are also carefully considered. The authors' main tools are the large deviation theory the centred limit theorem for stochastic processes, and the averaging principle - all presented in great detail. The results allow for explicit calculations of the asymptotics of many interesting characteristics of the perturbed system. Most of the results are closely connected with PDEs, and the authors' approach presents a powerful method for studying the asymptotic behavior of the solutions of initial-boundary value problems for corresponding PDEs. |
Table des matières
Introduction | 1 |
Random Perturbations 254 | 15 |
3 Wiener Process Stochastic Integral | 24 |
5 Diffusion Processes and Differential Equations | 34 |
CHAPTER 2 | 44 |
2 Expansion in Powers of a Small Parameter | 51 |
3 Elliptic and Parabolic Differential Equations with a Small Parameter at | 59 |
Action Functional 70 | 70 |
CHAPTER 7 | 212 |
2 The Averaging Principle when the Fast Motion is a Random Process | 216 |
3 Normal Deviations from an Averaged System | 219 |
4 Large Deviations from an Averaged System | 233 |
5 Large Deviations Continued | 241 |
6 The Behavior of the System on Large Time Intervals | 249 |
7 Not Very Large Deviations | 253 |
8 Examples | 257 |
3 Action Functional General Properties | 79 |
4 Action Functional for Gaussian Random Processes and Fields | 92 |
CHAPTER 4 | 103 |
3 Properties of the Quasipotential Examples | 118 |
5 Gaussian Perturbations of General Form | 132 |
CHAPTER 5 | 133 |
Perturbations Leading to Markov Processes | 136 |
2 Locally Infinitely Divisible Processes | 143 |
3 Special Cases Generalizations | 153 |
4 Consequences Generalization of Results of Chapter 4 | 157 |
CHAPTER 6 | 161 |
2 Markov Chains Connected with the Process X P | 168 |
3 Lemmas on Markov Chains | 176 |
4 The Problem of the Invariant Measure | 185 |
5 The Problem of Exit from a Domain | 192 |
6 Decomposition into Cycles Sublimit Distributions | 198 |
7 Eigenvalue Problems | 203 |
9 The Averaging Principle for Stochastic Differential Equations | 268 |
CHAPTER 8 | 283 |
2 Main Results | 295 |
3 Proof of Theorem 2 2 | 301 |
4 Proofs of Lemmas 3 1 to 3 4 | 312 |
5 Proof of Lemma 3 5 | 328 |
6 Proof of Lemma 3 6 | 338 |
7 Remarks and Generalizations | 344 |
CHAPTER 9 | 361 |
2 The Problem of Optimal Stabilization | 367 |
3 Examples | 373 |
CHAPTER 10 | 377 |
3 Processes with Small Diffusion with Reflection at the Boundary | 392 |
5 Random Perturbations of InfiniteDimensional Systems | 408 |
| 429 | |
Autres éditions - Tout afficher
Random Perturbations of Dynamical Systems Mark Freidlin,Alexander D. Wentzell Aucun aperçu disponible - 2013 |
Expressions et termes fréquents
absolutely continuous assume asymptotics behavior boundary bounded coefficients compact compactum consider constant continuous functions Cor(R corresponding curve cycle defined denote differential equations diffusion process distribution domain dynamical system eigenvalue English translation equal estimate example exists exit family of processes follows formula Freidlin Gaussian process graph H(Ok Hamiltonian implies inequality infimum interval invariant measure K₁ large deviations Legendre transform Lemma lower semicontinuous Markov process Markov property mathematical expectation matrix neighborhood o-algebra obtain operator probability problem proof prove random perturbations random process random variable respect satisfied small ɛ small parameter smooth solution space stochastic differential equation sufficiently small T₁ T₂ Theorem 2.1 Theory trajectories uniformly unique values vector VH(x Wentzell Wiener process x₁
Références à ce livre
Controlled Markov Processes and Viscosity Solutions Wendell H. Fleming,Halil Mete Soner Aucun aperçu disponible - 2006 |
Stochastic Approximation and Recursive Algorithms and Applications Harold Kushner,G. George Yin Aucun aperçu disponible - 2003 |