Random Perturbations of Dynamical Systems
Springer 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.
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absolutely continuous assertion assume asymptotics behavior boundary bounded Chebyshev's inequality choose coefficients compact compactum consider continuous functions converges to zero corresponding curve cycle defined denote density derivatives differential equations diffusion process distribution domain dynamical system eigenvalue equal estimate example exists exit family of processes family of random finite number follows formula Freidlin Gaussian process graph Hamiltonian implies inequality infimum initial point interval invariant measure large deviations Legendre transform Lemma Lipschitz condition lower semicontinuous Markov process Markov property mathematical expectation matrix neighborhood normalized action functional obtain operator problem prove random perturbations random process random variable respect right side small parameter smooth solution space stable equilibrium position stochastic differential equation sufficiently small Theorem 2.1 theory trajectories uniformly unique values vector vertex Wentzell Wiener process x e Rr
Page vi - PREFACE TO THE SECOND EDITION The first edition of this book was...
Page 420 - A mixed boundary value problem for elliptic differential equations of second order with a small parameter.
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