Skip to main content

2012 | Buch

Random Perturbations of Dynamical Systems

verfasst von: Mark I. Freidlin, Alexander D. Wentzell

Verlag: Springer Berlin Heidelberg

Buchreihe : Grundlehren der mathematischen Wissenschaften

insite
SUCHEN

Über dieses Buch

Many notions and results presented in the previous editions of this volume have since become quite popular in applications, and many of them have been “rediscovered” in applied papers.

In the present 3rd edition small changes were made to the chapters in which long-time behavior of the perturbed system is determined by large deviations. Most of these changes concern terminology. In particular, it is explained that the notion of sub-limiting distribution for a given initial point and a time scale is identical to the idea of metastability, that the stochastic resonance is a manifestation of metastability, and that the theory of this effect is a part of the large deviation theory. The reader will also find new comments on the notion of quasi-potential that the authors introduced more than forty years ago, and new references to recent papers in which the proofs of some conjectures included in previous editions have been obtained.

Apart from the above mentioned changes the main innovations in the 3rd edition concern the averaging principle. A new Section on deterministic perturbations of one-degree-of-freedom systems was added in Chapter 8. It is shown there that pure deterministic perturbations of an oscillator may lead to a stochastic, in a certain sense, long-time behavior of the system, if the corresponding Hamiltonian has saddle points. The usefulness of a joint consideration of classical theory of deterministic perturbations together with stochastic perturbations is illustrated in this section. Also a new Chapter 9 has been inserted in which deterministic and stochastic perturbations of systems with many degrees of freedom are considered. Because of the resonances, stochastic regularization in this case is even more important.

Inhaltsverzeichnis

Frontmatter
Chapter 1. Random Perturbations
Summary.
Chapter 1 serves as a preliminary one. Main notations are introduced here, and some results concerning Markov processes, stochastic differential equations, and their relation to partial differential equations are reminded.
Mark I. Freidlin, Alexander D. Wentzell
Chapter 2. Small Random Perturbations on a Finite Time Interval
Summary.
Chapter 2 treats the results of the law-of-large-numbers type and of the central-limit-theorem type for diffusion processes with small diffusion and similar families of stochastic processes with decreasing randomness. These results allow considering the perturbed dynamics on finite time intervals and some asymptotic problems for elliptic and parabolic partial differential equations.
The rest of the book is dealing mainly with long-time influence of small perturbations. The results of this type can be, roughly, divided in two parts: problems related to the large deviation theory (Chaps. 3–6, a part of Chap. 7, and Chap. 10), and problems where some version of the averaging principle can be applied (Chaps. 7–9). A special attention is given to applications of these results to second order partial differential equations of elliptic type with a small parameter in higher derivatives.
Mark I. Freidlin, Alexander D. Wentzell
Chapter 3. Action Functional
Summary.
In Chap. 3, the main notion of the large deviation theory, the action functional (rate function) for a family of probability measures (family of stochastic processes) is introduced and its properties are discussed in detail. The Wiener process with a small factor and its continuous transformations, in particular, Gaussian processes and diffusion processes with a constant diffusion matrix are considered as examples. General results concerning logarithmic asymptotics of the large deviation probabilities and the Laplace type asymptotics of expectations are presented in this chapter.
Mark I. Freidlin, Alexander D. Wentzell
Chapter 4. Gaussian Perturbations of Dynamical Systems. Neighborhood of an Equilibrium Point
Summary.
Perturbations of dynamical systems with a single asymptotically stable equilibrium (or limit cycle) by the white noise multiplied by a small factor are studied in Chap. 4. We calculate the action functional and introduce an important notion of quasi-potential. Properties of the quasi-potential are studied. One problem considered in this chapter (and in more general case, in Chap. 6) is the exit problem. The logarithmic asymptotics of the exit time and the most probable exit path (first exit from a domain containing the equilibrium point occurring near this path with probability close to 1 as the diffusion coefficient tends to zero) are calculated. The logarithmic asymptotics of the stationary distribution of the perturbed process is calculated in this chapter as well. The main tool for obtaining the results in Chap. 4, and also in Chap. 6, is considering cycles between successive reaching of a small neighborhood of the equilibrium point and leaving of a slightly larger neighborhood, then reaching the small neighborhood again, etc. Applications to the Dirichlet problem for elliptic equations with a small parameter are considered.
Mark I. Freidlin, Alexander D. Wentzell
Chapter 5. Perturbations Leading to Markov Processes
Summary.
In Chap. 5 the large-deviation results of Chap. 4 are generalized to include not only diffusion processes with small variable diffusion, but also to families of locally infinitely divisible Markov processes with small randomness. The action functional in these cases is expressed as the integral involving the test function and its first derivative, using the Legendre transformation of some characteristic of the Markov process in question. Some part of the proofs is based on a generalization of H. Cramér’s transformation the he applied to large deviations for sums of independent random variables.
Mark I. Freidlin, Alexander D. Wentzell
Chapter 6. Markov Perturbations on Large Time Intervals
Summary.
Perturbations of dynamical systems with multiple stable attractors are considered in Chap. 6. Random perturbations, in a long enough time, lead to transitions between the basins of attractors. Applying the results and constructions of Chaps. 3–5, we describe the most probable sequence of transitions and transition paths, calculate the logarithmic asymptotics of the transition times and the limiting behavior of the stationary distribution as the intensity of the noise tends to zero. A hierarchy of cycles and an important notion of metastable state (sublimit distribution) are introduced in Chap. 6. A special technique associated with finite graphs is developed in this chapter. This technique allows to express many results in an explicit form. The hierarchy of cycles and metastability are closely related to asymptotic problems for the eigenvalues and eigenfunctions of the generator of the perturbed system. It is explained in Chap. 6 that such an effect as stochastic resonance is a manifestation of metastability and should be considered within the framework of the large-deviation theory.
Mark I. Freidlin, Alexander D. Wentzell
Chapter 7. The Averaging Principle. Fluctuations in Dynamical Systems with Averaging
Summary.
In Chap. 7, rapidly oscillating stochastic perturbations of dynamical systems are considered. Results of the law-of-large-numbers type, those of central-limit theorem type, and large-deviation results are obtained. Those last results are applied to the asymptotics of the behavior of the perturbed dynamical system on growing time intervals.
Mark I. Freidlin, Alexander D. Wentzell
Chapter 8. Random Perturbations of Hamiltonian Systems
Summary.
In Chaps. 8–9 the situation is considered where the averaging is due to the mixing in the non-perturbed dynamical system, which is supposed to be a Hamiltonian one. In all averaging-principle problems we can single out in the perturbed system fast components and slow ones. Smallness of the perturbations must be considered as compared to the speed of the motion according to the dynamical system; so one possible model of small perturbations is considering the dynamical-system motion with velocities multiplied by a large parameter, the slow motion due to the perturbations being not too fast and not too slow. The fast motion is approximately the same as the sped-up non-perturbed motion—non-random, in fact; but the slow motion, considered in another time scale, remains random even when the randomness of the perturbation goes to zero. The question arises on what space should we consider this slow motion. If the non-perturbed system has no saddle-type points, we can, in the one-degree-of-freedom case, characterize the slow motion by the value of the Hamiltonian (in the multidimensional case with several first integrals, by the values of those first integrals). If there are saddle-type points, the slow motion should be considered on a graph in the one-degree-of-freedom case, and on an “open-book” type space in the multidimensional case. In the case of there being saddle-type points the randomness of the slow-motion component does not disappear as the diffusion coefficient of the perturbation goes to zero; and the limiting random process does not depend on the choice of the diffusion coefficients, that is on the truly random part of the perturbations—so it can be considered as the intrinsic property of the dynamical system with small non-random perturbations, due to high instability at the saddle-type points. The technique of proofs relies on martingale problems.
The one-degree-of-freedom case is considered in Chap. 8. In this case the slow motion is considered on a graph, and partial differential equations on it are reduced to several ordinary differential equations, whose solutions depend on a finite number of constants that is equal to the number of gluing conditions imposed on the solution. This takes care of the problem of existence and uniqueness of the solution of the martingale problem on the graph.
Mark I. Freidlin, Alexander D. Wentzell
Chapter 9. The Multidimensional Case
Summary.
In Chaps. 8–9 the situation is considered where the averaging is due to the mixing in the non-perturbed dynamical system, which is supposed to be a Hamiltonian one. In all averaging-principle problems we can single out in the perturbed system fast components and slow ones. Smallness of the perturbations must be considered as compared to the speed of the motion according to the dynamical system; so one possible model of small perturbations is considering the dynamical-system motion with velocities multiplied by a large parameter, the slow motion due to the perturbations being not too fast and not too slow. The fast motion is approximately the same as the sped-up non-perturbed motion—non-random, in fact; but the slow motion, considered in another time scale, remains random even when the randomness of the perturbation goes to zero. The question arises on what space should we consider this slow motion. If the non-perturbed system has no saddle-type points, we can, in the one-degree-of-freedom case, characterize the slow motion by the value of the Hamiltonian (in the multidimensional case with several first integrals, by the values of those first integrals). If there are saddle-type points, the slow motion should be considered on a graph in the one-degree-of-freedom case, and on an “open-book” type space in the multidimensional case. In the case of there being saddle-type points the randomness of the slow-motion component does not disappear as the diffusion coefficient of the perturbation goes to zero; and the limiting random process does not depend on the choice of the diffusion coefficients, that is on the truly random part of the perturbations—so it can be considered as the intrinsic property of the dynamical system with small non- random perturbations, due to high instability at the saddle-type points. The technique of proofs relies on martingale problems.
In Chap. 9 we consider the multidimensional case. Results about convergence of the slow motion are obtained in the case of the region without singular points—which is anyway the necessary preliminary thing for all cases to be considered—and in the case of weakly coupled oscillators, where the system is just several one-degree-of-freedom Hamiltonian systems tied up only by the small perturbations of the multidimensional system.
Mark I. Freidlin, Alexander D. Wentzell
Chapter 10. Stability Under Random Perturbations
Summary.
Stability of stochastically perturbed systems is considered in Chap. 10. If the nonperturbed system has an asymptotically stable equilibrium, and a bounded neighborhood of this equilibrium is such that the system is “destroyed” if the perturbed trajectory leaves that neighborhood, then the stability of the system can be characterized by the exit time from this neighborhood. If the noise is small, the main term of the asymptotics of the exit time characterizes the stability. Using the results obtained in previous chapters, one can calculate the exit time up to its main term. If a problem of optimal stabilization is considered and the stochastic perturbations are small, one should maximize the main term of the exit time. Results of this type are considered in this chapter.
Mark I. Freidlin, Alexander D. Wentzell
Chapter 11. Sharpenings and Generalizations
Summary.
Finally, in Chap. 11 some generalizations and further extensions are considered; among the results worth mentioning are: the application of large-deviation results of Chaps. 3–5 to wave propagation in semi-linear partial differential equations (in the spirit of the Kolmogorov–Petrovskii–Piskunov problem for reaction-diffusion equations); and large deviations for infinite-dimensional systems described by stochastic partial differential equations.
Mark I. Freidlin, Alexander D. Wentzell
Backmatter
Metadaten
Titel
Random Perturbations of Dynamical Systems
verfasst von
Mark I. Freidlin
Alexander D. Wentzell
Copyright-Jahr
2012
Verlag
Springer Berlin Heidelberg
Electronic ISBN
978-3-642-25847-3
Print ISBN
978-3-642-25846-6
DOI
https://doi.org/10.1007/978-3-642-25847-3