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1984 | Buch

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Random Perturbations of Dynamical Systems

verfasst von: M. I. Freidlin, A. D. Wentzell

Verlag: Springer US

Buchreihe : Grundlehren der mathematischen Wissenschaften

Enthalten in: Professional Book Archive

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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.

Inhaltsverzeichnis

Frontmatter

Introduction

Abstract

Let b(x) be a continuous vector field in R^r. First we discuss nonrandom perturbations of a dynamical system

$$\dot x_t \, = \,b(x_t ).$$

(1)

M. I. Freidlin, A. D. Wentzell

Chapter 1. Random Perturbations

Abstract

We shall assume known the basic facts of the Lebesgue integral and measure theory, as well as probability theory. The necessary information concerning these topics is contained, for example, in the corresponding chapters of the book by Kolmogorov and Fomin [1] and in the book by Gikhman and Skorokhod [1]. In this chapter we introduce notation and recall some information from the theory of stochastic processes in an appropriate form. We shall not provide proofs but rather references to the pertinent literature.

M. I. Freidlin, A. D. Wentzell

Chapter 2. Small Random Perturbations on a Finite Time Interval

Abstract

In the space R^r we consider the following system of ordinary differential equations:

$$X_t^\varepsilon \, = \,b(X_t^\varepsilon,\varepsilon \xi _t ),\quad X_0^\varepsilon \, = \,x.$$

(1.1)

M. I. Freidlin, A. D. Wentzell

Chapter 3. Action Functional

Abstract

We consider a random process $\mathop X\nolimits_t^\varepsilon \; = \;\mathop X\nolimits_t^\varepsilon (x)\;$ in the space R^r defined by the stochastic differential equation

$$\mathop {\dot X}\nolimits_t^\varepsilon \; = \;b(\mathop X\nolimits_t^\varepsilon )\; + \;\varepsilon \dot w_t,\quad \quad \mathop X\nolimits_0^\varepsilon \; = \;x.$$

(1.1)

M. I. Freidlin, A. D. Wentzell

Chapter 4. Gaussian Perturbations of Dynamical Systems. Neighborhood of an Equilibrium Point

Abstract

In this chapter we shall consider perturbations of a dynamical system

$$ {\dot x_t} = b\left( {{x_t}} \right),{\rm{ }}{x_0} = x,{\rm{ }}b\left( x \right) = \left( {{b^1}\left( x \right), \ldots ,{b^r}\left( x \right)} \right) $$

(1.1)

by a white noise process or by a Gaussian process in general. Unless otherwise stated, we shall assume that the functions bⁱ are bounded and satisfy a Lipschitz condition: |b(x) - b(y)| ≤ K|x - y|, |b(x)| ≤ K < ∞. Here we pay particular attention to the case where the perturbed process has the form

$$\mathop {\dot X}\nolimits_{t\;\;}^\varepsilon = \;b(\mathop X\nolimits_t^\varepsilon )\; + \;\varepsilon \dot w_t,\quad \mathop X\nolimits_{0\quad }^\varepsilon = \;x,$$

(1.2)

where W_t is an r-dimensional Wiener process.

M. I. Freidlin, A. D. Wentzell

Chapter 5. Perturbations Leading to Markov Processes

Abstract

In this chapter we shall consider theorems on the asymptotics of probabilities of large deviations for Markov random processes. These processes can be viewed as generalizations of the scheme of summing independent random variables; the constructions used in the study of large deviations for Markov processes generalize constructions encountered in the study of sums of independent terms.

M. I. Freidlin, A. D. Wentzell

Chapter 6. Markov Perturbations on Large Time Intervals

Abstract

In this chapter we consider families of diffusion processes $(X_t^\varepsilon,\,{\text{P}}_x^\varepsilon)$ on a connected manifold M. We shall assume that these families satisfy the hypotheses of Theorem 3.2 of Ch. 5 and the behavior of probabilities of large deviations from the “most probable” trajectory—the trajectory of the dynamical system $\dot x_t \, = \,b(x_t )$ —can be described as ε → 0, by the action functional $ \varepsilon ^{ - 2} S\left( \varphi \right)\, = \,\varepsilon ^{ - 2} S_{T_1 T_2 } \left( \varphi \right), $ where

$$ S(\varphi )\, = \,\frac{1} {2}\int_{T_1 }^{T_2 } {\mathop \sum \limits_{ij} } \,a_{ij} (\varphi _t )(\dot \varphi _t^i \, - \,b^i (\varphi _t ))(\dot \varphi _t^j \, - \,b^j (\varphi _t ))\,dt. $$

M. I. Freidlin, A. D. Wentzell

Chapter 7. The Averaging Principle. Fluctuations in Dynamical Systems with Averaging

Abstract

Let us consider the system

$$\dot Z_t^\varepsilon = \varepsilon b(Z_t^\varepsilon \xi _t ),\quad Z_0^\varepsilon = x$$

(1.1)

of ordinary differential equations in R^r, where ξ_t, t ≥ 0, is a function assuming values in R^l, ε is a small numerical parameter and

$$b(x,\,\,y) = (b^1 (x,\,y), \ldots,\,b^r (x,\,\,y)).$$

M. I. Freidlin, A. D. Wentzell

Chapter 8. Stability Under Random Perturbations

Abstract

In the theory of ordinary differential equations much work is devoted to the study of stability of solutions with respect to small perturbations of the initial conditions or of the right side of an equation. In this chapter we consider some problems concerning stability under random perturbations. First we recall the basic notions of classical stability theory. Let the dynamical system

$$\dot x_t \; = \;b(x_t )$$

(1.1)

in R ^r have an equilibrium position at the point O:b(O) = 0.

M. I. Freidlin, A. D. Wentzell

Chapter 9. Sharpenings and Generalizations

Abstract

In Chapters 3, 4, 5 and 7 we established limit theorems on large deviations, involving the rough asymptotics of probabilities of the type P{X^h ∈ A}There arises the following question: Is it possible to obtain subtler results for families of random processes (similar to those obtained for sums of independent random variables)—local limit theorems on large deviations and theorems on sharp asymptotics? There is some work in this direction; we give a survey of the results in this section.

M. I. Freidlin, A. D. Wentzell

Backmatter

Titel: Random Perturbations of Dynamical Systems
verfasst von: M. I. Freidlin
A. D. Wentzell
Verlag: Springer US
Electronic ISBN: 978-1-4684-0176-9
Print ISBN: 978-1-4684-0178-3
DOI: https://doi.org/10.1007/978-1-4684-0176-9

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Inhaltsverzeichnis

Frontmatter

Introduction

Chapter 1. Random Perturbations

Chapter 2. Small Random Perturbations on a Finite Time Interval

Chapter 3. Action Functional

Chapter 4. Gaussian Perturbations of Dynamical Systems. Neighborhood of an Equilibrium Point

Chapter 5. Perturbations Leading to Markov Processes

Chapter 6. Markov Perturbations on Large Time Intervals

Chapter 7. The Averaging Principle. Fluctuations in Dynamical Systems with Averaging

Chapter 8. Stability Under Random Perturbations

Chapter 9. Sharpenings and Generalizations

Backmatter