Convergence of Probability Measures. By P. Billingsley. Chichester, Sussex, Wiley, 1968. xii, 253 p. 9 1/4“. 117s.

Convergence of Probability Measures

“Multivalued Analysis” is the theory of set-valued maps (called multifonctions) and has important applications in many different areas. Multivalued analysis is a remarkable mixture of many different parts of mathematics such as point-set topology, measure theory and nonlinear functional analysis. It is also closely related to “Nonsmooth Analysis” (Chapter 5) and in fact one of the main motivations behind the development of the theory, was in order to provide necessary analytical tools for the study of problems in nonsmooth analysis. It is not a coincidence that the development of the two fields coincide chronologically and follow parallel paths. Today multivalued analysis is a mature mathematical field with its own methods, techniques and applications that range from social and economic sciences to biological sciences and engineering. There is no doubt that a modern treatise on “Nonlinear Functional Analysis” can not afford the luxury of ignoring multivalued analysis. The omission of the theory of multifunctions will drastically limit the possible applications.

SET-Valued Analysis

In order to provide a general framework within which the dispersal of cells or organisms can be studied, we introduce two stochastic processes that model the major modes of dispersal that are observed in nature. In the first type of movement, which we call the position jump or kangaroo process, the process comprises a sequence of alternating pauses and jumps. The duration of a pause is governed by a waiting time distribution, and the direction and distance traveled during a jump is fixed by the kernel of an integral operator that governs the spatial redistribution. Under certain assumptions concerning the existence of limits as the mean step size goes to zero and the frequency of stepping goes to infinity the process is governed by a diffusion equation, but other partial differential equations may result under different assumptions. The second major type of movement leads to what we call a velocity jump process. In this case the motion consists of a sequence of "runs" separated by reorientations, during which a new velocity is chosen. We show that under certain assumptions this process leads to a damped wave equation called the telegrapher's equation. We derive explicit expressions for the mean squared displacement and other experimentally observable quantities. Several generalizations, including the incorporation of a resting time between movements, are also studied. The available data on the motion of cells and other organisms is reviewed, and it is shown how the analysis of such data within the framework provided here can be carried out.

Models of dispersal in biological systems

The problem of stabilizing an unstable, time-invariant, discrete-time, linear system by means of state feedback when the measurements of the state are quantized is addressed. It is found that there is no control strategy that stabilizes the system in the traditional sense of making all closed-loop trajectories asymptotic to zero. If the system is not excessively unstable, feedback strategies that bring closed-loop trajectories arbitrarily close to zero for a long time can be implemented. It is also found that when the ordinary linear feedback of quantized state measurements is applied, the resulting closed-loop system behaves chaotically. The asymptotic pseudorandom closed-loop system dynamics differ substantially from what would be predicted by a conventional signal-with-noise analysis of the quantization's effects. Probabilistic reformulations of the stability problem in terms of the invariant measure are considered. >

Stabilizing a linear system with quantized state feedback

This chapter relates the notions of mutations with the concept of graphical derivatives of set-valued maps and more generally links the above results of morphological analysis with some basic facts of set-valued analysis that we shall recall.

Set-valued analysis

Preliminaries review of the ergodic theory the Frobenius-Perron operator absolutely continuous invariant measures other existence results spectral decomposition of the Frobenius-Perron operator properties of absolutely continuous invariant measures Markov transformations compactness theorem and approximation of invariant densities stability of invariant measures the inverse problem for the Frobenius-Perron equation applications. Solutions to all problems.

Laws of Chaos : Invariant Measures and Dynamical Systems in One Dimension

Laws of chaos

LetS be a bounded region inR
 N and let ℊ={S
 i}
 =1/
 be a partition ofS into a finite number of subsets having piecewiseC
 2 boundaries. We assume that whereC
 2 segments of the boundaries meet, the angle subtended by tangents to these segments at the point of contact is bounded away from 0. Letτ:S →S be piecewiseC
 2 on ℊ and expanding in the sense that there exists 0<σ< 1 such that for anyi=1, 2, ...,m, ‖Dτ
 −1
 ‖<σ, whereDτ
 −1
 is the derivative matrix ofτ
 −1
 and ‖ ‖ is the euclidean matrix norm. The main result provides an upper bound onσ which guarantees the existence of an absolutely continuous invariant measure forτ.

Absolutely continuous invariant measures for piecewise expandingC 2 transformations inR N

A class of piecewise C2 transformations from an interval into itself with slopes greater than 1 in absolute value, and having the property that it takes partition points into partition points is shown to have unique absolutely continuous invariant measures. For this class of functions, a central limit theorem holds for all real measurable functions. For the subclass of piecewise linear transformations having a fixed point, it is shown that the unique absolutely continuous invariant measures are piecewise constant.

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On a class of transformations which have unique absolutely continuous invariant measures

Let S be a bounded region in R N and let \(P = \left\{ {{S_I}} \right\}_{i - 1}^m\) be a partition of S into a finite number of subsets having piecewise C 2 boundaries. We assume that where C 2 segments of the boundaries meet, the angle subtended by tangents to these segments at the point of contact is bounded away from 0. Let τ: S → S be piecewise C 2 on P and expanding in the sense that there exists 0 < σ < 1 such that for any i = 1, 2,…,m, \( ||D{\tau _i}^{ - 1}||{\text{ < }}\sigma \) where Dτ i -1 is the derivative matrix of τ i -1 and || || is the euclidean matrix norm. The main result provides an upper bound on σ which guarantees the existence of an absolutely continuous invariant measure for τ.

Abraham Boyarsky

Papers

Laws of Chaos : Invariant Measures and Dynamical Systems in One Dimension

Laws of chaos

Absolutely continuous invariant measures for piecewise expandingC 2 transformations inR N

On a class of transformations which have unique absolutely continuous invariant measures

Absolutely Continuous Invariant Measures for Piecewise Expanding C 2 Transformations in R N