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Convex Analysisの二,三の進展について

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

Convergence of Probability Measures

N G van Kampen 1981 Amsterdam: North-Holland xiv + 419 pp price Dfl 180 This is a book which, at a lower price, could be expected to become an essential part of the library of every physical scientist concerned with problems involving fluctuations and stochastic processes, as well as those who just enjoy a beautifully written book. It provides an extensive graduate-level introduction which is clear, cautious, interesting and readable.

https://iopscience.iop.org/article/10.1088/0031-9112/34/4/040/pdf

Stochastic Processes in Physics and Chemistry

The author's preface gives an outline: "This book is about weakconvergence methods in metric spaces, with applications sufficient to show their power and utility. The Introduction motivates the definitions and indicates how the theory will yield solutions to problems arising outside it. Chapter 1 sets out the basic general theorems, which are then specialized in Chapter 2 to the space C[0, l ] of continuous functions on the unit interval and in Chapter 3 to the space D [0, 1 ] of functions with discontinuities of the first kind. The results of the first three chapters are used in Chapter 4 to derive a variety of limit theorems for dependent sequences of random variables. " The book develops and expands on Donsker's 1951 and 1952 papers on the invariance principle and empirical distributions. The basic random variables remain real-valued although, of course, measures on C[0, l ] and D[0, l ] are vitally used. Within this framework, there are various possibilities for a different and apparently better treatment of the material. More of the general theory of weak convergence of probabilities on separable metric spaces would be useful. Metrizability of the convergence is not brought up until late in the Appendix. The close relation of the Prokhorov metric and a metric for convergence in probability is (hence) not mentioned (see V. Strassen, Ann. Math. Statist. 36 (1965), 423-439; the reviewer, ibid. 39 (1968), 1563-1572). This relation would illuminate and organize such results as Theorems 4.1, 4.2 and 4.4 which give isolated, ad hoc connections between weak convergence of measures and nearness in probability. In the middle of p. 16, it should be noted that C*(S) consists of signed measures which need only be finitely additive if 5 is not compact. On p. 239, where the author twice speaks of separable subsets having nonmeasurable cardinal, he means "discrete" rather than "separable." Theorem 1.4 is Ulam's theorem that a Borel probability on a complete separable metric space is tight. Theorem 1 of Appendix 3 weakens completeness to topological completeness. After mentioning that probabilities on the rationals are tight, the author says it is an

Convergence of probability measures

These lecture notes provide an elementary introduction, within the framework of finite quantum systems, to recent developments in the theory of entropic fluctuations.

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Entropic Fluctuations in Quantum Statistical Mechanics. An Introduction

Given a W*-algebra ${\mathfrak M}$ with a W*-dynamics τ, we prove the existence of the perturbed W*-dynamics for a large class of unbounded perturbations. We compute its Liouvillean. If τ has a β-KMS state, and the perturbation satisfies some mild assumptions related to the Golden–Thompson inequality, we prove the existence of a β-KMS state for the perturbed W*-dynamics. These results extend the well known constructions due to Araki valid for bounded perturbations.

Perturbation Theory of W*-Dynamics, Liouvilleans and KMS-States

Using the scattering approach to the construction of Non-Equilibrium Steady States proposed by Ruelle we study the transport properties of systems of independent electrons. We show that Landauer-Buttiker and Green-Kubo formulas hold under very general conditions.

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Transport properties of quasi-free fermions

We study ergodic properties of Pauli-Fierz systems – W*-dynamical systems
often used to describe the interaction of a small quantum system with a
bosonic free field at temperature $ T \geq 0 $. We prove that, for a small coupling constant
uniform as the positive temperature $ T \downarrow 0 $, a large class of Pauli-Fierz systems
has the property of return to equilibrium. Most of our arguments are general and
deal with mathematical theory of Pauli-Fierz systems for an arbitrary density of
bosonic field.

https://link.springer.com/content/pdf/10.1007%2Fs00023-003-0146-4.pdf

Return to Equilibrium for Pauli-Fierz Systems

We study self adjoint operators of the form¶H ω = H 0 + ∑λω(n) δ n ,¶where the δ n ’s are a family of orthonormal vectors and the λω(n)’s are independently distributed random variables with absolutely continuous probability distributions. We prove a general structural theorem saying that for each pair (n,m), if the cyclic subspaces corresponding to the vectors δ n and δ m are not completely orthogonal, then the restrictions of H ω to these subspaces are unitarily equivalent (with probability one). This has some consequences for the spectral theory of such operators. In particular, we show that “well behaved” absolutely continuous spectrum of Anderson type Hamiltonians must be pure, and use this to prove the purity of absolutely continuous spectrum in some concrete cases.

Vojkan Jaksic

Papers

Entropic Fluctuations in Quantum Statistical Mechanics. An Introduction

Perturbation Theory of W*-Dynamics, Liouvilleans and KMS-States

Transport properties of quasi-free fermions

Return to Equilibrium for Pauli-Fierz Systems

Spectral structure of Anderson type Hamiltonians