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Book review

Author(s) Albeverio, S.
Blanchard, Ph.
Streit, L.
Title STOCHASTIC PROCESSES and their APPLICATIONS in MATHEMATICS and PHYSICS
Publisher Kluwer Academic Publishers
Year of publication 1990
   
Reviewed by S.V. Rogozin

This book is based on the articles presented on the II Symposium on "Stochastic Processes - Mathematics and Physics" organized by Bielefeld-Bochum Research Center Stochastic (BiBoS) in December 1985. These papers give the readers an imagination about the modern state in different branches of the theory of stochastic processes as well as about many of its applications. The precise contents of these articles will be described below.

G.F. Deil'Antonio ("STOCHASTIC STABILITY FOR VECTOR FIELDS WITH A MANIFOLD OF SINGULAR POINS, AND AN APPLICATION TO LATTICE GAUGE THEORY") has considered the diffusion <formula> on R^d which starts at some point <formula> and has a generator <formula> where Phi is a given smooth vector field in R^d. He has tried to find conditions on a and Phi under which the processes <formula> converge, when <formula>, to diffusion <formula> on closure of some open subset of the singular points set of Phi (<formula> is supposed to have a unique invariant measure independent of <formula>). This problem appears due to some questions from gauge field theory (in particular from that one on the lattices).

In the article of D. Bakry ("RICCI CURVATURE AND DIMENSION FOR DIFFUSION") is given a brief survey of some properties of the "iterated squared gradient" associated to some diffusion semigroups. It is introduced an "intrinsic" definition of the Ricci curvature and of the dimension of the semi-group and described some properties of diffusions with Ricci curvature bounded from below. Then it is obtained some extensions of these results for the case of diffusion with finite dimension.

The progress in the probabilistic study of time evolution of a particle whose wave function has multicomponents is reported in the papers of Ph. Blanchard, Ph. Combe, M. Sirugue, M. Sirugue-Collin ("THE ZITTERBEWEGUNG OF A DIRAC ELECTRON IN A CENTRAL FIELD"). Mainly they have delt with the Dirac equation in any space dimensions provided that the scalar potential is spherically symmetric.

E. Bolthausen has developed in the article "MAXIMUM ENTROPY PRINCIPLES FOR MARKOV PROCESSES" a general maximum entropy principle for Markov processes as a tool for studying the asymptotics for large times. More precisely on the discussion are what may be called the limit law of "typical paths" which contribute to an expression of the form <formula> for large T >= 0, where YT, T >= 0, are random elements with values in a suitable space and satisfy a large deviation principle.

The papers of A. Boutet de Monvel-Bertheir deal with the unique continuation property for Dirac operator <formula> in an open subset in R^3 ("AN OPTIMAL CARLEMAN TYPE INEQUALITY FOR THE DIRAC OPERATOR"). This is obtained for instance in the case when <formula> (mainly due to proved new inequality of the type <formula> which takes place for <formula>).

L. Boutet de Monvel ("TOEPLITZ OPERATORS - AN ASYMPTOTIC QUANTIZATION OF SYMMETRIC CONES") has described construction of Fourier integral operator with complex phase and system of first order pseudo-differential equations on a smooth compact manifold of odd dimension 2n-1. It is done due to some results on algebra of Toeplitz operators on a complex domain.

The motion of a quantum mechanical particle in a random potential on the v-dimensional lattice is on the discussion in the article "PERTURBATION THEORY FOR RANDOM DISODERED SYSTEMS" by F. Constantinescu and U. Scharffenberger. The main attention is paid to two types of perturbation expansions for the resolvents of the tight binding Hamiltonian, which described this kind of motion. Besides are given some applications of these expansions.

D. Dürr, N. Zanghi and H. Zessin ("ON RIGOROUS HYDRODYNAMICS, SELF-DIFFUSION AND GREEN KUBO FORMULAE") have considered the rigorous derivation of the hydrodynamics for deterministic microscopic mechanical model. They have started with Green-Kubo formulae and then investigated fluctuations of the locally conserved hydrodynamics fields.

In the article of D. Gandolfo, R. Hoegh-Krohn, R. Rodriguez "A STOCHASTIC MODEL FOR PLASMA DYNAMICS" on the consideration is the dynamics of stochastic, non-isotropic diffusion processes satisfied a stochastic differential equation of the form <formula>, where <formula> is the brownian motion, <formula> is a drift term determined by a stochastic dynamical assumptions, <formula> is a diffusion tensor. The main attention is paid to the diffusion of particles, transverse to the magnetic field, provided with the metrics <formula>, where <formula>.

R. Graham and T. Tél ("MACROSCOPIC POTENTIALS OF DISSIPATIVE DYNAMICAL SYSTEMS") have studied how the concept of coarse grained thermodynamic potentials can be extended for systems outside thermal equilibrium, more precisely for non-equilibrium systems with stable, generally time dependent, steady states. It is shown that the knowledge of the potential (inspite of the difficulties of its determination) can give some information about the probability of fluctuations in steady states with weak external noise generalizing the Boltzman-Einstein formula of thermodynamics. It is given a powerful method for analytic or numerical calculation of nonequilibrium potentials.

A review of recent resuls in the problem of certain random intersections is given by J. Hawkes ("RANDOM-PATH INTERSECTIONS IN GEOMETRY, PROBABILITY AND PHYSICS"). Besides are presented a number of results about this problem's applications in geometry, probability theory and physics. Mainly the author has interested in some new applicable methods and has explained some of the ideas behind a unified approach to the intersections' problem.

The main idea of the article "NONCOMMUTATIVE VERSION OF THE CENTRAL LIMIT THEOREM AND OF CRAMÉR'S THEOREM" of G.C. Hegerfeldt is to consider a random variable as a multiplication operator. This is realized in two close approaches. Due to constructive structure are obtained some algebraic results for such type of random variables (new version of Cramér's theorem and its applications are among them) as well as probabilistic ones (noncommutative Central Limit Theorem). These results are also extended to the corresponding properties of random operators.

In the paper of P. Krée ("DISTRIBUTION, SOBOLEV SPACES ON GAUSSIAN VECTOR SPACES AND ITO'S CALCULUS,) on the discussion are relations between two modern approaches to Ito's Calculus (the Hida's Analysis of the white noise and the Malliavin's Calculus presented in terms of Sobolev spaces). An attention is paid to distribution theory on Gaussian vector spaces and to differential calculus in Sobolev spaces.

M. Metivier ("ON PROBLEMS IN STOCHASTIC DIFFERENTIAL EQUATIONS CONNECTED WITH SOME PARTICULAR TYPE OF INTERACTING PARTICLES") has considered a model of interacting molecules of a given gas trown into a tube (under conditions that in the interior of this tube they have a diffusion movement with a drift and with possibility of temporary absorption by a liquid layer at the tube's boundary). Are stated some mathematical problems for this model and given some results on these problems.

Y. Ogura and M. Tomisaki ("ASYMPTOTIC BEHAVIOR OF MOMENTS FOR ONE-DIMENSIONAL GENERALIZED DIFFUSION PROCESSES") have considered a diffusion process [x(t), Px] with generator <formula>. They have tried to show how the coefficient of the diffusion operator such as Gu(x) above affects on the moment asymptotic formula. This is studied using the obtained general theorems, on the asymptotic behaviour of moments of one-dimensional generalized diffusion processes.

The paper of Y. Okabe ("LANGEVIN EQUATION AND FLUCTUATION-DISSIPATION THEOREM") is devoted to the extracting a mathematical structure of the fluctuation-dissipation theorem in statistical physics. The author has explained the course of generalization of the Langevin equation to KMO-Langevin equation. It is also given the clarification of the mathematical similarity and difference between white noise and Kubo noise. At last it is obtained the existence and uniqueness of solutions for the KMO-Langevin equation from viewpoint of theory of stochastic differential equations.

M. Röckner and B. Zegarlinski ("THE DIRICHLET PROBLEM FOR QUASI-LINEAR PARTIAL DIFFERENTIAL OPERATORS WITH BOUNDARY DATA GIVEN BY A DISTRIBUTION") have investigated the solution of the following problem: to find a distribution u on <formula>, satisfied an equation <formula>, where <formula> and <formula> such that u is represented by a function on the given open set <formula>.

In the paper of M. Scheutzow ("STATIONARY STOCHASTIC PERTURBATION OF A LINEAR DELAY EQUATION") is considered the family of one-dimensional affine stochastic delay equations <formula> where <formula> and <formula> is one-dimensional Wiener process. It is proved the weak convergence on C(R,R) the solutions of initial problem for this equation to a stationary Gaussian process.

Random lattice models are under consideration in G. Sobbota "RANDOM LATTICE MODELS". Are discussed general problems of quinched disodered systems. The thermodynamic description has been presented in three version. The most realistic one deals with the free energy for only one configuration of frozen in degrees of freedom. The most realizable one is formulated within an ensemble of configurations, over which the free energy is averaged. The less-known one is the generalized grand canonical description. In the article are presented some new theoretical results (obtained by the transfer tensor method) which are describing the non critical behaviour.

One more article is devoted to the investigation of quantum gauge fields (J.-M. Sourian "INTERACTIONS GALILEENNES AIMANTCHARGE"), more precisely to their prototype - classical electromagnetic fields. The author has studied the Galilean invariant interactions between charges and magnets.

H. Spohn ("THE POLARON FUNCTIONAL INTEGRAL") has paid the main attention in his article to the mathematical structure of the polaron functional integral and to its role in the statistical mechanics models and in random motion in random environments. Are given some new results and stated a number of open problems.

In the article of P.Di Vecchia ("THE BOSONIC STRING") is presented the review of modern state in the developing the bosonic strings model. The main attention is given to mathematical basis for this model (algebraic, analytic and so on).

The book is devoted to the specialists in stochastic analysis as well as to research mathematicians, mathematical and theoretical physicists whose work involves stochastic processes and their applications.