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Book review

Author(s) Prato, G. Da (ed.)
Tubaro, L. (ed.)
Title Stochastic partial differential equations and applications
Publisher Longman Scientific & Technical
Year of publication 1992
Reviewed by P.P. Zabrejko

This book is a collections of 21 articles that were presented at the third meeting on Stochastic Partial Differential Equations and Applications, held at Villa Madruzzo, Trento, in January 1990; the book is dedicated to Michael Métivier, whose untimely death did not allow him to take part in the meeting and who has contributed so much to the success of this meeting. Now Stochastic DIfferential Equations are on the one hand extensively developing theory and on the other are powerful and flexible instrument of analysis in such fields as Quantum Random Fields, Control Theory, Quantum Probability, Filtering Theory and many others. The articles represented in the book allow to become acquainted with broad spectrum of up-to-date results in the theory of stochastic differential equations of different types and theirs applications in different fields. To read this book is very simply for the short and exhausting abstracts of authors were preceded by the main text.

The article by S. Albeverio, K. Iwata, T. Kolsrud, Homogeneous Markov generalized vector fields and quantum fields over 4-dimenstonal space-time, presents the description and main properties of generalized vector random fields which are homogeneous with respect to the Euclidean group over the Euclidean 4-dimensional space. The article by A.V. Balakrishnan, Stochastic regulator theory for a class of hyperbolic systems, is devoted to abstract description of a special class of systems for which the state equation is a wave equation, the observed data are linked linearly with the state variable corrupted by noise and the cost is quadratic.

The article A. Bensoussan, Some existence results for stochastic partial differential equations presents some existence results for stochastic partial differential equations for which the drift part of the nonnonlinearity is a monotone operator of Leray-Lions type. The article by Z. Brzezniak, F. Flandoli, Regularity of solutions and random evolution operator for stochastic parabolic equations, describes a new class of stochastic parabolic equations for which the analysis of smoothness of solutions and the construction of Cauchy operator are given. The article by P. Cannarsa, G. Da Prato, Direct solution of a second order Hamilton-Jacobi equation in Hilbert spaces, presents the existence and uniqueness theorems of smooth solutions for Hamilton-Jacobi equation in infinite dimensions with locally Lipschitz nonlinear terms. The article by A.M. Chebotarev, F. Fagnola, P. Sundar, Towards a stochastic Stone's theorem, deals with quantum stochastic differential equations whose associated semigroup is a quantum generalization of the Markov semigroup of a pure birth or a birth-and-death process. The article by T.S. Chiang, G. Kallianpur, P. Sundar, Propagation of chaos for systems of interacting neurons, gives a existence and uniqueness theorem and theorem about asymptotic behavior as n -> for stochastic differential equation which describes a system of n interacting spatially extended neurons. The article by P.L. Chow, J.L. Jiang, J.L. Menaldi, Pathwise convergence of approximate solutions to Zakai's equation in a bounded domain, describes finite-dimensional approximations to the Zakai equation in nonlinear filtering and presents some results on the convergence and the rate of convergence of these approximations to exact solution. The article M.H.A. Davis, G. Burstein, On the relation between deterministic and stochastic optimal control, deals with the stochastic optimal control problem that is considered as a family of deterministic control problems parameterized by the paths of the driving Wiener process and of a newly introduced Lagrange multiplier stochastic process; the main result is a theorem that the value function in these problems is the unique global solution of a robust equation associated to a Hamilton-Jacobi-Bellman stochastic partial differential equation. The article by B. Fernandez, L.G. Gorostiza, Convergence of generalized semimartingales to a continuous process, presents a new result about the convergence of a sequence of semimartingales taking values in a space which is the dual of a countable Hilbert nuclear space to a continuous limit process; as an example a multiple particle system evolving by random migration, branching, mutation and immigration in space-time is considered. The article by P. Florchinger, Zakai equation of nonlinear filtering with unbounded coefficients, presents some results on existence and uniqueness of solutions to the Zakai equation associated with nonlinear filtering problems in which the observation coefficients are unbounded; these solutions are constructed as unnormalized filter associated with the considered system in two cases: with independent noises and with depended noises and a scalar observation process. The article by D. Gatarek, B. Goldys, On solving stochastic evolution equations by the change of drift with application to optimal control, deals with existence and uniqueness results for stochastic differential equations in Hilbert space with additive cylindrical white noise and nonlinear drift terms. The article by I. Gyongy, N.V. Krylov, On stochastic partial differential equations with unbounded coefficients, deals with existence, uniqueness and approximation results for stochastic partial differential equations. The article by U.G. Haussmann, Stochastic PDE's with unilateral constraints in higher dimensions, presents some existence and uniqueness results for stochastic partial differential equations in R^d in the case d = 1 that are similar to Haussmann and Pardoux ones that were obtained in the case d = 1 (under more general conditions). The article by D. Kannan, L. Hazareesingh, Multiplicative stochastic integral method for stochastic evolution equations, presents the generalization for infinite dimensional stochastic differential equations of the theory multiplicative integrals and corresponding formulas for solutions of equations. The article by P. Kotelenz, Positive solutions for a class of stochastic partial differential equations, is devoted to analysis of the conditions under these the stochastic partial differential equations in function spaces have positive solutions. The article by D. Ioffe, Large deviations for reaction-diffusion equation with rapidly oscillating random noise, deals with the averaging principle for random differential equation on the unit circle with a stationary in time random field. The article by S. Méléard, S. Roelly, Interacting branching measure processes, presents an existence theorem for a model of interacting measure-valued branching process. The article by S.K. Mitter, O. Zeitouni, An SPDE formulation for image segmentation, is devoted to some variational problem arising in the context of image segmentation; the solution to this problem is a density of a random field given by the solution of an stochastic partial differential equation. The article by B.L. Rosovski, Some results on a diffusion approximation in the induction equation, deals with the induction equation in the magnetic field, diffusion approximation to this equation and their Feynman-Kac representations. The last article by A.S. Ustinel A new class of stochastic partial differential equations on the Wiener space, deals with a new kind of stochastic partial differential equations in the distribution space on some Wiener space.

So the book constitutes a sizable row of new important results in stochastic partial differential equations and their applications to different problems in adjoining fields like control theory, filtering theory, quantum random fields an so on. The book undoubtedly is useful and interesting for specialists in probability theory and partial differential equations as well as in functional analysis and quantum physics.