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Stochastic Differential Equations in Infinite Dimensional Spaces (Lecture Notes-Monographs; Vol. 26): Gopinath Kallianpur, Jie Xiong

From the Introduction:

This volume is an expanded version of the lectures delivered by Professor Gopinath Kallianpur as part of the 1993 Barrett Lectures at the University of Tennessee, Knoxville, during March 25-27, 1993.

The tradition of the Barrett memorial Lectures began in 1970 to honor Professor John H. Barrett, who was the head of the mathematics department of the University of Tennessee at the time of his death in 1969. They have been given annually since 1972 by a succession of distinguished speakers in varied areas of vital current interest.

The main objective of the 1993 Barrett Lectures was to provide the participants the opportunity to gain a concise introduction to the important and rapidly developing fields of SDE's in infinite dimensional spaces and measure valued processes. To accomplish this objective, Professors Donald Dawson, Eugene Dynkin and Gopinath Kallianpur, three outstanding specialists in these fields, were invited. The format of the Lectures was so designed as to allow the speakers the opportunity to explore their respective subject in depth and also to leave time for informal discussions. Each Lecturer presented a series of three one-hour lectures. Professor Dawson presented ome recent developments of the long term behavior of measure wlued processes, while professor Dynkin talked about measure valued hranching processes and partial differential equations. Professor Kallianpur's lectures were devoted to the subject of infinite dimensional nuclear space valued SDE's. In order to make the planned activities accessible to a larger udience the Lecture Series was coordinated with the regional AMS meeting which took place at the same time at the University campus. The Lecture Series attracted more than one hundred prticipants. Though the majority of the participants were drawn from Universities and institutions of the outheastern United States, the lectures also attracted many participants from Canada, Mexico, France, Germany, Holland, Japan and Switzerland as well as from other parts of the United States.

From the Preface:

SDE's in infinite dimensional spaces occur as stochastic models in the investigation of problems that arise in a variety of disciplines such a seurophysiology, environmental pollution, chemical reaction diffusion, infinite particle systems and turbulence. Not alI of these applications are discussed in the monograph. In fact, ome of the problems such as a stochastic treatment of turbulence are subjects of intensive current research. The specific choice of topics for these lectures was motivated by the interest of one of us (G. K.) in problems of neuronal behavior which led to the development of the theory presented here. As a consequence, a greater emphasis is placed on SDE's and SPDE's driven by Poisson random measures (the latter being the simplest representative of a point process) and on approximations to diffusion equations. Stochastic models of this type occur also in other applications and are worthy of further study.

For the convenience of the reader, especially of the aspiring graduate student, we have tried to make the monograph as self-contained as possible by including three chapters containing topological and probabilistic preliminaries essential for the understanding of the later material. The first named author thanks Professors Balram Rajput and Jan Rosinski, and the University of Tennessee for the invitation to deliver the 1993 Barrett Lectures. Both of us thank the former and present editors of the IMS Monograph Series for their cooperation.

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