Overview

A careful and accessible exposition of functional analytic methods in stochastic analysis is provided in this book. It focuses on the interrelationship between three subjects in analysis: Markov processes, semi groups and elliptic boundary value problems. The author studies a general class of elliptic boundary value problems for second-order, Waldenfels integro-differential operators in partial differential equations and proves that this class of elliptic boundary value problems provides a general class of Feller semigroups in functional analysis. As an application, the author constructs a general class of Markov processes in probability in which a Markovian particle moves both by jumps and continuously in the state space until it 'dies' at the time when it reaches the set where the particle is definitely absorbed. Augmenting the 1st edition published in 2004, this edition includes four new chapters and eight re-worked and expanded chapters. It is amply illustrated and all chapters are rounded off with Notes and Comments where bibliographical references are primarily discussed. Thanks to the kind feedback from many readers, some errors in the first edition have been corrected. In order to keep the book up-to-date, new references have been added to the bibliography. Researchers and graduate students interested in PDEs, functional analysis and probability will find this volume useful.

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9783662436950

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Reviews

From the reviews: ""This book is devoted to the study of certain uniformly elliptic boundary value problems and associated semigroups. The main results are all laid out in the introduction, so it is always clear where the book is headed. For the probabilist, the book provides a good introduction to modern sophisticated results on analytical problems associated with diffusion processes with possibly additional Levy-type jumps. In some cases, the book may also be a useful reference ."" (Jan M. Swart, Jahresberichte der Deutschen Mathematiker Vereinigung, November, 2005) ""This book by Kazuaki Taira contains a detailed study of semigroups, elliptical boundary value problems, Markov processes and the relations between these mathematical concepts. The book grew out of a series of lectures and lecture notes; this facilitates its use for teaching at the graduate level. The presentation is detailed and clear . I would recommend the book for graduate students or researchers interested mainly in the analytical aspects of Markov process theory ."" (R. Frey, ZAA - Zeitschrift fur Analysis und ihre Anwendungen, Vol. 23 (3), 2004) ""In this book the author proposes the study of three interrelated subjects in analysis: semigroups, elliptic boundary value problems and Markov processes. The well chosen material given in an appropriate form and style makes the book very useful for the university students as well as for mathematicians with interests in probability theory, functional analysis and partial differential equations."" (Mikhail P. Moklyachuk, Zentralblatt MATH, Vol. 1035, 2004) ""The book is devoted to the generation of analytic Feller semigroups by operators corresponding to boundary value problems for second order elliptic differential and integro-differential equations. ... the present book is a valuable contribution to a rich field of mathematics emerging at the interface of functional analysis, partial differential equations, and stochastic processes."" (Anatoly N. Kochubei, Ma From the reviews: ""This book is devoted to the study of certain uniformly elliptic boundary value problems and associated semigroups. The main results are all laid out in the introduction, so it is always clear where the book is headed. For the probabilist, the book provides a good introduction to modern sophisticated results on analytical problems associated with diffusion processes with possibly additional Levy-type jumps. In some cases, the book may also be a useful reference ."" (Jan M. Swart, Jahresberichte der Deutschen Mathematiker Vereinigung, November, 2005) ""This book by Kazuaki Taira contains a detailed study of semigroups, elliptical boundary value problems, Markov processes and the relations between these mathematical concepts. The book grew out of a series of lectures and lecture notes; this facilitates its use for teaching at the graduate level. The presentation is detailed and clear . I would recommend the book for graduate students or researchers interested mainly in the analytical aspects of Markov process theory ."" (R. Frey, ZAA - Zeitschrift fur Analysis und ihre Anwendungen, Vol. 23 (3), 2004) ""In this book the author proposes the study of three interrelated subjects in analysis: semigroups, elliptic boundary value problems and Markov processes. The well chosen material given in an appropriate form and style makes the book very useful for the university students as well as for mathematicians with interests in probability theory, functional analysis and partial differential equations."" (Mikhail P. Moklyachuk, Zentralblatt MATH, Vol. 1035, 2004) ""The book is devoted to the generation of analytic Feller semigroups by operators corresponding to boundary value problems for second order elliptic differential and integro-differential equations. ... the present book is a valuable contribution to a rich field of mathematics emerging at the interface of functional analysis, partial differential equations, and stochastic processes."" (

From the reviews: This book is devoted to the study of certain uniformly elliptic boundary value problems and associated semigroups. The main results are all laid out in the introduction, so it is always clear where the book is headed. For the probabilist, the book provides a good introduction to modern sophisticated results on analytical problems associated with diffusion processes with possibly additional Levy-type jumps. In some cases, the book may also be a useful reference . (Jan M. Swart, Jahresberichte der Deutschen Mathematiker Vereinigung, November, 2005) This book by Kazuaki Taira contains a detailed study of semigroups, elliptical boundary value problems, Markov processes and the relations between these mathematical concepts. The book grew out of a series of lectures and lecture notes; this facilitates its use for teaching at the graduate level. The presentation is detailed and clear . I would recommend the book for graduate students or researchers interested mainly in the analytical aspects of Markov process theory . (R. Frey, ZAA - Zeitschrift fur Analysis und ihre Anwendungen, Vol. 23 (3), 2004) In this book the author proposes the study of three interrelated subjects in analysis: semigroups, elliptic boundary value problems and Markov processes. The well chosen material given in an appropriate form and style makes the book very useful for the university students as well as for mathematicians with interests in probability theory, functional analysis and partial differential equations. (Mikhail P. Moklyachuk, Zentralblatt MATH, Vol. 1035, 2004) The book is devoted to the generation of analytic Feller semigroups by operators corresponding to boundary value problems for second order elliptic differential and integro-differential equations. ... the present book is a valuable contribution to a rich field of mathematics emerging at the interface of functional analysis, partial differential equations, and stochastic processes. (

Author Information

Professor Kazuaki TAIRA, born in Tokyo, Japan in 1946, served for many years as a Professor of Mathematics at the University of Tsukuba (1998-2009). He received his Bachelor of Science in 1969 from the University of Tokyo, Japan and his Master of Science in 1972 from Tokyo Institute of Technology, Japan, where he served as an assistant from 1972 to 1978. The Doctor of Science degree was awarded to him in 1976 by the University of Tokyo and in 1978 the Doctorat d'Etat degree was given to him by Universite de Paris-Sud, France. He held a French government scholarship from 1976 to 1978. He has also served as a member of the Institute for Advanced Study, USA (1980-1981), as an Associate Professor at the University of Tsukuba (1981-1995) and as a Professor at Hiroshima University, Japan (1995-1998). In 1998, he accepted an invitation from the University of Tsukuba to teach there again as a Professor. Since 2009, he has also been a part-time Professor at Waseda University. His current research interests are in the study of three interrelated subjects in analysis: semi groups, elliptic boundary value problems and Markov processes.

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