Ebook: Semigroups, Boundary Value Problems and Markov Processes
Author: Kazuaki Taira (auth.)
- Tags: Functional Analysis, Abstract Harmonic Analysis, Partial Differential Equations, Probability Theory and Stochastic Processes
- Series: Springer Monographs in Mathematics
- Year: 2004
- Publisher: Springer-Verlag Berlin Heidelberg
- Edition: 1
- Language: English
- pdf
The purpose of this book is to provide a careful and accessible account along modern lines of the subject wh ich the title deals, as weIl as to discuss prob lems of current interest in the field. Unlike many other books on Markov processes, this book focuses on the relationship between Markov processes and elliptic boundary value problems, with emphasis on the study of analytic semigroups. More precisely, this book is devoted to the functional analytic approach to a class of degenerate boundary value problems for second-order elliptic integro-differential operators, called Waldenfels operators, whi:h in cludes as particular cases the Dirichlet and Robin problems. We prove that this class of boundary value problems provides a new example of analytic semi groups both in the LP topology and in the topology of uniform convergence. As an application, we construct a strong Markov process corresponding to such a physical phenomenon that 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. The approach here is distinguished by the extensive use of the techniques characteristic of recent developments in the theory of partial differential equa tions. The main technique used is the calculus of pseudo-differential operators which may be considered as a modern theory of potentials.
The purpose of this book is to provide a careful and accessible account along modern lines of the subject which the title deals, as well as to discuss problems of current interest in the field. More precisely this book is devoted to the functional-analytic approach to a class of degenerate boundary value problems for second-order elliptic integro-differential operators which includes as particular cases the Dirichlet and Robin problems. This class of boundary value problems provides a new example of analytic semigroups. As an application, we construct a strong Markov process corresponding to such a diffusion phenomenon that 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.
The purpose of this book is to provide a careful and accessible account along modern lines of the subject which the title deals, as well as to discuss problems of current interest in the field. More precisely this book is devoted to the functional-analytic approach to a class of degenerate boundary value problems for second-order elliptic integro-differential operators which includes as particular cases the Dirichlet and Robin problems. This class of boundary value problems provides a new example of analytic semigroups. As an application, we construct a strong Markov process corresponding to such a diffusion phenomenon that 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.
Content:
Front Matter....Pages I-XI
Introduction and Main Results....Pages 1-17
Theory of Semigroups....Pages 19-45
Markov Processes and Semigroups....Pages 47-72
Theory of Distributions....Pages 73-96
Theory of Pseudo-Differential Operators....Pages 97-120
Elliptic Boundary Value Problems....Pages 121-134
Elliptic Boundary Value Problems and Feller Semigroups....Pages 135-168
Proof of Theorem 1.1....Pages 169-185
Proof of Theorem 1.2....Pages 187-189
A Priori Estimates....Pages 191-196
Proof of Theorem 1.3....Pages 197-202
Proof of Theorem 1.4, Part (i)....Pages 203-212
Proofs of Theorem 1.5 and Theorem 1.4, Part (ii)....Pages 213-243
Boundary Value Problems for Waldenfels Operators....Pages 245-270
Back Matter....Pages 271-340
The purpose of this book is to provide a careful and accessible account along modern lines of the subject which the title deals, as well as to discuss problems of current interest in the field. More precisely this book is devoted to the functional-analytic approach to a class of degenerate boundary value problems for second-order elliptic integro-differential operators which includes as particular cases the Dirichlet and Robin problems. This class of boundary value problems provides a new example of analytic semigroups. As an application, we construct a strong Markov process corresponding to such a diffusion phenomenon that 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.
Content:
Front Matter....Pages I-XI
Introduction and Main Results....Pages 1-17
Theory of Semigroups....Pages 19-45
Markov Processes and Semigroups....Pages 47-72
Theory of Distributions....Pages 73-96
Theory of Pseudo-Differential Operators....Pages 97-120
Elliptic Boundary Value Problems....Pages 121-134
Elliptic Boundary Value Problems and Feller Semigroups....Pages 135-168
Proof of Theorem 1.1....Pages 169-185
Proof of Theorem 1.2....Pages 187-189
A Priori Estimates....Pages 191-196
Proof of Theorem 1.3....Pages 197-202
Proof of Theorem 1.4, Part (i)....Pages 203-212
Proofs of Theorem 1.5 and Theorem 1.4, Part (ii)....Pages 213-243
Boundary Value Problems for Waldenfels Operators....Pages 245-270
Back Matter....Pages 271-340
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