Boundary Value Problems And Markov Processes
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Author | : Kazuaki Taira |
Publisher | : Springer Science & Business Media |
Total Pages | : 196 |
Release | : 2009-06-30 |
Genre | : Mathematics |
ISBN | : 3642016766 |
This is a thorough and accessible exposition on the functional analytic approach to the problem of construction of Markov processes with Ventcel’ boundary conditions in probability theory. It presents new developments in the theory of singular integrals.
Author | : Kazuaki Taira |
Publisher | : Springer Nature |
Total Pages | : 502 |
Release | : 2020-07-01 |
Genre | : Mathematics |
ISBN | : 3030487881 |
This 3rd edition provides an insight into the mathematical crossroads formed by functional analysis (the macroscopic approach), partial differential equations (the mesoscopic approach) and probability (the microscopic approach) via the mathematics needed for the hard parts of Markov processes. It brings these three fields of analysis together, providing a comprehensive study of Markov processes from a broad perspective. The material is carefully and effectively explained, resulting in a surprisingly readable account of the subject. The main focus is on a powerful method for future research in elliptic boundary value problems and Markov processes via semigroups, the Boutet de Monvel calculus. A broad spectrum of readers will easily appreciate the stochastic intuition that this edition conveys. In fact, the book will provide a solid foundation for both researchers and graduate students in pure and applied mathematics interested in functional analysis, partial differential equations, Markov processes and the theory of pseudo-differential operators, a modern version of the classical potential theory.
Author | : Kazuaki Taira |
Publisher | : Springer |
Total Pages | : 724 |
Release | : 2014-08-07 |
Genre | : Mathematics |
ISBN | : 3662436965 |
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.
Author | : Mark I. Freidlin |
Publisher | : Birkhäuser |
Total Pages | : 155 |
Release | : 2012-12-06 |
Genre | : Mathematics |
ISBN | : 3034891911 |
Probabilistic methods can be applied very successfully to a number of asymptotic problems for second-order linear and non-linear partial differential equations. Due to the close connection between the second order differential operators with a non-negative characteristic form on the one hand and Markov processes on the other, many problems in PDE's can be reformulated as problems for corresponding stochastic processes and vice versa. In the present book four classes of problems are considered: - the Dirichlet problem with a small parameter in higher derivatives for differential equations and systems - the averaging principle for stochastic processes and PDE's - homogenization in PDE's and in stochastic processes - wave front propagation for semilinear differential equations and systems. From the probabilistic point of view, the first two topics concern random perturbations of dynamical systems. The third topic, homog- enization, is a natural problem for stochastic processes as well as for PDE's. Wave fronts in semilinear PDE's are interesting examples of pattern formation in reaction-diffusion equations. The text presents new results in probability theory and their applica- tion to the above problems. Various examples help the reader to understand the effects. Prerequisites are knowledge in probability theory and in partial differential equations.
Author | : Kazuaki Taira |
Publisher | : |
Total Pages | : 132 |
Release | : 1991 |
Genre | : Boundary value problems |
ISBN | : 9780387549965 |
Author | : Karl K. Sabelfeld |
Publisher | : Walter de Gruyter GmbH & Co KG |
Total Pages | : 208 |
Release | : 2016-09-26 |
Genre | : Mathematics |
ISBN | : 3110479451 |
This monograph is devoted to random walk based stochastic algorithms for solving high-dimensional boundary value problems of mathematical physics and chemistry. It includes Monte Carlo methods where the random walks live not only on the boundary, but also inside the domain. A variety of examples from capacitance calculations to electron dynamics in semiconductors are discussed to illustrate the viability of the approach. The book is written for mathematicians who work in the field of partial differential and integral equations, physicists and engineers dealing with computational methods and applied probability, for students and postgraduates studying mathematical physics and numerical mathematics. Contents: Introduction Random walk algorithms for solving integral equations Random walk-on-boundary algorithms for the Laplace equation Walk-on-boundary algorithms for the heat equation Spatial problems of elasticity Variants of the random walk on boundary for solving stationary potential problems Splitting and survival probabilities in random walk methods and applications A random WOS-based KMC method for electron–hole recombinations Monte Carlo methods for computing macromolecules properties and solving related problems Bibliography
Author | : Kazuaki Taira |
Publisher | : Cambridge University Press |
Total Pages | : 348 |
Release | : 2016-04-28 |
Genre | : Mathematics |
ISBN | : 1316620867 |
This second edition explores the relationship between elliptic and parabolic initial boundary value problems, for undergraduate and graduate students.
Author | : Vassili N. Kolokoltsov |
Publisher | : Walter de Gruyter |
Total Pages | : 449 |
Release | : 2011 |
Genre | : Mathematics |
ISBN | : 3110250101 |
This work offers a highly useful, well developed reference on Markov processes, the universal model for random processes and evolutions. The wide range of applications, in exact sciences as well as in other areas like social studies, require a volume that offers a refresher on fundamentals before conveying the Markov processes and examples for
Author | : Kazuaki Taira |
Publisher | : Springer Nature |
Total Pages | : 749 |
Release | : |
Genre | : |
ISBN | : 9819736595 |
Author | : Karl K. Sabelfeld |
Publisher | : Walter de Gruyter |
Total Pages | : 148 |
Release | : 2013-07-05 |
Genre | : Mathematics |
ISBN | : 311094202X |
This monograph presents new probabilistic representations for classical boundary value problems of mathematical physics and is the first book devoted to the walk on boundary algorithms. Compared to the well-known Wiener and diffusion path integrals, the trajectories of random walks in this publication are simlated on the boundary of the domain as Markov chains generated by the kernels of the boundary integral equations equivalent to the original boundary value problem. The book opens with an introduction for solving the interior and exterior boundary values for the Laplace and heat equations, which is followed by applying this method to all main boundary value problems of the potential and elasticity theories.