Brownian Motion And Potential Theory Modern And Classical
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Brownian Motion And Potential Theory Modern And Classical
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Author : Palle Jorgensen
language : en
Publisher: World Scientific
Release Date : 2024-10-29
Brownian Motion And Potential Theory Modern And Classical written by Palle Jorgensen and has been published by World Scientific this book supported file pdf, txt, epub, kindle and other format this book has been release on 2024-10-29 with Mathematics categories.
In this book, potential theory is presented in an inclusive and accessible manner, with the emphasis reaching from classical to modern, from analytic to probabilistic, and from Newtonian to abstract or axiomatic potential theory (including Dirichlet spaces). The reader is guided through stochastic analysis featuring Brownian motion in its early chapters to potential theory in its latter sections. This path covers the following themes: martingales, diffusion processes, semigroups and potential operators, analysis of super harmonic functions, Dirichlet problems, balayage, boundaries, and Green functions.The wide range of applications encompasses random walk models, especially reversible Markov processes, and statistical inference in machine learning models. However, the present volume considers the analysis from the point of view of function space theory, using Dirchlet energy as an inner product. This present volume is an expanded and revised version of an original set of lectures in the Aarhus University Mathematics Institute Lecture Note Series.
Brownian Motion And Potential Theory Modern And Classical
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Author : Palle Jorgensen
language : en
Publisher:
Release Date : 2024-11-09
Brownian Motion And Potential Theory Modern And Classical written by Palle Jorgensen and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2024-11-09 with Mathematics categories.
In this book, potential theory is presented in an inclusive and accessible manner, with the emphasis reaching from classical to modern, from analytic to probabilistic, and from Newtonian to abstract or axiomatic potential theory (including Dirichlet spaces). The reader is guided through stochastic analysis featuring Brownian motion in its early chapters to potential theory in its latter sections. This path covers the following themes: martingales, diffusion processes, semigroups and potential operators, analysis of super harmonic functions, Dirichlet problems, balayage, boundaries, and Green functions. The wide range of applications encompasses random walk models, especially reversible Markov processes, and statistical inference in machine learning models. However, the present volume considers the analysis from the point of view of function space theory, using Dirchlet energy as an inner product. This present volume is an expanded and revised version of an original set of lectures in the Aarhus University Mathematics Institute Lecture Note Series.
Classical Potential Theory And Its Probabilistic Counterpart
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Author : J. L. Doob
language : en
Publisher: Springer Science & Business Media
Release Date : 2012-12-06
Classical Potential Theory And Its Probabilistic Counterpart written by J. L. Doob and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-12-06 with Mathematics categories.
Potential theory and certain aspects of probability theory are intimately related, perhaps most obviously in that the transition function determining a Markov process can be used to define the Green function of a potential theory. Thus it is possible to define and develop many potential theoretic concepts probabilistically, a procedure potential theorists observe withjaun diced eyes in view of the fact that now as in the past their subject provides the motivation for much of Markov process theory. However that may be it is clear that certain concepts in potential theory correspond closely to concepts in probability theory, specifically to concepts in martingale theory. For example, superharmonic functions correspond to supermartingales. More specifically: the Fatou type boundary limit theorems in potential theory correspond to supermartingale convergence theorems; the limit properties of monotone sequences of superharmonic functions correspond surprisingly closely to limit properties of monotone sequences of super martingales; certain positive superharmonic functions [supermartingales] are called "potentials," have associated measures in their respective theories and are subject to domination principles (inequalities) involving the supports of those measures; in each theory there is a reduction operation whose properties are the same in the two theories and these reductions induce sweeping (balayage) of the measures associated with potentials, and so on.
Classical Fine Potential Theory
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Author : Mohamed El Kadiri
language : en
Publisher: Springer Nature
Release Date : 2025-04-03
Classical Fine Potential Theory written by Mohamed El Kadiri and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2025-04-03 with Mathematics categories.
This comprehensive book explores the intricate realm of fine potential theory. Delving into the real theory, it navigates through harmonic and subharmonic functions, addressing the famed Dirichlet problem within finely open sets of R^n. These sets are defined relative to the coarsest topology on R^n, ensuring the continuity of all subharmonic functions. This theory underwent extensive scrutiny starting from the 1970s, particularly by Fuglede, within the classical or axiomatic framework of harmonic functions. The use of methods from fine potential theory has led to solutions of important classical problems and has allowed the discovery of elegant results for extension of classical holomorphic function to wider classes of “domains”. Moreover, this book extends its reach to the notion of plurisubharmonic and holomorphic functions within plurifinely open sets of C^n and its applications to pluripotential theory. These open sets are defined by coarsest topology that renders all plurisubharmonic functions continuous on C^n. The presentation is meticulously crafted to be largely self-contained, ensuring accessibility for readers at various levels of familiarity with the subject matter. Whether delving into the fundamentals or seeking advanced insights, this book is an indispensable reference for anyone intrigued by potential theory and its myriad applications. Organized into five chapters, the first four unravel the intricacies of fine potential theory, while the fifth chapter delves into plurifine pluripotential theory.
Foundations Of Modern Probability
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Author : Olav Kallenberg
language : en
Publisher: Springer Nature
Release Date : 2021-02-07
Foundations Of Modern Probability written by Olav Kallenberg and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2021-02-07 with Mathematics categories.
The first edition of this single volume on the theory of probability has become a highly-praised standard reference for many areas of probability theory. Chapters from the first edition have been revised and corrected, and this edition contains four new chapters. New material covered includes multivariate and ratio ergodic theorems, shift coupling, Palm distributions, Harris recurrence, invariant measures, and strong and weak ergodicity.
Potential Theory
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Author : Jürgen Bliedtner
language : en
Publisher: Springer Science & Business Media
Release Date : 2012-12-06
Potential Theory written by Jürgen Bliedtner and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-12-06 with Mathematics categories.
During the last thirty years potential theory has undergone a rapid development, much of which can still only be found in the original papers. This book deals with one part of this development, and has two aims. The first is to give a comprehensive account of the close connection between analytic and probabilistic potential theory with the notion of a balayage space appearing as a natural link. The second aim is to demonstrate the fundamental importance of this concept by using it to give a straight presentation of balayage theory which in turn is then applied to the Dirichlet problem. We have considered it to be beyond the scope of this book to treat further topics such as duality, ideal boundary and integral representation, energy and Dirichlet forms. The subject matter of this book originates in the relation between classical potential theory and the theory of Brownian motion. Both theories are linked with the Laplace operator. However, the deep connection between these two theories was first revealed in the papers of S. KAKUTANI [1], [2], [3], M. KAC [1] and J. L. DO DB [2] during the period 1944-54: This can be expressed by the·fact that the harmonic measures which occur in the solution of the Dirichlet problem are hitting distri butions for Brownian motion or, equivalently, that the positive hyperharmonic func tions for the Laplace equation are the excessive functions of the Brownian semi group.
Markov Processes And Potential Theory
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Author : Robert McCallum Blumenthal
language : en
Publisher: Courier Corporation
Release Date : 2007-01-01
Markov Processes And Potential Theory written by Robert McCallum Blumenthal and has been published by Courier Corporation this book supported file pdf, txt, epub, kindle and other format this book has been release on 2007-01-01 with Mathematics categories.
This graduate-level text explores the relationship between Markov processes and potential theory, in addition to aspects of the theory of additive functionals. Topics include Markov processes, excessive functions, multiplicative functionals and subprocesses, and additive functionals and their potentials. A concluding chapter examines dual processes and potential theory. 1968 edition.
Potential Theory Selected Topics
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Author : Hiroaki Aikawa
language : en
Publisher: Springer
Release Date : 2006-11-14
Potential Theory Selected Topics written by Hiroaki Aikawa and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2006-11-14 with Mathematics categories.
The first part of these lecture notes is an introduction to potential theory to prepare the reader for later parts, which can be used as the basis for a series of advanced lectures/seminars on potential theory/harmonic analysis. Topics covered in the book include minimal thinness, quasiadditivity of capacity, applications of singular integrals to potential theory, L(p)-capacity theory, fine limits of the Nagel-Stein boundary limit theorem and integrability of superharmonic functions. The notes are written for an audience familiar with the theory of integration, distributions and basic functional analysis.
Function Spaces And Potential Theory
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Author : David R. Adams
language : en
Publisher: Springer Science & Business Media
Release Date : 2012-12-06
Function Spaces And Potential Theory written by David R. Adams and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-12-06 with Mathematics categories.
Function spaces, especially those spaces that have become known as Sobolev spaces, and their natural extensions, are now a central concept in analysis. In particular, they play a decisive role in the modem theory of partial differential equations (PDE). Potential theory, which grew out of the theory of the electrostatic or gravita tional potential, the Laplace equation, the Dirichlet problem, etc. , had a fundamen tal role in the development of functional analysis and the theory of Hilbert space. Later, potential theory was strongly influenced by functional analysis. More re cently, ideas from potential theory have enriched the theory of those more general function spaces that appear naturally in the study of nonlinear partial differential equations. This book is motivated by the latter development. The connection between potential theory and the theory of Hilbert spaces can be traced back to C. F. Gauss [181], who proved (with modem rigor supplied almost a century later by O. Frostman [158]) the existence of equilibrium potentials by minimizing a quadratic integral, the energy. This theme is pervasive in the work of such mathematicians as D. Hilbert, Ch. -J. de La Vallee Poussin, M. Riesz, O. Frostman, A. Beurling, and the connection was made particularly clear in the work of H. Cartan [97] in the 1940's. In the thesis of J. Deny [119], and in the subsequent work of J. Deny and J. L.
Quantum Potential Theory
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Author : Philippe Biane
language : en
Publisher: Springer Science & Business Media
Release Date : 2008-09-23
Quantum Potential Theory written by Philippe Biane and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2008-09-23 with Mathematics categories.
This book offers the revised and completed notes of lectures given at the 2007 conference, "Quantum Potential Theory: Structures and Applications to Physics." These lectures provide an introduction to the theory and discuss various applications.