Classical Potential Theory
Title | Classical Potential Theory PDF eBook |
Author | David H. Armitage |
Publisher | Springer Science & Business Media |
Total Pages | 343 |
Release | 2012-12-06 |
Genre | Mathematics |
ISBN | 1447102339 |
A long-awaited, updated introductory text by the world leaders in potential theory. This essential reference work covers all aspects of this major field of mathematical research, from basic theory and exercises to more advanced topological ideas. The largely self-contained presentation makes it basically accessible to graduate students.
Classical Potential Theory and Its Probabilistic Counterpart
Title | Classical Potential Theory and Its Probabilistic Counterpart PDF eBook |
Author | J. L. Doob |
Publisher | Springer Science & Business Media |
Total Pages | 865 |
Release | 2012-12-06 |
Genre | Mathematics |
ISBN | 1461252083 |
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.
Potential Theory
Title | Potential Theory PDF eBook |
Author | Lester L. Helms |
Publisher | Springer Science & Business Media |
Total Pages | 494 |
Release | 2014-04-10 |
Genre | Mathematics |
ISBN | 1447164229 |
Potential Theory presents a clear path from calculus to classical potential theory and beyond, with the aim of moving the reader into the area of mathematical research as quickly as possible. The subject matter is developed from first principles using only calculus. Commencing with the inverse square law for gravitational and electromagnetic forces and the divergence theorem, the author develops methods for constructing solutions of Laplace's equation on a region with prescribed values on the boundary of the region. The latter half of the book addresses more advanced material aimed at those with the background of a senior undergraduate or beginning graduate course in real analysis. Starting with solutions of the Dirichlet problem subject to mixed boundary conditions on the simplest of regions, methods of morphing such solutions onto solutions of Poisson's equation on more general regions are developed using diffeomorphisms and the Perron-Wiener-Brelot method, culminating in application to Brownian motion. In this new edition, many exercises have been added to reconnect the subject matter to the physical sciences. This book will undoubtedly be useful to graduate students and researchers in mathematics, physics and engineering.
Potential Theory on Locally Compact Abelian Groups
Title | Potential Theory on Locally Compact Abelian Groups PDF eBook |
Author | C. van den Berg |
Publisher | Springer Science & Business Media |
Total Pages | 205 |
Release | 2012-12-06 |
Genre | Mathematics |
ISBN | 3642661289 |
Classical potential theory can be roughly characterized as the study of Newtonian potentials and the Laplace operator on the Euclidean space JR3. It was discovered around 1930 that there is a profound connection between classical potential 3 theory and the theory of Brownian motion in JR . The Brownian motion is determined by its semigroup of transition probabilities, the Brownian semigroup, and the connection between classical potential theory and the theory of Brownian motion can be described analytically in the following way: The Laplace operator is the infinitesimal generator for the Brownian semigroup and the Newtonian potential kernel is the" integral" of the Brownian semigroup with respect to time. This connection between classical potential theory and the theory of Brownian motion led Hunt (cf. Hunt [2]) to consider general "potential theories" defined in terms of certain stochastic processes or equivalently in terms of certain semi groups of operators on spaces of functions. The purpose of the present exposition is to study such general potential theories where the following aspects of classical potential theory are preserved: (i) The theory is defined on a locally compact abelian group. (ii) The theory is translation invariant in the sense that any translate of a potential or a harmonic function is again a potential, respectively a harmonic function; this property of classical potential theory can also be expressed by saying that the Laplace operator is a differential operator with constant co efficients.
Brownian Motion and Classical Potential Theory
Title | Brownian Motion and Classical Potential Theory PDF eBook |
Author | Sidney Port |
Publisher | Academic Press |
Total Pages | 264 |
Release | 1978-09-28 |
Genre | Mathematics |
ISBN |
Brownian Motion and Classical Potential Theory is a six-chapter text that discusses the connection between Brownian motion and classical potential theory. The first three chapters of this book highlight the developing properties of Brownian motion with results from potential theory. The subsequent chapters are devoted to the harmonic and superharmonic functions, as well as the Dirichlet problem. These topics are followed by a discussion on the transient potential theory of Green potentials, with an emphasis on the Newtonian potentials, as well as the recurrent potential theory of logarithmic potentials. The last chapters deal with the application of Brownian motion to obtain the main theorems of classical potential theory. This book will be of value to physicists, chemists, and biologists.
Foundations of Potential Theory
Title | Foundations of Potential Theory PDF eBook |
Author | Oliver Dimon Kellogg |
Publisher | Read Books Ltd |
Total Pages | 329 |
Release | 2011-03-23 |
Genre | Philosophy |
ISBN | 1446547833 |
The present volume gives a systematic treatment of potential functions. It takes its origin in two courses, one elementary and one advanced, which the author has given at intervals during the last ten years, and has a two-fold purpose first, to serve as an introduction for students whose attainments in the Calculus include some knowledge of partial derivatives and multiple and line integrals and secondly, to provide the reader with the fundamentals of the subject, so that he may proceed immediately to the applications, or to - the periodical literature of the day. It is inherent in the nature of the subject that physical intuition and illustration be appealed to freely, and this has been done. However, in order that the ok may present sound ideals to the student, and also serve the mathematician, both for purposes of reference and as a basis for further developments, the proofs have been given by rigorous methods. This has led, at a number of points, to results either not found elsewhere, or not readily accessible. Thus, Chapter IV contains a proof for the general regular region of the divergence theorem Gauss, or Greens theorem on the reduction of volume to surface integrals. The treatment of the fundamental existence theorems in Chapter XI by means of integral equations meets squarely the difficulties incident to the discontinuity of the kernel, and the same chapter gives an account of the most recent developments with respect to the Pirichlet problem. Exercises are introduced in the conviction that no mastery of a mathematical subject is possible without working with it. They are designed primarily to illustrate or extend the theory, although the desirability of requiring an occasional concrete numerical result has not been lost sight of.
Classical Potential Theory and Its Probabilistic Counterpart
Title | Classical Potential Theory and Its Probabilistic Counterpart PDF eBook |
Author | Joseph L. Doob |
Publisher | Springer Science & Business Media |
Total Pages | 866 |
Release | 2012-12-06 |
Genre | Mathematics |
ISBN | 3642565735 |
From the reviews: "Here is a momumental work by Doob, one of the masters, in which Part 1 develops the potential theory associated with Laplace's equation and the heat equation, and Part 2 develops those parts (martingales and Brownian motion) of stochastic process theory which are closely related to Part 1". --G.E.H. Reuter in Short Book Reviews (1985)