Search Results for potential-theory-on-locally-compact-abelian-groups

Classical potential theory can be roughly characterized as the study of Newtonian potentials and the Laplace operator on the Euclidean space JR3.

Author: C. van den Berg

Publisher: Springer Science & Business Media

ISBN: 9783642661280

Category: Mathematics

Page: 200

View: 398

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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.
2012-12-06 By C. van den Berg

Author: Christian Berg

Publisher:

ISBN: OCLC:848229534

Category: Locally compact Abelian groups

Page: 197

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1975 By Christian Berg

Chapter 7 Some thoughts on probability and analysis on locally compact groups We consider some problems concerning probabilities and potentials on locally compact groups. The three main objects of investigation are Gaussian measures and ...

Author: Alexander Bendikov

Publisher: Walter de Gruyter

ISBN: 9783110876840

Category: Mathematics

Page: 190

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The series is devoted to the publication of monographs and high-level textbooks in mathematics, mathematical methods and their applications. Apart from covering important areas of current interest, a major aim is to make topics of an interdisciplinary nature accessible to the non-specialist. The works in this series are addressed to advanced students and researchers in mathematics and theoretical physics. In addition, it can serve as a guide for lectures and seminars on a graduate level. The series de Gruyter Studies in Mathematics was founded ca. 30 years ago by the late Professor Heinz Bauer and Professor Peter Gabriel with the aim to establish a series of monographs and textbooks of high standard, written by scholars with an international reputation presenting current fields of research in pure and applied mathematics. While the editorial board of the Studies has changed with the years, the aspirations of the Studies are unchanged. In times of rapid growth of mathematical knowledge carefully written monographs and textbooks written by experts are needed more than ever, not least to pave the way for the next generation of mathematicians. In this sense the editorial board and the publisher of the Studies are devoted to continue the Studies as a service to the mathematical community. Please submit any book proposals to Niels Jacob.
1995-01-01 By Alexander Bendikov

87 Classical potential theory can be roughly characterized as the study of Newtonian potentials and the Laplace operator ... of classical potential theory are preserved : ( i ) The theory is defined on a locally compact abelian group .

Author: Christian Berg

Publisher: Springer

ISBN: 0387072497

Category: Locally compact Abelian groups

Page: 197

View: 302

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1975 By Christian Berg

Z. 148, 141–146 (1976) Berg, Ch., Forst, G.: Potential theory on locally compact Abelian groups. Berlin-HeidelbergNew York: Springer 1975 Berglund, J.F., Hofmann, K.H. : Compact semitopological semigroups and weakly almost periodic ...

Author: H. Heyer

Publisher: Springer Science & Business Media

ISBN: 9783642667060

Category: Mathematics

Page: 532

View: 410

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Probability measures on algebraic-topological structures such as topological semi groups, groups, and vector spaces have become of increasing importance in recent years for probabilists interested in the structural aspects of the theory as well as for analysts aiming at applications within the scope of probability theory. In order to obtain a natural framework for a first systematic presentation of the most developed part of the work done in the field we restrict ourselves to prob ability measures on locally compact groups. At the same time we stress the non Abelian aspect. Thus the book is concerned with a set of problems which can be regarded either from the probabilistic or from the harmonic-analytic point of view. In fact, it seems to be the synthesis of these two viewpoints, the initial inspiration coming from probability and the refined techniques from harmonic analysis which made this newly established subject so fascinating. The goal of the presentation is to give a fairly complete treatment of the central limit problem for probability measures on a locally compact group. In analogy to the classical theory the discussion is centered around the infinitely divisible probability measures on the group and their relationship to the convergence of infinitesimal triangular systems.
2012-12-06 By H. Heyer

[23] Berg C. and Forst G. Potential theory on locally compact abelian groups. Springer, New York–Heidelberg, 1975. [24] Bliedtner J. Harmonische Gruppen und Huntsche Faltungskerne. In: Seminar über Potentialtheorie, Lecture Notes in ...

Author: Pascal Auscher

Publisher: American Mathematical Soc.

ISBN: 9780821833834

Category: Mathematics

Page: 423

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This volume contains the expanded lecture notes of courses taught at the Emile Borel Centre of the Henri Poincare Institute (Paris) on heat kernels, random walks, and analysis on manifolds and graphs. In the book, leading experts introduce recent research in their fields. The unifying theme is the study of heat kernels in various situations using related geometric and analytic tools. Topics include analysis of complex-coefficient elliptic operators, diffusions on fractals and on infinite-dimensional groups, heat kernel and isoperimetry on Riemannian manifolds, heat kernels and infinite dimensional analysis, diffusions and Sobolev-type spaces on metric spaces, quasi-regular mappings and $p$-Laplace operators, heat kernel and spherical inversion on $SL_2(C)$, random walks and spectral geometry on crystal lattices, isoperimetric and isocapacitary inequalities, and generating function techniques for random walks on graphs. This volume is suitable for graduate students and research mathematicians interested in random processes and analysis on manifolds.
2003 By Pascal Auscher

Bibliography [ 1 ] D. L. Armacost , The structure of locally compact abelian groups . Monographs Textbooks Pure Appl . Math . ... [ 6 ] A. Bendikov , Potential theory on infinite - dimensional abelian groups . De Gruyter Stud . Math .

Author: Gennadiĭ Mikhaĭlovich Felʹdman

Publisher: European Mathematical Society

ISBN: 3037190450

Category: Mathematics

Page: 256

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This book deals with the characterization of probability distributions. It is well known that both the sum and the difference of two Gaussian independent random variables with equal variance are independent as well. The converse statement was proved independently by M. Kac and S. N. Bernstein. This result is a famous example of a characterization theorem. In general, characterization problems in mathematical statistics are statements in which the description of possible distributions of random variables follows from properties of some functions in these variables. In recent years, a great deal of attention has been focused upon generalizing the classical characterization theorems to random variables with values in various algebraic structures such as locally compact Abelian groups, Lie groups, quantum groups, or symmetric spaces. The present book is aimed at the generalization of some well-known characterization theorems to the case of independent random variables taking values in a locally compact Abelian group $X$. The main attention is paid to the characterization of the Gaussian and the idempotent distribution (group analogs of the Kac-Bernstein, Skitovich-Darmois, and Heyde theorems). The solution of the corresponding problems is reduced to the solution of some functional equations in the class of continuous positive definite functions defined on the character group of $X$. Group analogs of the Cramer and Marcinkiewicz theorems are also studied. The author is an expert in algebraic probability theory. His comprehensive and self-contained monograph is addressed to mathematicians working in probability theory on algebraic structures, abstract harmonic analysis, and functional equations. The book concludes with comments and unsolved problems that provide further stimulation for future research in the theory.

Its importance for potential theory on locally compact abelian groups is worked out in C. BERG – G. FORST [1]. Section 5. (5.1) is a generalization of the LAX-MILGRAM lemma due to G. STAMPACCHIA [1]. CHAPTER II Section 1.

Author: Jürgen Bliedtner

Publisher: Springer Science & Business Media

ISBN: 9783642711312

Category: Mathematics

Page: 435

View: 742

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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.
2012-12-06 By Jürgen Bliedtner

Berg, C., Forst, G. : Potential theory on locally compact abelian groups. Ergebn. d. Math. 87, Springer-Verlag (1975). Bliedtner, J. : Harmonische Gruppen und Huntsche Faltungskerne. In : "Seminar über Potential theorie", LN in Math.

Author: Josef Kral

Publisher: Springer

ISBN: 9783540459521

Category: Mathematics

Page: 278

View: 399

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The volume comprises eleven survey papers based on survey lectures delivered at the Conference in Prague in July 1987, which covered various facets of potential theory, including its applications in other areas. The survey papers deal with both classical and abstract potential theory and its relations to partial differential equations, stochastic processes and other branches such as numerical analysis and topology. A collection of problems from potential theory, compiled on the occasion of the conference, is included, with additional commentaries, in the second part of this volume.
2007-02-08 By Josef Kral

[13] Bercovici, H., Vittoriono, P.: Stable laws and domains of attraction in free probability theory. Ann. Math. 149, 1023–1060 (1999) (Cited on page 570.) [14] Berg, Ch., Forst, G.: Potential Theory on Locally Compact Abelian Groups.

Author: Wilfried Hazod

Publisher: Springer Science & Business Media

ISBN: 9789401730617

Category: Mathematics

Page: 612

View: 647

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Generalising classical concepts of probability theory, the investigation of operator (semi)-stable laws as possible limit distributions of operator-normalized sums of i.i.d. random variable on finite-dimensional vector space started in 1969. Currently, this theory is still in progress and promises interesting applications. Parallel to this, similar stability concepts for probabilities on groups were developed during recent decades. It turns out that the existence of suitable limit distributions has a strong impact on the structure of both the normalizing automorphisms and the underlying group. Indeed, investigations in limit laws led to contractable groups and - at least within the class of connected groups - to homogeneous groups, in particular to groups that are topologically isomorphic to a vector space. Moreover, it has been shown that (semi)-stable measures on groups have a vector space counterpart and vice versa. The purpose of this book is to describe the structure of limit laws and the limit behaviour of normalized i.i.d. random variables on groups and on finite-dimensional vector spaces from a common point of view. This will also shed a new light on the classical situation. Chapter 1 provides an introduction to stability problems on vector spaces. Chapter II is concerned with parallel investigations for homogeneous groups and in Chapter III the situation beyond homogeneous Lie groups is treated. Throughout, emphasis is laid on the description of features common to the group- and vector space situation. Chapter I can be understood by graduate students with some background knowledge in infinite divisibility. Readers of Chapters II and III are assumed to be familiar with basic techniques from probability theory on locally compact groups.
2013-03-14 By Wilfried Hazod