Advertisement

Potential Theory on Locally Compact Abelian Groups

  • Christian Berg
  • Gunnar Forst

Part of the Ergebnisse der Mathematik und ihrer Grenzgebiete book series (MATHE2, volume 87)

Table of contents

  1. Front Matter
    Pages I-VII
  2. Christian Berg, Gunnar Forst
    Pages 1-38
  3. Christian Berg, Gunnar Forst
    Pages 39-96
  4. Christian Berg, Gunnar Forst
    Pages 97-190
  5. Back Matter
    Pages 191-200

About this book

Introduction

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.

Keywords

Differential operator Finite Invariant Potentialtheorie function

Authors and affiliations

  • Christian Berg
    • 1
  • Gunnar Forst
    • 1
  1. 1.University of KøbenhavnKøbenhavnDenmark

Bibliographic information

  • DOI https://doi.org/10.1007/978-3-642-66128-0
  • Copyright Information Springer-Verlag Berlin Heidelberg 1975
  • Publisher Name Springer, Berlin, Heidelberg
  • eBook Packages Springer Book Archive
  • Print ISBN 978-3-642-66130-3
  • Online ISBN 978-3-642-66128-0
  • Series Print ISSN 0071-1136
  • Buy this book on publisher's site