Ebook: 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. 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.;I. Harmonic Analysis -- § 1. Notation and Preliminaries -- § 2. Some Basic Results From Harmonic Analysis -- § 3. Positive Definite Functions -- § 4. Fourier Transformation of Positive Definite Measures -- § 5. Positive Definite Functions on ? -- § 6. Periodicity -- II. Negative Definite Functions and Semigroups -- § 7. Negative Definite Functions -- § 8. Convolution Semigroups -- § 9. Completely Monotone Functions and Bernstein Functions -- § 10. Examples of Negative Definite Functions and Convolution Semigroups -- § 11. Contraction Semigroups -- § 12. Translation Invariant Contraction Semigroups -- III. Potential Theory for Transient Convolution Semigroups -- § 13. Transient Convolution Semigroups -- § 14. Transient Convolution Semigroups on the Half-Axis and Integrals of Convolution Semigroups -- § 15. Convergence Lemmas and Potential Theoretic Principles -- § 16. Excessive Measures -- § 17. Fundamental Families Associated With Potential Kernels -- § 18. The Lévy Measure for a Convolution Semigroup -- Symbols -- General Index.