Classical potential theory and its probabilistic counterpart.

*(English)*Zbl 0549.31001
Grundlehren der Mathematischen Wissenschaften, 262. New York etc.: Springer-Verlag., XXIII, 846 p. DM 168.00; $ 62.70 (1984).

From the introduction: ”The purpose of this book is to develop the correspondence between potential theory and probability theory by examining in detail classical potential theory, that is, the potential theory of Laplace’s equation, together with the corresponding probability theory, that is, martingale theory. The joining link which makes this correspondence especially perspicuous is the Brownian motion process, so this process is studied as needed. In order to carry through this program it is necessary to study parabolic potential theory, that is, the potential theory of the heat equation, and the corresponding process of space time Brownian motion. No knowledge of potential theory is presupposed but it is assumed that the reader is familiar with basic probability concepts through conditional expectations. The necessary lattice theory, analytic set theory and capacity theory are covered in the Appendices.”

”One natural criticism of this project is that there is no reason to treat the very special potential theories of the Laplace and heat equations rather than general axiomatic potential theory. Another criticism is that there is no reason to treat potential theory other than as a special subhead of Markov process theory. In the author’s opinion, however, classical potential theory is too important to serve merely as a source of illustrations of axiomatic potential theory, which theory in turn is too important in its own right to be left to the probabilists.”

The author made the effort to make this book into an encyclopedia.

”One natural criticism of this project is that there is no reason to treat the very special potential theories of the Laplace and heat equations rather than general axiomatic potential theory. Another criticism is that there is no reason to treat potential theory other than as a special subhead of Markov process theory. In the author’s opinion, however, classical potential theory is too important to serve merely as a source of illustrations of axiomatic potential theory, which theory in turn is too important in its own right to be left to the probabilists.”

The author made the effort to make this book into an encyclopedia.

Reviewer: A.Spătaru

##### MSC:

31-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to potential theory |

31A05 | Harmonic, subharmonic, superharmonic functions in two dimensions |

31D05 | Axiomatic potential theory |

60J45 | Probabilistic potential theory |