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**Classical potential theory and its probabilistic counterpart.
Reprint of the 1984 edition.**
*(English)*
Zbl 0990.31001

Classics in Mathematics. Berlin: Springer. xxiii, 846 p. (2001).

It is good news that Doob’s monumental book is now available at a very reasonable price. The impressive volume (846 pages!) is still the only book concentrating on a thorough presentation of the potential theory of the Laplace operator (about 260 pages) and the potential theory of the heat operator (about 120 pages, developed in parallel by adding dots) and their probabilistic counterparts (in the second half of the book). The choice of the material is motivated by the author’s opinion that “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 potential theory of the Laplace operator covers familiar topics such as general properties of harmonic and superharmonic functions, the reduction operation, Riesz decomposition of a superharmonic function, polar sets, Green functions, Dirichlet problems, balayage of measures, fine topology, Martin boundary, energy, and capacity. A significant difference to other textbooks is the systematic treatment of potential theory relative to a strictly positive superharmonic function.

The probabilistic part starts out more generally presenting first some fundamental theory of stochastic processes (assuming that the reader is familiar with basic concepts of probability theory). After a short discussion of optional times and associated notions martingales and supermartingales are studied in great detail. General Markov processes are considered very quickly, before Brownian motion and the Itô integral are treated. After these preparations the following chapters show in detail how Brownian motion is related to (analytic) potential theory. The material in the chapters on conditional Brownian motion and Brownian motion on the Martin space cannot easily be found in that depth elsewhere.

A long appendix on various topics (more than 50 pages) and many historical notes complete this great “encyclopedia”.

The potential theory of the Laplace operator covers familiar topics such as general properties of harmonic and superharmonic functions, the reduction operation, Riesz decomposition of a superharmonic function, polar sets, Green functions, Dirichlet problems, balayage of measures, fine topology, Martin boundary, energy, and capacity. A significant difference to other textbooks is the systematic treatment of potential theory relative to a strictly positive superharmonic function.

The probabilistic part starts out more generally presenting first some fundamental theory of stochastic processes (assuming that the reader is familiar with basic concepts of probability theory). After a short discussion of optional times and associated notions martingales and supermartingales are studied in great detail. General Markov processes are considered very quickly, before Brownian motion and the Itô integral are treated. After these preparations the following chapters show in detail how Brownian motion is related to (analytic) potential theory. The material in the chapters on conditional Brownian motion and Brownian motion on the Martin space cannot easily be found in that depth elsewhere.

A long appendix on various topics (more than 50 pages) and many historical notes complete this great “encyclopedia”.

Reviewer: Wolfhard Hansen (Bielefeld)

### MSC:

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

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

31B05 | Harmonic, subharmonic, superharmonic functions in higher dimensions |

60J45 | Probabilistic potential theory |

31D05 | Axiomatic potential theory |

31C35 | Martin boundary theory |

31C40 | Fine potential theory; fine properties of sets and functions |