Potential theory. An analytic and probabilistic approach to balayage.

*(English)*Zbl 0706.31001
Universitext. Berlin etc.: Springer-Verlag. xi, 434 p. DM 98.00 (1986).

There are two kinds of advanced mathematical books. Some of them make available to the public recent progress in rapidly growing fields, and some organize vast areas of knowledge, to prevent it from being lost when the thread connecting papers disseminated in tens of journals will be forgotten. This excellent book belongs to the second kind, and seems to be the only work in print that gives a general account of modern potential theory.

Besides that, this book contains a large amount of introductory material of great interest to the general reader. It starts with a short introduction to classical potential theory and its relation with Brownian motion. Chapter I contains useful functional analytic tools, among which some are now standard, like analytic sets and capacities, the Choquet boundary, or Stampacchia’s theorem on coercive bilinear forms, but at least one, Choquet’s theory of adapted cones, is of great importance and doesn’t seem to have ever appeared in book form. Chapter II and IV are quick and self-contained introductions to semigroups and resolvents, and to the theory of Markov processes. All this is concise and elegant. Again, in the last chapters there is much introductory material of high value, concerning the Dirichlet problem or hypoelliptic second order partial differential equations. The last chapter contains, among other results, Bony’s theorem that a harmonic space whose harmonic functions are smooth is associated with a second order differential operator.

Readers of modern expositions of potential theory sometimes complain that they are too abstract, and leave it to the reader to work out the applications for himself. This isn’t the case here. Several examples are followed throughout the book with explicit computations, among which the Newton and Green potentials of the classical theory, the Riesz potentials, and the heat equation. Essential results of classical potential theory are included, though they may be a little difficult to find because they are disseminated. To compensate for this, a guide to standard examples has been included at the end.

The main topic of the book is the construction (due to the authors themselves) of a system of analytic axioms (“balayage spaces”) for potential theory, whose generality is comparable to that of the probabilistic theory (in contrast to this, the theory of harmonic spaces corresponds to a probabilistic theory excluding jumps). The main restriction is the fact that all excessive (positive superharmonic) functions are lower semicontinuous as in the classical theory, and sufficiently many are continuous. The proofs are analytical, though probabilistic interpretations are given. The authors’ theorem that such axioms lead to nice semigroups and to Hunt processes is fully proved. Other pieces of the authors’ own research are included, like their beautiful results on semi-polar sets.

The literature on modern potential theory is scanty. The extant books either have been written by probabilists (sometimes even for probabilists), or are devoted to more special subjects: the classical theory, harmonic spaces, Dirichlet forms. Thus it seems that Bliedtner and Hansen’s book will become a basic reference on the subject. It also seems very appropriate for teaching.

Besides that, this book contains a large amount of introductory material of great interest to the general reader. It starts with a short introduction to classical potential theory and its relation with Brownian motion. Chapter I contains useful functional analytic tools, among which some are now standard, like analytic sets and capacities, the Choquet boundary, or Stampacchia’s theorem on coercive bilinear forms, but at least one, Choquet’s theory of adapted cones, is of great importance and doesn’t seem to have ever appeared in book form. Chapter II and IV are quick and self-contained introductions to semigroups and resolvents, and to the theory of Markov processes. All this is concise and elegant. Again, in the last chapters there is much introductory material of high value, concerning the Dirichlet problem or hypoelliptic second order partial differential equations. The last chapter contains, among other results, Bony’s theorem that a harmonic space whose harmonic functions are smooth is associated with a second order differential operator.

Readers of modern expositions of potential theory sometimes complain that they are too abstract, and leave it to the reader to work out the applications for himself. This isn’t the case here. Several examples are followed throughout the book with explicit computations, among which the Newton and Green potentials of the classical theory, the Riesz potentials, and the heat equation. Essential results of classical potential theory are included, though they may be a little difficult to find because they are disseminated. To compensate for this, a guide to standard examples has been included at the end.

The main topic of the book is the construction (due to the authors themselves) of a system of analytic axioms (“balayage spaces”) for potential theory, whose generality is comparable to that of the probabilistic theory (in contrast to this, the theory of harmonic spaces corresponds to a probabilistic theory excluding jumps). The main restriction is the fact that all excessive (positive superharmonic) functions are lower semicontinuous as in the classical theory, and sufficiently many are continuous. The proofs are analytical, though probabilistic interpretations are given. The authors’ theorem that such axioms lead to nice semigroups and to Hunt processes is fully proved. Other pieces of the authors’ own research are included, like their beautiful results on semi-polar sets.

The literature on modern potential theory is scanty. The extant books either have been written by probabilists (sometimes even for probabilists), or are devoted to more special subjects: the classical theory, harmonic spaces, Dirichlet forms. Thus it seems that Bliedtner and Hansen’s book will become a basic reference on the subject. It also seems very appropriate for teaching.

Reviewer: P.A.Meyer

##### MSC:

31-02 | Research exposition (monographs, survey articles) pertaining to potential theory |

31B35 | Connections of harmonic functions with differential equations in higher dimensions |

60J45 | Probabilistic potential theory |

60J25 | Continuous-time Markov processes on general state spaces |

65H10 | Numerical computation of solutions to systems of equations |