Positive harmonic functions and diffusion: an integrated analytic and probabilistic approach.

*(English)*Zbl 0858.31001
Cambridge Studies in Advanced Mathematics. 45. Cambridge: Cambridge University Press. xvi, 474 p. (1995).

The author gives a thorough account of the theory of positive harmonic functions for second order elliptic operators, using an integrated probabilistic and analytic approach. Many results that form the folklore of the subject or are published by the author and many others somewhere during the last decades are given here a rigorous and largely self-contained exposition. At quite a number of places, the author presents his own proof, and some of the results are entirely new.

The book begins with a treatment of the construction and basic properties of diffusion processes (Itô processes, stochastic differential equations, Stroock-Varadhan solution of the martingale problem, Feynman-Kac formula, transience and recurrence). This might make the book accessible also to analysts having little probabilistic background.

Having recalled a number of standard results from functional analysis and elliptic PDE theory, the author discusses the (generalized) principal eigenvalue \(\lambda_c\) for elliptic operators \(L= {1\over 2} \nabla \cdot a \nabla+b \cdot \nabla+V\) on arbitrary domains \(D\) in \(\mathbb{R}^d\) (assuming local Hölder continuity of \(V\) and the partial derivatives of \(a_{ij}\) and \(b_i)\). This includes a characterization by the existence of Green’s function for \(L-\lambda\), \(\lambda > \lambda_c\), on \(D\). A small chapter is devoted to the one-dimensional case and the radially symmetric multidimensional case. Another chapter deals with criteria for transience or recurrence and explosion or non-explosion of the associated diffusion (Lyapunov method, variational approach).

Of course, the associated Martin boundary theory forms an essential part of the book. In particular, it is shown that the Martin boundary for \(L\) on a domain \(D\) coincides with the exterior harmonic measure boundary for the adjoint \(\widetilde L\) on \(D\). For several classes of operators, the Martin boundary is explicitly calculated. A short discussion of the theory of bounded harmonic functions and Brownian motion on manifolds of negative curvature finishes the book.

Each chapter is complemented by many exercises. In some of them the reader is asked to supply a proof or a step of a proof omitted from the text. For that reason, generous hints are given. Each chapter ends with historical notes giving many references.

Apparently, the book is written with great care, and it is a pleasure to read it. I have no doubt that it will be an excellent guide for those beginning to work in the field. And for the specialist it will offer new details and may serve as a useful reference.

The book begins with a treatment of the construction and basic properties of diffusion processes (Itô processes, stochastic differential equations, Stroock-Varadhan solution of the martingale problem, Feynman-Kac formula, transience and recurrence). This might make the book accessible also to analysts having little probabilistic background.

Having recalled a number of standard results from functional analysis and elliptic PDE theory, the author discusses the (generalized) principal eigenvalue \(\lambda_c\) for elliptic operators \(L= {1\over 2} \nabla \cdot a \nabla+b \cdot \nabla+V\) on arbitrary domains \(D\) in \(\mathbb{R}^d\) (assuming local Hölder continuity of \(V\) and the partial derivatives of \(a_{ij}\) and \(b_i)\). This includes a characterization by the existence of Green’s function for \(L-\lambda\), \(\lambda > \lambda_c\), on \(D\). A small chapter is devoted to the one-dimensional case and the radially symmetric multidimensional case. Another chapter deals with criteria for transience or recurrence and explosion or non-explosion of the associated diffusion (Lyapunov method, variational approach).

Of course, the associated Martin boundary theory forms an essential part of the book. In particular, it is shown that the Martin boundary for \(L\) on a domain \(D\) coincides with the exterior harmonic measure boundary for the adjoint \(\widetilde L\) on \(D\). For several classes of operators, the Martin boundary is explicitly calculated. A short discussion of the theory of bounded harmonic functions and Brownian motion on manifolds of negative curvature finishes the book.

Each chapter is complemented by many exercises. In some of them the reader is asked to supply a proof or a step of a proof omitted from the text. For that reason, generous hints are given. Each chapter ends with historical notes giving many references.

Apparently, the book is written with great care, and it is a pleasure to read it. I have no doubt that it will be an excellent guide for those beginning to work in the field. And for the specialist it will offer new details and may serve as a useful reference.

Reviewer: W.Hansen (Bielefeld)

##### MSC:

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

31C05 | Harmonic, subharmonic, superharmonic functions on other spaces |

31C35 | Martin boundary theory |

60G44 | Martingales with continuous parameter |

35J25 | Boundary value problems for second-order elliptic equations |

60H05 | Stochastic integrals |

60H15 | Stochastic partial differential equations (aspects of stochastic analysis) |

60J60 | Diffusion processes |