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Random thoughts on reversible potential theory. (English) Zbl 0757.31007

This is a discursive discussion of several subjects in potential theory and probability theory and their interrelations. It is part of the author’s praiseworthy purpose to bring closer together the point of view of axiomatic potential theory and techniques of analysis such as Sobolev type inequalities, heat kernels, elliptic operators, harmonic functions, boundary behaviour,…. It is humanly impossible to render coherent such a multitude of concepts and ideas and the extensive theories built upon them in less than several volumes. For this reason the present series of lectures is unavoidably bewildering, at least for most readers, this reviewer believes.
An idea of the extensive areas of mathematics involved might be obtained from the References at the end of the article and from the following list: axiomatic potential theory (SĂ©minaire Brelot, Cartan, Choquet, Deny,…), Dirichlet spaces (Beurling and Deny), reversible Markov processes, harmonic functions and boundary theory (Martin), Sobolev inequalities and pointwise estimates on heat kernels (Varopoulos and co- workers), information theory (R. M. Fano) and entropy (D. Sullivan), hyperbolic groups in the sense of Gromov, harmonic functions on groups, Riemannian manifolds, ... and much more. The final two chapters deal with examples, counter-examples and future directions.
Relatively few theorems are stated, sometimes with references to the proofs. The conversational style of exposition is quite different from the usual Def., Thm., Proof.

MSC:

31C35 Martin boundary theory
60J45 Probabilistic potential theory
43A35 Positive definite functions on groups, semigroups, etc.
31-02 Research exposition (monographs, survey articles) pertaining to potential theory
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