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**Diffusions on fractals.**
*(English)*
Zbl 0916.60069

Barlow, Martin T. et al., Lectures on probability theory and statistics. Ecole d’Eté de probabilités de Saint-Flour XXV - 1995. Lectures given at the summer school in Saint-Flour, France, July 10-26, 1995. Berlin: Springer. Lect. Notes Math. 1690, 1-121 (1998).

This paper relies on the lectures the author held at St. Flour in 1995. It is a course that the author presents in a very detailed and logically strengthened way. It contains eight chapters and has totally 121 pages. The purpose of this work is to present the most important results on the diffusion on finitely ramified fractals, more precisely to give a studious glimpse to the diffusion on some classes of regular self-similar sets. Some classical examples, such as diffusion on Sierpiński gasket among others are analysed in a deep focus. A class of well-behaving diffusions on metric spaces, namely “fractional diffusion” is defined and some remarkable results are presented. The tome also offers a brief introduction to the theory of Dirichlet form viewing its connection toward electrical resistances.

Using the analytical “Japanese” approach developed by Kusuoka, Kigami, Fukushima and others the author edifies the construction and enlights some properties of diffusion on a class of finitely ramified regular fractals. This book is on the highlights of actuality in its field of research, offering a program on the fundamental principles of this theory, and presenting, as its sparking issues, new and important results, in an elegantly composed style, which also perfectly suits the extreme accuracy of the mathematical demonstration. The rich bibliography has also to be mentioned. It contains most of the mathematical works in this area, even some of those would happen not to be mentioned in the text.

For the entire collection see [Zbl 0894.00045].

Using the analytical “Japanese” approach developed by Kusuoka, Kigami, Fukushima and others the author edifies the construction and enlights some properties of diffusion on a class of finitely ramified regular fractals. This book is on the highlights of actuality in its field of research, offering a program on the fundamental principles of this theory, and presenting, as its sparking issues, new and important results, in an elegantly composed style, which also perfectly suits the extreme accuracy of the mathematical demonstration. The rich bibliography has also to be mentioned. It contains most of the mathematical works in this area, even some of those would happen not to be mentioned in the text.

For the entire collection see [Zbl 0894.00045].

Reviewer: I.C.Vladimirescu (Craiova)