Random Fields and Geometry

This self-contained monograph focuses on recent important developments in the study of random fields, stochastic processes defined over high dimensional parameter spaces. While it replaces Adler's 1981 classic The Geometry of Random Fields, this is not an update, but a completely new work with a completely new way of handling both the Geometry and the Probability that are its central themes.There are three quite distinct parts to the monograph. Part I provides a comprehensive background to the general theory of Gaussian random fields, treating classical topics such as continuity and boundedness, entropy and majorising measures, Borell and Slepian inequalities. The treatment is didactic and user-friendly. Part II is about Geometry, both integral and Riemannian, and the material included here is what is needed for the over-riding probabilistic theme of the book. It contains a quick review of both these geometric settings, followed by carefully presented introductions to topics such as Crofton formulae, curvature measures for stratified manifolds, critical point theory and tube formulae. This is the only place in which all these topics, necessary for the study of random fields can be found in a concise, self-contained, treatment. The most important part of the book is in Part III, which is about the geometry of excursion sets of random fields and the related Euler characteristic approach to extremal probabilities. This part contains path-breaking material of both theoretical and practical importance and is unique in the way in which it intertwines probabilistic and geometric problems.Applications of this theory, which are significant and cover areas as widespread as brain imaging, physical oceanography and astrophysics, will be treated in a separate volume with Keith Worsley.This monograph will be of interest to probabilists and statisticians, both applied and theoretical, along with mathematicians interested in learning about new relationships between geometry and probability. It is also a basic reference text for those who will eventually be interested mainly in the companion volume of applications. Given the clear and pedagogical style of the book and the current importance of research in random fields this comprehensive and definitive work will serve as an indispensable reference work, while at the same time being an excellent text for self study and graduate courses in probability, statistics, analysis and geometry. This monograph is devoted to a completely new approach to geometric problems arising in the study of random fields. The groundbreaking material in Part III, for which the background is carefully prepared in Parts I and II, is of both theoretical and practical importance, and striking in the way in which problems arising in geometry and probability are beautifully intertwined.§§The three parts to the monograph are quite distinct. Part I presents a user-friendly yet comprehensive background to the general theory of Gaussian random fields, treating classical topics such as continuity and boundedness, entropy and majorizing measures, Borell and Slepian inequalities. Part II gives a quick review of geometry, both integral and Riemannian, to provide the reader with the material needed for Part III, and to give some new results and new proofs of known results along the way. Topics such as Crofton formulae, curvature measures for stratified manifolds, critical point theory, and tube formulae are covered. In fact, this is the only concise, self-contained treatment of all of the above topics, which are necessary for the study of random fields. The new approach in Part III is devoted to the geometry of excursion sets of random fields and the related Euler characteristic approach to extremal probabilities.§§"Random Fields and Geometry" will be useful for probabilists and statisticians, and for theoretical and applied mathematicians who wish to learn about new relationships between geometry and probability. It will be helpful for graduate students in a classroom setting, or for self-study. Finally, this text will serve as a basic reference for all those interested in the companion volume of the applications of the theory. These applications, to appear in a forthcoming volume, will cover areas as widespread as brain imaging, physical oceanography, and astrophysics.