Stochastic and Integral Geometry

ISBN-13: 9783540788584 / Angielski / Twarda / 2008 / 693 str.

Rolf Schneider; Wolfgang Weil

Opis

Stochastic geometry deals with models for random geometric structures. Its early beginnings are found in playful geometric probability questions, and it has vigorously developed during recent decades, when an increasing number of real-world applications in various sciences required solid mathematical foundations. Integral geometry studies geometric mean values with respect to invariant measures and is, therefore, the appropriate tool for the investigation of random geometric structures that exhibit invariance under translations or motions. Stochastic and Integral Geometry provides the mathematically oriented reader with a rigorous and detailed introduction to the basic stationary models used in stochastic geometry - random sets, point processes, random mosaics - and to the integral geometry that is needed for their investigation. The interplay between both disciplines is demonstrated by various fundamental results. A chapter on selected problems about geometric probabilities and an outlook to non-stationary models are included, and much additional information is given in the section notes.

Recenzje

From the reviews:

"In 'Stochastic and Integral Geometry,' R. Schneider and W. Weil give priority to the basic concepts in stochastic geometry ... while keeping from integral geometry only what is relevant for applications in stochastic geometry. ... Each chapter section is concluded by notes in which the main references are cited and numerous possible extensions are discussed. ... Stochastic and Integral Geometry is a profound work by two eminent specialists which is essential reading for those willing to learn deep theory." (Pierre Calka, Mathematical Geosciences, Vol. 45, 2013)

"This book ... provides the systematic and exhaustive account of mathematical foundations of stochastic geometry with particular emphasis on tools from convex geometry. ... The thorough and up-to-date presentation in this text makes it an invaluable source for researchers pursuing studies not only in stochastic geometry, but also in convex geometry and various applications ... . an absolutely indispensable part of all mathematical libraries. ... also beneficial for personal collections of all mathematicians who ever deal with probability measures on spaces of geometric objects." (Ilya S. Molchanov, Zentralblatt MATH, Vol. 1175, 2010)

"The book presents a number of results that are otherwise scattered among an immense number of research papers and mostly provides full proofs for them. ... The most remarkable aspect of the book is the reader-friendly structure and the style in which it has been written. The book is also worth owning not only for those working in stochastic geometry and immediately related fields of theoretical and applied probability and spatial statistics. ... This book ... will be an essential part of every mathematical library." (V. K. Oganyan, Mathematical Reviews, Issue 2010 g)

Spis treści

Foundations of Stochastic Geometry.- Prolog.- Random Closed Sets.- Point Processes.- Geometric Models.- Integral Geometry.- Averaging with Invariant Measures.- Extended Concepts of Integral Geometry.- Integral Geometric Transformations.- Selected Topics from Stochastic Geometry.- Some Geometric Probability Problems.- Mean Values for Random Sets.- Random Mosaics.- Non-stationary Models.- Facts from General Topology.- Invariant Measures.- Facts from Convex Geometry.

O autorze

Rolf Schneider: Born 1940, Studies of Mathematics and Physics in Frankfurt/M, Diploma 1964, PhD 1967 (Frankfurt), Habilitation 1969 (Bochum), 1970 Professor TU Berlin, 1974 Professor Univ. Freiburg, 2003 Dr. h.c. Univ. Salzburg, 2005 Emeritus

Wolfgang Weil: Born 1945, Studies of Mathematics and Physics in Frankfurt/M, Diploma 1968, PhD 1971 (Frankfurt), Habilitation 1976 (Freiburg), 1978 Akademischer Rat Univ. Freiburg, 1980 Professor Univ. Karlsruhe

Blurb

Stochastic geometry has in recent years experienced considerable progress, both in its applications to other sciences and engineering, and in its theoretical foundations and mathematical expansion. This book, by two eminent specialists of the subject, provides a solid mathematical treatment of the basic models of stochastic geometry -- random sets, point processes of geometric objects (particles, flats), and random mosaics. It develops, in a measure-theoretic setting, the integral geometry for the motion and the translation group, as needed for the investigation of these models under the usual invariance assumptions. A characteristic of the book is the interplay between stochastic and geometric arguments, leading to various major results. Its main theme, once the foundations have been laid, is the quantitative investigation of the basic models. This comprises the introduction of suitable parameters, in the form of functional densities, relations between them, and approaches to their estimation. Much additional information on stochastic geometry is collected in the section notes.

As a combination of probability theory and geometry, the volume is intended for readers from either field. Probabilists with interest in random spatial structures, or motivated by the prospect of applications, will find an in-depth presentation of the geometric background. Geometers can see integral geometry "at work" and may be surprised to learn how classical results from convex geometry have elegant applications in a stochastic setting.