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Random Fields and Geometry de R. J. Adler

Random Fields and Geometry: Springer Monographs in Mathematics

Autor R. J. Adler, Jonathan E. Taylor
en Limba Engleză Hardback – 12 iun 2007

Descoperim în Random Fields and Geometry o lucrare de referință semnată de R. J. Adler și Jonathan E. Taylor, cercetători a căror autoritate în analiza stocastică fundamentează un volum dens, publicat în prestigioasa serie Springer Monographs in Mathematics. Autorii propun o delimitare riguroasă a conceptului de „câmp aleatoriu”, tratându-l ca un proces stocastic definit pe spații parametrice cu cel puțin o dimensiune, cu un accent deosebit pe varietăți. Suntem de părere că această abordare teoretică este esențială pentru înțelegerea geometriei funcțiilor aleatorii, oferind instrumentele matematice necesare pentru a analiza comportamentul proceselor în spații euclidiene și non-euclidiene.

Recomandăm acest volum pentru rigoarea cu care este structurat, pornind de la fundamentele proceselor și câmpurilor gaussiene, trecând prin inegalități și expansiuni ortogonale, până la o explorare profundă a geometriei integrale și diferențiale. Progresia materialului este logică: după stabilirea cadrului probabilistic, autorii introduc teoria punctelor critice și volumele tuburilor, culminând cu geometria câmpurilor aleatorii pe varietăți și volumele intrinseci medii. Acoperă aceeași arie tematică precum Stochastic Geometry, Spatial Statistics and Random Fields de Evgeny Spodarev, dar cu o abordare mult mai axată pe fundamentele matematice pure și pe geometria fină a suprafețelor gaussiene, lăsând aplicațiile statistice pentru volumul companion.

Spre deosebire de Topological Complexity of Smooth Random Functions, care oferă o introducere mai intuitivă și mai puțin tehnică, ediția de față reprezintă monografia completă, oferind demonstrații detaliate și un tratament exhaustiv al probabilităților de excursie pentru câmpuri netede. Este o piesă centrală pentru orice bibliotecă de matematică avansată, definind standardul în studiul intersecției dintre probabilități și geometrie.

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Specificații

ISBN-13: 9780387481128
ISBN-10: 0387481125
Pagini: 472
Ilustrații: XVIII, 454 p. 21 illus.
Dimensiuni: 160 x 241 x 31 mm
Greutate: 0.87 kg
Ediția:2007
Editura: Springer
Colecția Springer Monographs in Mathematics
Seria Springer Monographs in Mathematics

Locul publicării:New York, NY, United States

Public țintă

Research

De ce să citești această carte

Recomandăm această carte cercetătorilor și doctoranzilor în matematică care doresc să stăpânească teoria riguroasă a câmpurilor aleatorii. Cititorul câștigă o înțelegere profundă a modului în care geometria diferențială interacționează cu procesele stocastice. Este volumul fundamental pentru oricine studiază probabilitățile de excursie și topologia funcțiilor aleatorii, oferind baza teoretică necesară înainte de a trece la aplicații practice în fizică sau cosmologie.

Despre autor

R. J. Adler este un profesor și cercetător recunoscut la nivel internațional pentru contribuțiile sale în domeniul probabilităților și statisticii, fiind o figură centrală în dezvoltarea teoriei câmpurilor aleatorii. Alături de Jonathan E. Taylor, profesor la Stanford University specializat în statistică și geometrie diferențială, a format o echipă de cercetare care a redefinit intersecția dintre topologie și procesele stocastice. Taylor este cunoscut pentru aplicarea metodelor geometrice în analiza datelor de imagistică cerebrală, expertiză care infuzează rigoarea matematică a acestui volum cu o perspectivă modernă asupra structurilor complexe.

Descriere scurtă

Since the term “random ?eld’’ has a variety of different connotations, ranging from agriculture to statistical mechanics, let us start by clarifying that, in this book, a random ?eld is a stochastic process, usually taking values in a Euclidean space, and de?ned over a parameter space of dimensionality at least 1. Consequently, random processes de?ned on countable parameter spaces will not 1 appear here. Indeed, even processes on R will make only rare appearances and, from the point of view of this book, are almost trivial. The parameter spaces we like best are manifolds, although for much of the time we shall require no more than that they be pseudometric spaces. With this clari?cation in hand, the next thing that you should know is that this book will have a sequel dealing primarily with applications. In fact, as we complete this book, we have already started, together with KW (Keith Worsley), on a companion volume [8] tentatively entitled RFG-A,or Random Fields and Geometry: Applications. The current volume—RFG—concentrates on the theory and mathematical background of random ?elds, while RFG-A is intended to do precisely what its title promises. Once the companion volume is published, you will ?nd there not only applications of the theory of this book, but of (smooth) random ?elds in general.

Cuprins

Gaussian Processes.- Gaussian Fields.- Gaussian Inequalities.- Orthogonal Expansions.- Excursion Probabilities.- Stationary Fields.- Geometry.- Integral Geometry.- Differential Geometry.- Piecewise Smooth Manifolds.- Critical Point Theory.- Volume of Tubes.- The Geometry of Random Fields.- Random Fields on Euclidean Spaces.- Random Fields on Manifolds.- Mean Intrinsic Volumes.- Excursion Probabilities for Smooth Fields.- Non-Gaussian Geometry.

Recenzii

From the reviews:
Developing good bounds for the distribution of the suprema of a Gaussian field $f$, i.e., for the quantity $\Bbb{P}\{\sup_{t\in M}f(t)\ge u}$, has been for a long time both a difficult and an interesting subject of research. A thorough presentation of this problem is the main goal of the book under review, as is stated by the authors in its preface. The authors develop their results in the context of smooth Gaussian fields, where the parameter spaces $M$ are Riemannian stratified manifolds, and their approach is of a geometrical nature. The book is divided into three parts. Part I is devoted to the presentation of the necessary tools of Gaussian processes and fields. Part II concisely exposes the required prerequisites of integral and differential geometry. Finally, in part III, the kernel of the book, a formula for the expectation of the Euler characteristic function of an excursion set and its approximation to the distribution of the maxima of the field, is precisely established. The book is written in an informal style, which affords a very pleasant reading. Each chapter begins with a presentation of the matters to be addressed, and the footnotes, located throughout the text, serve as an indispensable complement and many times as historical references. The authors insist on the fact that this book should not only be considered as a theoretical adventure and they recommend a second volume where they develop indispensable applications which highlight all the power of their results. (José Rafael León for Mathematical Reviews)
"This book presents the modern theory of excursion probabilities and the geometry of excursion sets for … random fields defined on manifolds. ... The book is understandable for students … with a good background in analysis. ... The interdisciplinary nature of this book, the beauty and depth of the presented mathematical theory make it an indispensable part of every mathematical library and a bookshelfof all probabilists interested in Gaussian processes, random fields and their statistical applications." (Ilya S. Molchanov, Zentralblatt MATH, Vol. 1149, 2008)

Textul de pe ultima copertă

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.

Caracteristici

Recasts old topics in random fields by following a completely new way of handling both geometry and probability Significant exposition of the work of others in the field Excellent reference work as well as excellent work for self study