EAN: 9783642367380

FRACTIONAL FIELDS AND APPLICATIONS - SERGE COHE...

This book focuses mainly on fractional Brownian fields and their extensions. It has been used
to teach graduate students at Grenoble and Toulouse's Universities. It is as self-contained as
possible and contains numerous exercises  with solutions in an appendix. After a foreword by
Stéphane Jaffard  a long first chapter is devoted to classical results from stochastic fields
and fractal analysis. A central notion throughout this book is self-similarity  which is dealt
with in a second chapter with a particular emphasis on the celebrated Gaussian self-similar
fields  called fractional Brownian fields after Mandelbrot and Van Ness's seminal paper.
Fundamental properties of fractional Brownian fields are then stated and proved. The second
central notion of this book is the so-called local asymptotic self-similarity (in short lass) 
which is a local version of self-similarity  defined in the third chapter. A lengthy study is
devoted to lass fields with finite variance. Among these lass fields  we find both Gaussian
fields and non-Gaussian fields  called Lévy fields. The Lévy fields can be viewed as bridges
between fractional Brownian fields and stable self-similar fields. A further key issue concerns
the identification of fractional parameters. This is the raison d'être of the statistics
chapter  where generalized quadratic variations methods are mainly used for estimating
fractional parameters. Last but not least  the simulation is addressed in the last chapter.
Unlike the previous issues  the simulation of fractional fields is still an area of ongoing
research. The algorithms presented in this chapter are efficient but do not claim to close the
debate.

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