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Providing a novel approach to sparsity, this comprehensive book presents the theory of stochastic processes that are ruled by linear stochastic differential equations, and that admit a parsimonious representation in a matched wavelet-like basis. Two key themes are the statistical property of infinite divisibility, which leads to two distinct types of behaviour - Gaussian and sparse - and the structural link between linear stochastic processes and spline functions, which is exploited to simplify the mathematical analysis. The core of the book is devoted to investigating sparse processes, including a complete description of their transform-domain statistics. The final part develops practical signal-processing algorithms that are based on these models, with special emphasis on biomedical image reconstruction. This is an ideal reference for graduate students and researchers with an interest in signal/image processing, compressed sensing, approximation theory, machine learning, or statistics.

Stochastic differential equations. --- Random fields. --- Gaussian processes.

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Many areas of continuum physics pose a challenge to physicists. What are the most general, admissible statistically homogeneous and isotropic tensor-valued random fields (TRFs)? Previously, only the TRFs of rank 0 were completely described. This book assembles a complete description of such fields in terms of one- and two-point correlation functions for tensors of ranks 1 through 4. Working from the standpoint of invariance of physical laws with respect to the choice of a coordinate system, spatial domain representations, as well as their wavenumber domain counterparts are rigorously given in full detail. The book also discusses, an introduction to a range of continuum theories requiring TRFs, an introduction to mathematical theories necessary for the description of homogeneous and isotropic TRFs, and a range of applications including a strategy for simulation of TRFs, ergodic TRFs, scaling laws of stochastic constitutive responses, and applications to stochastic partial differential equations. It is invaluable for mathematicians looking to solve problems of continuum physics, and for physicists aiming to enrich their knowledge of the relevant mathematical tools.

Field theory (Physics) --- Geometry, Differential. --- Tensor fields. --- Random fields.

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This volume contains refereed research or review papers presented at the 6th Seminar on Stochastic Processes, Random Fields and Applications, which took place at the Centro Stefano Franscini (Monte Verità) in Ascona, Switzerland, in May 2008. The seminar focused mainly on stochastic partial differential equations, especially large deviations and control problems, on infinite dimensional analysis, particle systems and financial engineering, especially energy markets and climate models. The book will be a valuable resource for researchers in stochastic analysis and professionals interested in stochastic methods in finance. Contributors: S. Albeverio S. Ankirchner V. Bogachev R. Brummelhuis Z. Brzeźniak R. Carmona C. Ceci J.M. Corcuera A.B. Cruzeiro G. Da Prato M. Fehr D. Filipović B. Goldys M. Hairer E. Hausenblas F. Hubalek H. Hulley P. Imkeller A. Jakubowski A. Kohatsu-Higa A. Kovaleva E. Kyprianou C. Léonard J. Lörinczi A. Malyarenko B. Maslowski J.C. Mattingly S. Mazzucchi L. Overbeck E. Platen M. Röckner M. Romito T. Schmidt R. Sircar W. Stannat K.-T. Sturm A. Toussaint L. Vostrikova J. Woerner Y. Xiao J.-C. Zambrini.

Random fields -- Congresses. --- Random fields. --- Stochastic analysis -- Congresses. --- Stochastic analysis. --- Stochastic analysis --- Random fields --- Mathematics --- Distribution (Probability theory) --- Physical Sciences & Mathematics --- Mathematical Statistics --- Fields, Random --- Mathematics. --- Probabilities. --- Probability Theory and Stochastic Processes. --- Stochastic processes --- Distribution (Probability theory. --- Distribution functions --- Frequency distribution --- Characteristic functions --- Probabilities --- Probability --- Statistical inference --- Combinations --- Chance --- Least squares --- Mathematical statistics --- Risk

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Transfer functions --- Fonctions de transfert. --- Fourier transformations --- Fourier, Transformations de. --- Markov random fields --- Markov, Champs aléatoires de. --- Markov, Champs aléatoires de.

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This is the second volume in a subseries of the Lecture Notes in Mathematics called Lévy Matters, which is published at irregular intervals over the years. Each volume examines a number of key topics in the theory or applications of Lévy processes and pays tribute to the state of the art of this rapidly evolving subject with special emphasis on the non-Brownian world.   The expository articles in this second volume cover two important topics in the area of Lévy processes. The first article by Serge Cohen reviews the most important findings on fractional Lévy fields to date in a self-contained piece, offering a theoretical introduction as well as possible applications and simulation techniques. The second article, by Alexey Kuznetsov, Andreas E. Kyprianou, and Victor Rivero, presents an up to date account of the theory and application of scale functions for spectrally negative Lévy processes, including an extensive numerical overview.

Lâevy processes --- Random fields --- Mathematics --- Physical Sciences & Mathematics --- Mathematical Theory --- Mathematical Statistics --- Lévy processes. --- Random fields. --- Mathematics. --- Probabilities. --- Mathematics, general. --- Probability Theory and Stochastic Processes. --- Probability --- Statistical inference --- Combinations --- Chance --- Least squares --- Mathematical statistics --- Risk --- Math --- Science --- Fields, Random --- Stochastic processes --- Random walks (Mathematics) --- Distribution (Probability theory. --- Distribution functions --- Frequency distribution --- Characteristic functions --- Probabilities --- Levy processes.