Quantitative Stochastic Homogenization and Large-Scale Regularity

by Scott Armstrong, Tuomo Kuusi, Jean-Christophe Mourrat

Overview

The focus of this book is the large-scale statistical behavior of solutions of divergence-form elliptic equations with random coefficients, which is closely related to the long-time asymptotics of reversible diffusions in random media and other basic models of statistical physics. Of particular interest is the quantification of the rate at which solutions converge to those of the limiting, homogenized equation in the regime of large scale separation, and the description of their fluctuations around this limit. This self-contained presentation gives a complete account of the essential ideas and fundamental results of this new theory of quantitative stochastic homogenization, including the latest research on the topic, and is supplemented with many new results. The book serves as an introduction to the subject for advanced graduate students and researchers working in partial differential equations, statistical physics, probability and related fields, as well as a comprehensive reference for experts in homogenization. Being the first text concerned primarily with stochastic (as opposed to periodic) homogenization and which focuses on quantitative results, its perspective and approach are entirely different from other books in the literature.

Product Details

ISBN-13:	9783030155476
Publisher:	Springer-Verlag New York, LLC
Publication date:	06/29/2020
Series:	Grundlehren der mathematischen Wissenschaften , #352
Edition description:	1st ed. 2019
Pages:	518
Product dimensions:	6.10(w) x 9.25(h) x (d)

About the Author

S. Armstrong: Currently Associate Professor at the Courant Institute at NYU. Received his PhD from University of California, Berkeley, in 2009 and previously held positions at Louisiana State University, The University of Chicago, Univ. of Wisconsin-Madison and the University of Paris-Dauphine with the CNRS.

T. Kuusi: Currently Professor at the University of Helsinki. He previously held positions at the University of Oulu and Aalto University. Received his PhD from Aalto University in 2007.

J.-C. Mourrat: Currently CNRS research scientist at Ecole Normale Supérieure in Paris. Previously held positions at ENS Lyon and EPFL in Lausanne. Received his PhD in 2010 jointly from Aix-Marseille University and PUC in Santiago, Chile.

Table of Contents

Preface.- Assumptions and examples.- Frequently asked questions.- Notation.- Introduction and qualitative theory.- Convergence of the subadditive quantities.- Regularity on large scales.- Quantitative description of first-order correctors.- Scaling limits of first-order correctors.- Quantitative two-scale expansions.- Calderon-Zygmund gradient Lsubp estimates.- Estimates for parabolic problems.- Decay of the parabolic semigroup.- Linear equations with nonsymmetric coefficients.- Nonlinear equations.- Appendices: A.The O_s notation.- B.Function spaces and elliptic equations on Lipschitz domains.- C.The Meyers Lsub{2+\delta} estimate.- D. Sobolev norms and heat flow.- Parabolic Green functions.- Bibliography.- Index.

Editorial Reviews

The text presents a nice collection of important results in the theory of shastic homogenization and regularity theory. … The text is a very well-written piece of work that is pleasant to read. It is an excellent resource both for experts and beginners in field of shastic homogenization.” (Alpár R. Mészáros, zbMATH 1482.60001, 2022)

From the Publisher