Quantitative Stochastic Homogenization and Large-Scale Regularity

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.

First book focusing on stochastic (as opposed to periodic) homogenization, presenting the quantitative theory, and exposing the renormalization approach to stochastic homogenization Collects the essential ideas and results of the theory of quantitative stochastic homogenization, including the optimal error estimates and scaling limit of the first-order correctors to a variant of the Gaussian free field Proves for the first time important new results, including optimal estimates for the first-order correctors in negative Sobolev spaces, optimal error estimates for Dirichlet and Neumann problems and the optimal quantitative description of the parabolic and elliptic Green functions Contains an original construction and interpretation of the Gaussian free field and the functional spaces to which it belongs, and an elementary new derivation of the heat kernel formulation of Sobolev space norms

Auteur
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.

Contenu
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 L^p 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 L^{2+\delta} estimate.- D. Sobolev norms and heat flow.- Parabolic Green functions.- Bibliography.- Index.