Quantitative Stochastic Homogenization and Large-Scale Regularity
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Beschreibung
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.
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
Autor*in
Scott Armstrong
Themen in »Quantitative Stochastic Homogenization and Large-Scale Regularity«
stochastic homogenization large-scale regularity theory optimal error estimates Gaussian free field rates of convergence two-scale expansion calculus of variations 35B27, 60F17, 35B65 renormalization random walk in random environment random conductance model divergence-form elliptic equation invariance principle Green function partial differential equations
Stimmen zu »Quantitative Stochastic Homogenization and Large-Scale Regularity«
“The text presents a nice collection of important results in the theory of stochastic 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 stochastic homogenization.” (Alpár R. Mészáros, zbMATH 1482.60001, 2022)
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