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作者 Funaki, Tadahisa, author
書名 Lectures on Random Interfaces / by Tadahisa Funaki
出版項 Singapore : Springer Singapore : Imprint : Springer, 2016
國際標準書號 9789811008498
9811008493
9789811008481
9811008485
國際標準號碼 10.1007/978-981-10-0849-8
book jacket
館藏地 索書號 處理狀態 OPAC 訊息 條碼
 數學所圖書室  QA274 F863 2016    在架上    30340200562571
說明 XII, 138 pages 44 illustrations, 9 illustrations in color.
系列 SpringerBriefs in Probability and Mathematical Statistics, 2365-4333
SpringerBriefs in Probability and Mathematical Statistics, 2365-4333
附註 Includes bibliographical references and index
1. Scaling limits for pinned gaussian random interfaces in the presence of two possible candidates -- 2. Dynamic young diagrams -- 3. Stochastic partial differential equations -- 4. Sharp interface limits for a stochastic Allen-Cahn equation -- 5. KPZ equation
Interfaces are created to separate two distinct phases in a situation in which phase coexistence occurs. This book discusses randomly fluctuating interfaces in several different settings and from several points of view: discrete/continuum, microscopic/macroscopic, and static/dynamic theories. The following four topics in particular are dealt with in the book. Assuming that the interface is represented as a height function measured from a fixed-reference discretized hyperplane, the system is governed by the Hamiltonian of gradient of the height functions. This is a kind of effective interface model called ∇?-interface model. The scaling limits are studied for Gaussian (or non-Gaussian) random fields with a pinning effect under a situation in which the rate functional of the corresponding large deviation principle has non-unique minimizers. Young diagrams determine decreasing interfaces, and their dynamics are introduced. The large-scale behavior of such dynamics is studied from the points of view of the hydrodynamic limit and non-equilibrium fluctuation theory. Vershik curves are derived in that limit. A sharp interface limit for the Allen-Cahn equation, that is, a reaction-diffusion equation with bistable reaction term, leads to a mean curvature flow for the interfaces. Its stochastic perturbation, sometimes called a time-dependent Ginzburg-Landau model, stochastic quantization, or dynamic P(?)-model, is considered. Brief introductions to Brownian motions, martingales, and stochastic integrals are given in an infinite dimensional setting. The regularity property of solutions of stochastic PDEs (SPDEs) of a parabolic type with additive noises is also discussed. The Kardar-Parisi-Zhang (KPZ) equation , which describes a growing interface with fluctuation, recently has attracted much attention. This is an ill-posed SPDE and requires a renormalization. Especially its invariant measures are studied.
主題 Mathematics
Differential equations, Partial
Probabilities
Mathematical physics
Mathematics
Probability Theory and Stochastic Processes
Partial Differential Equations
Mathematical Physics
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