- Shows that the microscopic point of view is useful in choosing a real minimizer of a variational problem that determines an interface shape
- Is the first book to discuss the stochastic extension of the Sharp interface limit for non-random PDEs
- Is one of the few books dealing with the KPZ equation, a recent hot topic in probability theory
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- About this book
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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 ∇φ-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(φ)-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.
- Reviews
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“The book at hand discusses various aspects of random interfaces, both in static and in dynamic settings, from various points of view. … the book may serve as a good introductory text to several aspects of random interfaces.” (Leonid Petrov, Mathematical Reviews, February, 2018)
- Table of contents (5 chapters)
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Scaling Limits for Pinned Gaussian Random Interfaces in the Presence of Two Possible Candidates
Pages 1-28
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Dynamic Young Diagrams
Pages 29-79
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Stochastic Partial Differential Equations
Pages 81-92
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Sharp Interface Limits for a Stochastic Allen-Cahn Equation
Pages 93-110
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KPZ Equation
Pages 111-124
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Table of contents (5 chapters)
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Bibliographic Information
- Bibliographic Information
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- Book Title
- Lectures on Random Interfaces
- Authors
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- Tadahisa Funaki
- Series Title
- SpringerBriefs in Probability and Mathematical Statistics
- Copyright
- 2016
- Publisher
- Springer Singapore
- Copyright Holder
- The Author(s)
- eBook ISBN
- 978-981-10-0849-8
- DOI
- 10.1007/978-981-10-0849-8
- Softcover ISBN
- 978-981-10-0848-1
- Series ISSN
- 2365-4333
- Edition Number
- 1
- Number of Pages
- XII, 138
- Number of Illustrations
- 35 b/w illustrations, 9 illustrations in colour
- Topics
