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1.

図書

図書
Vassili N. Kolokoltsov
出版情報: Cambridge, UK ; Tokyo : Cambridge University Press, 2010  xvii, 375 p. ; 23 cm
シリーズ名: Cambridge tracts in mathematics ; 182
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Preface
Basic definitions, notation and abbreviations
Introduction / 1:
Nonlinear Markov chains / 1.1:
Examples: replicator dynamics, the Lotka-Volterra equations, epidemics, coagulation / 1.2:
Interacting-particle approximation for discrete mass-exchange processes / 1.3:
Nonlinear Lévy processes and semigroups / 1.4:
Multiple coagulation, fragmentation and collisions; extended Smoluchovski and Boltzmann models / 1.5:
Replicator dynamics of evolutionary game theory / 1.6:
Interacting Markov processes; mean field and kth-order interactions / 1.7:
Classical kinetic equations of statistical mechanics: Vlasov, Boltzmann, Landau / 1.8:
Moment measures, correlation functions and the propagation of chaos / 1.9:
Nonlinear Markov processes and semigroups; nonlinear martingale problems / 1.10:
Tools from Markov process theory / Part I:
Probability and analysis / 2:
Semigroups, propagators and generators / 2.1:
Feller processes and conditionally positive operators / 2.2:
Jump-type Markov processes / 2.3:
Connection with evolution equations / 2.4:
Probabilistic constructions / 3:
Stochastic integrals and SDEs driven by nonlinear Lévy noise / 3.1:
Nonlinear version of Ito's approach to SDEs / 3.2:
Homogeneous driving noise / 3.3:
An alternative approximation scheme / 3.4:
Regularity of solutions / 3.5:
Coupling of Lévy processes / 3.6:
Analytical constructions / 4:
Comparing analytical and probabilistic tools / 4.1:
Integral generators: one-barrier case / 4.2:
Integral generators: two-barrier case / 4.3:
Generators of order at most one: well-posedness / 4.4:
Generators of order at most one: regularity / 4.5:
Further techniques: martingale problem, Sobolev spaces, heat kernels etc. / 4.6:
Unbounded coefficients / 5:
A growth estimate for Feller processes / 5.1:
Extending Feller processes / 5.2:
Invariant domains / 5.3:
Nonlinear Markov processes and semigroups / Part II:
Integral generators / 6:
Overview / 6.1:
Bounded generators / 6.2:
Additive bounds for rates: existence / 6.3:
Additive bounds for rates: well-posedness / 6.4:
A tool for proving uniqueness / 6.5:
Multiplicative bounds for rates / 6.6:
Another existence result / 6.7:
Conditional positivity / 6.8:
Generators of Lévy-Khintchine type / 7:
Variable coefficients via fixed-point arguments / 7.1:
Nonlinear SDE construction / 7.3:
Smoothness with respect to initial data / 7.4:
Motivation and plan; a warm-up result / 8.1:
Lévy-Khintchine-type generators / 8.2:
Jump-type models / 8.3:
Estimates for Smoluchovski's equation / 8.4:
Propagation and production of moments for the Boltzmann equation / 8.5:
Estimates for the Boltzmann equation / 8.6:
Applications to interacting particles / Part III:
The dynamic law of large numbers / 9:
Manipulations with generators / 9.1:
Interacting diffusions, stable-like and Vlasov processes / 9.2:
Pure jump models: probabilistic approach / 9.3:
Rates of convergence for Smoluchovski coagulation / 9.4:
Rates of convergence for Boltzmann collisions / 9.5:
The dynamic central limit theorem / 10:
Generators for fluctuation processes / 10.1:
Weak CLT with error rates: the Smoluchovski and Boltzmann models, mean field limits and evolutionary games / 10.2:
Summarizing the strategy followed / 10.3:
Infinite-dimensional Ornstein-Uhlenbeck processes / 10.4:
Full CLT for coagulation processes (a sketch) / 10.5:
Developments and comments / 11:
Measure-valued processes as stochastic dynamic LLNs for interacting particles; duality of one-dimensional processes / 11.1:
Discrete nonlinear Markov games and controlled processes; the modeling of deception / 11.2:
Nonlinear quantum dynamic semigroups and the nonlinear Schrödinger equation / 11.3:
Curvilinear Ornstein-Uhlenbeck processes (linear and nonlinear) and stochastic geodesic flows on manifolds / 11.4:
The structure of generators / 11.5:
Bibliographical comments / 11.6:
Appendices
Distances on measures / A:
Topology on càdlàg paths / B:
Convergence of processes in Skorohod spaces / C:
Vector-valued ODEs / D:
Pseudo-differential operator notation / E:
Variational derivatives / F:
Geometry of collisions / G:
A combinatorial lemma / H:
Approximation of infinite-dimensional functions / I:
Bogolyubov chains, generating functionals and Fock-space calculus / J:
Infinite-dimensional Riccati equations / K:
References
Index
Preface
Basic definitions, notation and abbreviations
Introduction / 1:
2.

図書

図書
Sigurd Assing, Wolfgang M. Schmidt
出版情報: Berlin : Springer, c1998  xii, 135 p. ; 24 cm
シリーズ名: Lecture notes in mathematics ; 1688
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3.

図書

図書
Р.Л. Стратонович
出版情報: Москва : Изд-во Московского университета., 1966  318 p. ; 22 cm
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4.

図書

図書
Е.Б. Дынкин
出版情報: Москва : Гос. изд-во физико-математической лит-ры, 1963  859 p. ; 21 cm
シリーズ名: Теория вероятностей и математическая статистика
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5.

図書

図書
Claude Kipnis, Claudio Landim
出版情報: Berlin : Springer-Verlag, c1999  xvi, 442 p. ; 25 cm
シリーズ名: Die Grundlehren der mathematischen Wissenschaften ; 320
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6.

図書

図書
Gregory F. Lawler
出版情報: Providence, R.I. : American Mathematical Society, c2005  xi, 242 p. ; 27 cm
シリーズ名: Mathematical surveys and monographs ; v. 114
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7.

図書

図書
Onésimo Hernández-Lerma, Jean B. Lasserre
出版情報: New York ; Tokyo : Springer, c1999  xii, 276 p. ; 24 cm
シリーズ名: Applications of mathematics ; 42
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8.

図書

図書
Harry Kesten (editor)
出版情報: Berlin ; Tokyo : Springer, c2004  vi, 351 p. ; 25 cm
シリーズ名: Encyclopaedia of mathematical sciences / editor-in-chief, R.V. Gamkrelidze ; v. 110 . Probability theory ; 1
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The Random-Cluster Model / Geoffrey Grimmett
Models of First-Passage Percolation / C. Douglas Howard
Relaxation Times of Markov Chains in Statistical Mechanics and Combinatorial Structures / Fabio Martinelli
Random Walks on Finite Groups / Laurent Saloff-Coste
Index
Introduction / 1:
A Motivating Example: the Assignment Problem / 1.1:
A Stalking Horse: the Partial Matching Problem / 1.2:
Organization of the Survey / 1.3:
Geometric Graphs and Local Weak Convergence / 2:
Geometric Graphs / 2.1:
<$>{\cal G}^\ast<$> as a Metric Space / 2.2:
Local Weak Convergence / 2.3:
The Standard Construction / 2.4:
A Prototype: The Limit of Uniform Random Trees / 2.5:
Maximal Weight Partial Matching on Random Trees / 3:
Weighted Matchings of Graphs in General / 3.1:
Our Case: Random Trees with Random Edge Weights / 3.2:
Two Obvious Guesses: One Right, One Wrong / 3.3:
Not Your Grandfather's Recursion / 3.4:
A Direct and Intuitive Plan / 3.5:
Characterization of the Limit of <$>B(T_n^{small})<$> / 3.6:
Characterization of the Limit of <$>B(T_n^{big})<$> / 3.7:
The Limit Theorem for Maximum Weight Partial Matchings / 3.8:
Closing the Loop: Another Probabilistic Solution of a Fixed-Point Equation / 3.9:
From Coupling to Stability - Thence to Convergence / 3.10:
Looking Back: Perspective on a Case Study / 3.11:
The Mean-Field Model of Distance / 4:
From Poisson Points in <$>{\op R}^{d}<$> to a Simple Distance Model / 4.1:
The Poisson Weighted Infinite Tree - or, the PWIT / 4.2:
The Cut-off Components of a Weighted Graph and a PWIT / 4.3:
The Minimum Spanning Forests of an Infinite Graph / 4.4:
The Average Length Per Vertex of the MSF of a PWIT / 4.5:
The Connection to Frieze's ζ(3) Theorem / 4.6:
Minimal Cost Perfect Matchings / 5:
A Natural Heuristic - Which Fails for a Good Reason / 5.1:
Involution Invariance and the Standard Construction / 5.2:
Involution Invariance and the Convergence of MSTs / 5.3:
A Heuristic That Works by Focusing on the Unknown / 5.4:
A Distributional Identity with a Logistic Solution / 5.5:
A Stochastic Process that Constructs a Matching / 5.6:
Calculation of a Limiting Constant: <$>\pi^{2}/6<$> / 5.7:
Passage from a PWIT Matching to a Kn Matching / 5.8:
Finally - Living Beyond One's Means / 5.9:
Problems in Euclidean Space / 6:
A Motivating Problem / 6.1:
Far Away Places and Their Influence / 6.2:
Euclidean Methods and Some Observations in Passing / 6.3:
Recurrence of Random Walks in Limits of Planar Graphs / 6.4:
Limitations, Challenges, and Perspectives / 7:
References
Potts and random-cluster processes
Random-cluster measures
Ising and Potts models
Random-cluster and Ising-Potts coupled
The limit as <$>q \downarrow 0<$>
Rank-generating functions
Infinite-volume random-cluster measures
Stochastic ordering
A differential formula
Conditional probabilities
Infinite-volume weak limits
Random-cluster measures on infinite graphs
The case q < 1
Phase transition, the big picture
Infinite open clusters
First- and second-order phase transition
General results in d (≥ 2) dimensions
The subcritical phase, p < pc(q)
The supercritical phase, p > pc(q)
Near the critical point, p ≃ pc(q)
In two dimensions
Graphical duality
Value of the critical point
First-order phase transition
SLE limit when q ≤ 4
On complete graphs and trees
On complete graphs / 7.1:
On trees and non-amenable graphs / 7.2:
Time-evolutions of random-cluster models / 8:
Reversible dynamics / 8.1:
Coupling from the past / 8.2:
Swendsen-Wang dynamics / 8.3:
The Basic Model and Some Fundamental Questions
Notation
The Time Constant
The Fundamental Processes of Hammersley and Welsh
About μ
Minimizing Paths
Asymptotic Shape and Shape Fluctuations
Shape Theorems for Standard FPP
About the Asymptotic Shape for Lattice FPP
FPP Based on Poisson Point Processes
Upper Bounds on Shape Fluctuations
Some Related Longitudinal Fluctuation Exponents
Monotonicity
Transversal Fluctuations and the Divergence of Shape Fluctuations
Transversal Fluctuation Exponents
Upper Bounds on <$>\xi<$>
Lower Bounds on <$>\chi<$>
Lower Bounds on <$>\xi<$>
Fluctuations for Other Related Models
Infinite Geodesics and Spanning Trees
Semi-Infinite Geodesics and Spanning Trees
Coalescence and Another Spanning Tree in 2 Dimensions
Doubly-Infinite Geodesics
Summary of Some Open Problems
Mixing times for reversible, continuous-time Markov chains
Analytic methods
Tensorization of the Poincaré and logarithmic Sobolev inequalities
Geometric tools
Comparison methods
Coupling methods and block dynamics
Statistical mechanics models in <$>{\op Z}^d<$>
Grand canonical Gibbs measures
Mixing conditions and absence of long-range order
Canonical Gibbs measures for lattice gases
The ferromagnetic Ising and Potts models
FK representation of Potts models
Antiferromagnetic models on an arbitrary graph: Potts and hard-core models
Model with random interactions
Unbounded spin systems
Ground states of certain quantum Heisenberg models as classical Gibbs measures
Glauber dynamics in <$>{\op Z}^d<$>
The dynamics in a finite volume
The dynamics in an infinite volume
Graphical construction
Uniform ergodicity and logarithmic Sobolev constant
Mixing property versus logarithmic Sobolev constant in <$>{\op Z}^d<$>
The auxiliary chain and sweeping out relations method
The renormalization group approach
The martingale method
The recursive analysis
Rapid mixing for unbounded spin systems
Torpid mixing in the phase coexistence region
Torpid mixing for the Ising model in <$>\Lambda \subset {\op Z}^{d}<$> with free boundary conditions
Interface driven mixing inside one phase
Torpid mixing for Potts model in <$>{\op Z}^d<$>
Glauber dynamics for certain random systems in <$>{\op Z}^d<$>
Combination of torpid and rapid mixing: the dilute Ising model
Relaxation to equilibrium for spin glasses
Glauber dynamics for more general structures
Glauber dynamics on trees and hyperbolic graphs
Glauber dynamics for the hard-core model
Cluster algorithms: the Swendsen-Wang dynamics for Potts models
Mixing time for conservative dynamics / 9:
Random transposition, Bernoulli-Laplace and symmetric simple exclusion / 9.1:
The asymmetric simple exclusion / 9.2:
The Kac model for the Boltzmann equation / 9.3:
Adsorbing staircase walks / 9.4:
Kawasaki dynamics for lattice gases / 10:
Diffusive scaling of the mixing time in the one-phase region / 10.1:
Background and Notation / 10.2:
Finite Markov Chains
Invariant Markov Chains on Finite Groups
Shuffling Cards and the Cut-off Phenomenon
Three Examples of Card Shuffling
Exact Computations
The Cut-off Phenomenon
Probabilistic Methods
Coupling
Strong Stationary Times
Spectrum and Singular Values
General Finite Markov Chains
The Random Walk Case
Lower Bounds
Eigenvalue Bounds Using Paths
Cayley Graphs
The Second Largest Eigenvalue
The Lowest Eigenvalue
Diameter Bounds, Isoperimetry and Expanders
Results Involving Volume Growth Conditions
Moderate Growth
Nilpotent Groups
Nilpotent Groups with many Generators / 7.3:
Representation Theory for Finite Groups
The General Set-up
Abelian Examples
Random Random Walks
Central Measures and Bi-invariant Walks
Characters and Bi-invariance
Random Transposition on the Symmetric Group
Walks Based on Conjugacy Classes of the Symmetric Group
Finite Classical Groups
Fourier Analysis for Non-central Measures / 9.5:
Comparison Techniques
The min-max Characterization of Eigenvalues
Comparing Dirichlet Forms Using Paths
Comparison for Non-symmetric Walks / 10.3:
The Random-Cluster Model / Geoffrey Grimmett
Models of First-Passage Percolation / C. Douglas Howard
Relaxation Times of Markov Chains in Statistical Mechanics and Combinatorial Structures / Fabio Martinelli
9.

図書

図書
R. Daniel Mauldin and Mariusz Urbański
出版情報: Cambridge : Cambridge University Press, 2003  xi, 281 p. ; 24 cm
シリーズ名: Cambridge tracts in mathematics ; 148
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Introduction
Symbolic dynamics / 1:
Hölder families of functions / 3:
Conformal graph directed Markov systems / 4:
Examples of graph directed Markov systems / 5:
Conformal iterated function systems / 6:
Dynamical rigidity of conformal iterated function systems / 7:
Parabolic iterated function systems / 8:
Parabolic systems: Hausdorff and packing measures / 9:
Introduction
Symbolic dynamics / 1:
Hölder families of functions / 3:
10.

図書

図書
by P.L. Antonelli and T.J. Zastawniak
出版情報: Dordrecht : Kluwer Academic, c1999  vii, 205 p. ; 25 cm
シリーズ名: Fundamental theories of physics ; v. 101
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