| Description |
1 online resource |
| Series |
Grundlehren der mathematischen Wissenschaften, 0072-7830 ; 345 |
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Grundlehren der mathematischen Wissenschaften ; 345
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| Contents |
880-01 Part 1. General Theory -- A Warming-Up Example -- Central Limit Theorems -- Random Walks in Random Environment -- Bounds and Variational Principles for the Asymptotic Variance -- Part 2. Simple Exclusion Processes -- The Simple Exclusion Process -- Self-diffusion -- Equilibrium Fluctuations of the Density Field -- Regularity of the Asymptotic Variance -- Part 3. Diffusions in Random Environments -- Diffusions in Random Environments -- Variational Principles for the Limiting Variance -- Diffusions with Divergence Free Drifts -- Diffusions with Gaussian Drifts -- Ornstein-Uhlenbeck Process with a Random Potential -- Analytic Methods in Homogenization Theory |
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880-01/(S Machine generated contents note: pt. I General Theory -- 1. Warming-Up Example -- 1.1. Ergodic Markov Chains -- 1.2. Almost Sure Central Limit Theorem for Ergodic Markov Chains -- 1.3. Central Limit Theorem for Martingales -- 1.4. Time-Variance in Reversible Markov Chains -- 1.5. Central Limit Theorem for Reversible Markov Chains -- 1.6. Space of Finite Time-Variance Functions -- 1.6.1. Spaced H1 -- 1.6.2. Space H-1 -- 1.7. Comments and References -- References -- 2. Central Limit Theorems -- 2.1. Central Limit Theorem for Continuous Time Martingales -- 2.2. Spaces H1 and H-1 -- 2.3. Resolvent Equation -- 2.4. Dynkin's Martingales -- 2.5. H-1 Estimates of the Time-Variance -- 2.6. Central Limit Theorem for Markov Processes -- 2.7. Some Examples -- 2.7.1. Reversibility -- 2.7.2. Spectral Gap -- 2.7.3. Sector Condition -- 2.7.4. Graded Sector Condition -- 2.7.5. Perturbations of Normal Operators -- 2.8. Invariance Principles in the Multidimensional Case -- 2.8.1. Martingales with Stationary Increments -- 2.8.2. Additive Functional of Markov Processes -- 2.9. Comments and References -- References -- 3. Random Walks in Random Environment -- 3.1. Random Walks with Random Conductances -- 3.2. Doubly Stochastic Random Walks -- 3.3. Cyclic Random Walks -- 3.4. Random Walks with Drift in H-1 -- 3.4.1. Corrector Field -- 3.4.2. Elliptic Equation for the Corrector Field -- 3.4.3. Energy Identity -- 3.5. Random Walks in Mixing Environments -- 3.6. Doubly Stochastic Random Walks in Dimension d = 1 -- 3.7. Symmetric Random Walks -- 3.8. Comments and References -- References -- 4. Bounds and Variational Principles for the Asymptotic Variance -- 4.1. Quadratic Functional of the Resolvent -- 4.2. Bounds and Variational Formulas for the Variance -- 4.3. Variational Principles in the Graded Sector Context -- 4.4. Estimates of the Variance -- 4.5. Comments and References -- References -- pt. II Simple Exclusion Processes -- 5. Simple Exclusion Process -- 5.1. Exclusion Processes -- 5.2. Central Limit Theorems for Additive Functionals -- 5.3. Mean Zero Asymmetric Case -- 5.4. Duality -- 5.5. Asymmetric Case, α = 1/2 -- 5.6. Asymmetric Case, α [≠] 1/2 -- 5.7. Transient Markov Processes -- 5.8. Comments and References -- References -- 6. Self-diffusion -- 6.1. Exclusion Process as Seen from a Tagged Particle -- 6.2. Elementary Martingales -- 6.3. Spaces H1 and H-1 -- 6.4. Law of Large Numbers -- 6.5. Central Limit Theorem -- 6.6. Mean Zero Asymmetric Case -- 6.7. Duality -- 6.8. Asymmetric Case in Dimension d [≥] 3 -- 6.9. Self-diffusion Matrix -- 6.10. Comments and References -- References -- 7. Equilibrium Fluctuations of the Density Field -- 7.1. Duality -- 7.2. Approximations in H0.- 1 -- 7.3. Fluctuation-Dissipation Theorem -- 7.4. Second Class Particle -- 7.5. Estimates on the Operators θ,2,3, a and ± -- 7.6. Comments and References -- References -- 8. Regularity of the Asymptotic Variance -- 8.1. Resolvent Equation -- 8.2. Symmetric Case -- 8.3. Mean Zero Case -- 8.4. Asymmetric Case in d [≥] 3 -- 8.5. Regularity of the Diffusion Coefficients -- 8.6. Comments and References -- References -- pt. III Diffusions in Random Environments -- 9. Diffusions in Random Environments -- 9.1. Diffusions with Periodic Coefficients -- 9.2. Remark About the Quasi-periodic Case -- 9.3. Diffusions with Stationary Coefficients -- 9.3.1. Preliminaries on Stationary Environments -- 9.3.2. Spaces of Smooth Functions -- 9.3.3. Ito Equations with Stationary Coefficients -- 9.4. Environment Process and Its Properties -- 9.5. Martingale Decomposition and Central Limit Theorem -- 9.6. Homogenization of Solutions of Parabolic Partial Differential Equations -- 9.6.1. Random Coefficient Case -- 9.6.2. Periodic Case -- 9.7. Proofs of Propositions 9.8 and 9.9 -- 9.7.1. Proof of Proposition 9.8 -- 9.7.2. Proof of Proposition 9.9 -- 9.8. One-Dimensional Case -- 9.9. Diffusions with Time Dependent Coefficients -- 9.9.1. Space-Time Stationary Environments -- 9.9.2. Central Limit Theorem -- 9.10. Comments and References -- References -- 10. Variational Principles for the Limiting Variance -- 10.1. Spaces of Vector Fields -- 10.2. Upper Bound -- 10.3. Lower Bound -- 10.4. Spectral Representation of Homogeneous Fields -- 10.5. Comments and References -- References -- 11. Diffusions with Divergence Free Drifts -- 11.1. Passive Tracer Model -- 11.2. Properties of the Flow and the Definition of the Stream Matrix -- 11.3. Central Limit Theorem for a Diffusion with Bounded Stream -- Matrix -- 11.4. Convection Enhanced Diffusions -- 11.5. Time Dependent Flows with Finite Peclet Number -- 11.6. Proof of Theorem 11.4 -- 11.6.1. Notation -- 11.6.2. Statements of Some Technical Results -- 11.6.3. Properties of the Environment Process -- 11.6.4. Properties of the H1-Norm -- 11.6.5. Construction of the Corrector Field -- 11.6.6. Proof of the Energy Identity -- 11.7. Proofs of the Technical Results -- 11.7.1. Proof of Proposition 11.6 -- 11.7.2. Proof of Proposition 11.7 -- 11.7.3. Proof of Proposition 11.8 -- 11.7.4. Proof of Proposition 11.9 -- 11.7.5. Ergodic Theorem -- 11.8. Comments and References -- References -- 12. Diffusions with Gaussian Drifts -- 12.1. Stationary Gaussian Fields -- 12.2. Hermite Polynomials and Graded Structure of L2(Q) -- 12.3. Environment Process and Its Properties -- 12.4. Central Limit Theorem -- 12.5. Proofs of Technical Results -- 12.5.1. Proof of Estimate (12.2) -- 12.5.2. Proofs of Theorem 12.3 and Proposition 12.4 -- 12.6. Superdiffusive Transport in a Flow with Infinite Peclet Number -- 12.6.1. Homogeneous, Isotropic Gaussian Flows -- 12.6.2. Flows with Infinite Peclet Numbers -- 12.7. Central Limit Theorem for Diffusions in Gaussian and Markovian Flows -- 12.8. Markovian Dynamics of the Environment -- 12.8.1. Hermite Polynomials -- 12.8.2. Definition of the Transition Semigroup -- 12.8.3. Properties of the Generator -- 12.8.4. More General Formulation of the Markov Property of the Environment Process -- 12.9. Periodic Approximation of the Flow -- 12.10. Environment Process -- 12.11. Proof of Part (1) of Theorem 12.13 -- 12.12. On Superdiffusive Behavior of a Tracer in an Isotropic Flow -- 12.13. Proof of Part (2) of Theorem 12.13 -- 12.14. Proofs of the Results from Sect. 12.8 -- 12.14.1. Construction of the Semigroup -- 12.14.2. Proof of Proposition 12.16 -- 12.14.3. Proof of Proposition 12.17 -- 12.15. Proofs of the Results from Sect. 12.10 -- 12.15.1. Proof of Proposition 12.19 -- 12.15.2. Proof of Proposition 12.20 -- 12.16. Appendix: Some Auxiliary Results About Gaussian Random Fields -- 12.16.1. Multiple Stochastic Integrals -- 12.16.2. Some Properties of Hermite Polynomials -- 12.17. Comments and References -- References -- 13. Ornstein-Uhlenbeck Process with a Random Potential -- 13.1. Random Diffusion of a Particle with Inertia -- 13.2. Proof of the Central Limit Theorem -- 13.3. Proof of Proposition 13.2 -- 13.4. Gaussian Bounds on Transition Probability Densities -- 13.5. Comments and References -- References -- 14. Analytic Methods in Homogenization Theory -- 14.1. G-Convergence of Operators -- 14.2. Λ-Convergence of Quadratic Forms -- 14.3. G-Convergence of Matrix Valued Functions -- 14.4. Application to Homogenization of Diffusions in Random -- Media -- 14.5. Appendix: Ellipticity of the Coefficient Matrix of a Coercive Form -- 14.6. Comments and References -- References |
| Summary |
Diffusive phenomena in statistical mechanics and in other fields arise from markovian modeling and their study requires sophisticated mathematical tools. In infinite dimensional situations, time symmetry properties can be exploited in order to make martingale approximations, along the lines of the seminal work of Kipnis and Varadhan. The present volume contains the most advanced theories on the martingale approach to central limit theorems. Using the time symmetry properties of the Markov processes, the book develops the techniques that allow us to deal with infinite dimensional models that appear in statistical mechanics and engineering (interacting particle systems, homogenization in random environments, and diffusion in turbulent flows, to mention just a few applications). The first part contains a detailed exposition of the method, and can be used as a text for graduate courses. The second concerns application to exclusion processes, in which the duality methods are fully exploited. The third part is about the homogenization of diffusions in random fields, including passive tracers in turbulent flows (including the superdiffusive behavior). ¡ There are no other books in the mathematical literature that deal with this kind of approach to the problem of the central limit theorem. Hence, this volume meets the demand for a monograph on this powerful approach, now widely used in many areas of probability and mathematical physics. The book also covers the connections with and application to hydrodynamic limits and homogenization theory, so besides probability researchers it will also be of interest to mathematical physicists and analysts |
| Analysis |
Distribution (Probability theory) |
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Probability Theory and Stochastic Processes |
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Mathematical Physics |
| Bibliography |
Includes bibliographical references and index |
| Subject |
Markov processes.
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Fluctuations (Physics)
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Martingales (Mathematics)
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Markov Chains
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MATHEMATICS -- Applied.
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MATHEMATICS -- Probability & Statistics -- General.
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Martingalas (Matemáticas)
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Markov, Procesos de
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Fluctuations (Physics)
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Markov processes
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Martingales (Mathematics)
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| Form |
Electronic book
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| Author |
Landim, Claudio, 1965-
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Olla, Stefano, 1959-
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| ISBN |
9783642298806 |
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364229880X |
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