《套用隨機分析（影印版）》是高等教育出版社出版圖書。
基本介紹
- 中文名：套用隨機分析（影印版）
- 作者：Weinan E，Tiejun Li 著
- 出版時間：2021年3月1日
- 出版社：高等教育出版社
- 頁數：336 頁
- ISBN：9787040556391
- 開本：16 開
- 裝幀：精裝
- 版次：1
- 商品編碼：13168916
- 用紙：膠版紙
內容簡介,目錄,
內容簡介
本書是套用數學專業高年級本科生和研究生的入門教材。它介紹了隨機分析基本的數學基礎（機率論和隨機過程），以及一些重要的實用工具和套用（如與微分方程、數值方法、路徑積分、隨機場、統計物理、化學動力學和罕見事件的聯繫）。本書在數學形式主義和直覺論證之間找到了一個很好的平衡，這種風格***套用數學家。讀者可以學習隨機分析的嚴格處理，以及在建模和仿真中的實際套用。本書提供的大量練習是主要內容的有益補充。
目錄
Introduction to the Series
Preface
Notation
Part 1. Fundamentals
Chapter 1. Random Variables
§1.1. Elementary Examples
§1.2. Probability Space
§1.3. Conditional Probability
§1.4. Discrete Distributions
§1.5. Continuous Distributions
§1.6. Independence
§1.7. Conditional Expectation
§1.8. Notions of Convergence
§1.9. Characteristic Function
§1.10. Generating Function and Cumulants
§1.11. The Borel-Cantelli Lemma
Exercises
Notes
Chapter 2. Limit Theorems
§2.1. The Law of Large Numbers
§2.2. Central Limit Theorem
§2.3. Cramér's Theorem for Large Deviations
§2.4. Statistics of Extrema
Exercises
Notes
Chapter 3. Markov Chains
§3.1. Discrete Time Finite Markov Chains
§3.2. Invariant Distribution
§3.3. Ergodic Theorem for Finite Markov Chains
§3.4. Poisson Processes
§3.5. Q-processes
§3.6. Embedded Chain and Irreducibility
§3.7. Ergodic Theorem for Q-processes
§3.8. Time Reversal
§3.9. Hidden Markov Model
§3.10. Networks and Markov Chains
Exercises
Notes
Chapter 4. Monte Carlo Methods
§4.1. Numerical Integration
§4.2. Generation of Random Variables
§4.3. Variance Reduction
§4.4. The Metropolis Algorithm
§4.5. Kinetic Monte Carlo
§4.6. Simulated Tempering
§4.7. Simulated Annealing
Exercises
Notes
Chapter 5. Stochastic Processes
§5.1. Axiomatic Construction of Stochastic Process
§5.2. Filtration and Stopping Time
§5.3. Markov Processes
§5.4. Gaussian Processes
Exercises
Notes
Chapter 6. Wiener Process
§6.1. The Diffusion Limit of Random Walks
§6.2. The Invariance Principle
§6.3. Wiener Process as a Gaussian Process
§6.4. Wiener Process as a Markov Process
§6.5. Properties of the Wiener Process
§6.6. Wiener Process under Constraints
§6.7. Wiener Chaos Expansion
Exercises
Notes
Chapter 7. Stochastic Differential Equations
§7.1. Ito Integral
§7.2. Ito's Formula
§7.3. Stochastic Differential Equations
§7.4. Stratonovich Integral
§7.5. Numerical Schemes and Analysis
§7.6. Multilevel Monte Carlo Method
Exercises
Notes
Chapter 8. Fokker-Planck Equation
§8.1. Fokker-Planck Equation
§8.2. Boundary Condition
§8.3. The Backward Equation
§8.4. Invariant Distribution
§8.5. The Markov Semigroup
§8.6. Feynman-Kac Formula
§8.7. Boundary Value Problems
§8.8. Spectral Theory
§8.9. Asymptotic Analysis of SDEs
§8.10. Weak Convergence
Exercises
Notes
Part 2. Advanced Topics
Chapter 9. Path Integral
§9.1. Formal Wiener Measure
§9.2. Girsanov Transformation
§9.3. Feynman-Kac Formula Revisited
Exercises
Notes
Chapter 10. Random Fields
§10.1. Examples of Random Fields
§10.2. Gaussian Random Fields
§10.3. Gibbs Distribution and Markov Random Fields
Exercise
Notes
Chapter 11. Introduction to Statistical Mechanics
§11.1. Thermodynamic Heuristics
§11.2. Equilibrium Statistical Mechanics
§11.3. Generalized Langevin Equation
§11.4. Linear Response Theory
§11.5. The Mori-Zwanzig Reduction
§11.6. Kac-Zwanzig Model
Exercises
Notes
Chapter 12. Rare Events
§12.1. Metastability and Transition Events
§12.2. WKB Analysis
§12.3. Transition Rates
§12.4. Large Deviation Theory and Transition Paths
§12.5. Computing the Minimum Energy Paths
§12.6. Quasipotential and Energy Landscape
Exercises
Notes
Chapter 13. Introduction to Chemical Reaction Kinetics
§13.1. Reaction Rate Equations
§13.2. Chemical Master Equation
§13.3. Stochastic Differential Equations
§13.4. Stochastic Simulation Algorithm
§13.5. The Large Volume Limit
§13.6. Diffusion Approximation
§13.7. The Tau-leaping Algorithm
§13.8. Stationary Distribution
§13.9. Multiscale Analysis of a Chemical Kinetic System
Exercises
Notes
Appendix
A. Laplace Asymptotics and Varadhan's Lemma
B. Gronwall's Inequality
C. Measure and Integration
D. Martingales
E. Strong Markov Property
F. Semigroup of Operators
Bibliography
Index
