本书是应用数学专业高年级本科生和研究生的入门教材。它介绍了随机分析基本的数学基础（概率论和随机过程），以及一些重要的实用工具和应用（如与微分方程、数值方法、路径积分、随机场、统计物理、化学动力学和罕见事件的联系）。本书在数学形式主义和直觉论证之间找到了一个很好的平衡，这种风格最适合应用数学家。读者可以学习随机分析的严格处理，以及在建模和仿真中的实际应用。本书提供的大量练习是主要内容的有益补充。

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