内容简介
本书是应用数学专业高年级本科生和研究生的入门教材。它介绍了分析基本的数学基础(概率论和过程)，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、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衡，这种风格适合应用数学家。读者可以学分析的严格处理，以及在建模和仿真中的实际应用。本书提供的大量练要内容的有益补充。
目录
Introduction to the SeriesPrefaceNotationPart 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. Embed Chain and Irreducibility §3.7. Ergodic Theorem for Q-processes §3.8. Time Reversal §3.9. Hen 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. Kiic 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 NotesPart 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. metaility 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 Kiics §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 Kiic System Exercises NotesAppendix A. Laplace Asymptotics and Varadhan's Lemma B. Gronwall's Inequality C. Measure and Integration D. Martingales E. Strong Markov Property F. Semigroup of OperatorsBibliographyIndex