Probability II
Lectures held on Tsinghua Campus, Teaching Building 4, room 4206, Thursday 9:50-11:25 and Friday 13:30-15:05, starting February 26 for 12 weeks.
Grading based on: Homework (15%), Lecture writeup (20%), Midterm (30%), Final Exam (35%).
Grading based on: Homework (15%), Lecture writeup (20%), Midterm (30%), Final Exam (35%).
讲师
日期
2026年02月26日 至 06月27日
位置
| Weekday | Time | Venue | Online | ID | Password |
|---|---|---|---|---|---|
| 周四 | 09:50 - 11:25 | Tsinghua Building 4, Room 4206 | ZOOM 02 | 518 868 7656 | BIMSA |
| 周五 | 13:30 - 15:05 | Tsinghua Building 4, Room 4206 | ZOOM 02 | 518 868 7656 | BIMSA |
修课要求
Law of Large numbers, Central Limit Theorem, Borel-Cantelli Lemma, Conditional expectation (Kolmogorov definition), Definition of a martingale.
课程大纲
Topics Covered:
1. Applications of Martingales: 1.1 Branching processes: classification 1.2 Growth and structure of Galton-Watson trees 1.3 Polya Urns
2. Finite Markov chains: 2.1 Irreducibility, periodicity, reversibility 2.2 Stationary measure 2.3 Total variation distance 2.4 Convergence theorem 2.5 Mixing time
3. Markov chains on a countable state space: 3.1 Transience, recurrence and positive recurrence 3.2 Random walk on regular trees and lattices
4. Random walk on networks: 4.1 Voltage, flow 4.2 Effective resistance 4.3 Energy of a flow 4.4 infinite networks
5. Continuous time Markov processes: 5.1 Definitions 5.2 Poisson process on a line 5.3 Infinitesimal generator 5.4 Stationary distribution 5.5 Queuing
6. Spatial processes: 6.1 Percolation on a tree and a lattice 6.2 Poisson process in space 6.3 Voronoi tessellation 6.4 Buffon needle
7. Brownian motion (BM): 7.1 Gaussian vectors 7.2 Construction and continuity 7.3 Reflection principle 7.4. Strong Markov property 7.5 BM in d dimensions.
1. Applications of Martingales: 1.1 Branching processes: classification 1.2 Growth and structure of Galton-Watson trees 1.3 Polya Urns
2. Finite Markov chains: 2.1 Irreducibility, periodicity, reversibility 2.2 Stationary measure 2.3 Total variation distance 2.4 Convergence theorem 2.5 Mixing time
3. Markov chains on a countable state space: 3.1 Transience, recurrence and positive recurrence 3.2 Random walk on regular trees and lattices
4. Random walk on networks: 4.1 Voltage, flow 4.2 Effective resistance 4.3 Energy of a flow 4.4 infinite networks
5. Continuous time Markov processes: 5.1 Definitions 5.2 Poisson process on a line 5.3 Infinitesimal generator 5.4 Stationary distribution 5.5 Queuing
6. Spatial processes: 6.1 Percolation on a tree and a lattice 6.2 Poisson process in space 6.3 Voronoi tessellation 6.4 Buffon needle
7. Brownian motion (BM): 7.1 Gaussian vectors 7.2 Construction and continuity 7.3 Reflection principle 7.4. Strong Markov property 7.5 BM in d dimensions.
参考资料
Williams: Probability with Martingales
Karlin and Taylor: A first course in stochastic processes
Levin and Peres: Markov chains and mixing times (Second edition, 2017)
Moerters and Peres: Brownian motion
Karlin and Taylor: A first course in stochastic processes
Levin and Peres: Markov chains and mixing times (Second edition, 2017)
Moerters and Peres: Brownian motion
听众
Advanced Undergraduate
, Graduate
视频公开
公开
笔记公开
公开
语言
英文
讲师介绍
Yuval Peres 1990年于耶路撒冷希伯来大学博士毕业，并先后于斯坦福大学和耶鲁大学任职博士后。此后，他在美国加州大学伯克利分校和耶路撒冷希伯来大学担任数学和统计学教授，并任微软公司首席研究员。Peres在概率论的大部分领域总共发表过超过350篇论文，包括随机游走，布朗运动，渗流和随机图。与他人著有多本专题专著：《概率与分析中的分形》，《布朗运动》，《高斯解析函数的零点与行列式点过程》，《马尔可夫链与混合时间》，《树图与网络中的概率论》，《博弈论》，并被美国数学会和剑桥大学出版社出版。专著涉及马可夫链、概率图、博弈论和布朗运动等方向，具体信息可以在以下网址查找：https://www.yuval-peres-books.com/ . 他的报告可在以下网址查找：https://yuval-peres-presentations.com/。
Peres是Rollo Davidson奖和Loeve奖得主，是2002年北京国际数学家大会、2008年欧洲数学家年会、2017年美国数学家年会邀请报告人，并于2016年当选美国科学院院士。他指导过21名博士，包括Elchanan Mossel (美国麻省理工大学教授, 美国数学家学会会士), 丁剑 (北京大学, 国际华人数学家大会金奖、Rollo Davidson奖得主), 以及Balint Virag和Gabor Pete (Rollo Davidson奖得主).