Speaker: Sergei Kuksin
Title ：Random processes and equations with randomness(6 lectures).
Place: 双清综合楼 B1010
Times:
Lecture 1 and 2: Thursday, May 28, 3:20-4:55 pm,
Lecture 3 and 4: Thursday, June 4, 3:20-4:55 pm,
Lecture 5 and 6: Thursday, June 11, 3:20-4:55 pm,
PLAN
1. Motivation: random dynamical systems with discrete and continuous time.
2. Quick preliminaries: random variables and processes, law of a random variable, independence, Polish spaces.
3. Spaces of measures on Polish spaces and metrics on them. Dobrushin's lemma.
4. Conditional expectations and conditional probabilities.
5. Transition probabilities. Regular conditional probabilities. Disintegration of measures.
6. Filtered probability spaces. Adapted random processes. Markov processes and Markov families.
7. Kolmogorov-Chapman relations. Markov semi-groups. Feller Markov processes. Stationary processes and stationary Markov processes.
8. Discrete time random dynamical systems (RDS), stirred by discrete time white noises. The Markov processes which they define.
9. Mixing for the RDS in item 8. First examples. Some properties of mixing RDS. Discussing of the mixing.
10. RDS, stirred by bounded mixing stationary processes. Their Markovian liftings.
11. Idea of Doeblin's coupling as a tool to prove the mixing for RDE as in items 9 and 10.
REFERENCES
For 1-7 I will refer to the following books:
[Doob] Doob "Random Processes".
[Dud] Dudley "Real Analysis and Probability".
[KS] Koralov, Sinai "Theory of Probability and Random Processes".
主讲人介绍：
Sergei Kuksin是一位在哈密顿偏微分方程、随机偏微分方程和湍流理论等领域享有极高声誉的俄罗斯数学家。
Kuksin教授是俄罗斯斯捷克洛夫数学研究所的首席科学家，同时在法国巴黎的多所大学担任高级研究员。他的学术成就获得了广泛认可，曾获得俄罗斯科学院颁发的李雅普诺夫奖，并在1998年受邀在柏林国际数学家大会作报告。
他的研究兴趣广泛且深入，主要贡献在三个核心领域：
哈密顿偏微分方程：无穷维哈密顿系统的视角来研究演化偏微分方程，是这一领域的领军人物。他发展了一套将经典力学中KAM理论（科尔莫戈罗夫-阿诺尔德-莫泽理论）应用于偏微分方程的精细分析技术，用于构建和维持这些方程的复杂拟周期解（即不变环面）。
随机偏微分方程与湍流：他利用随机Burgers方程成功构建了一个严格描述流体湍流的数学模型。
随机动力系统：近期，他与合作者研究了一类更具一般性的系统，其随机驱动力不必是“白噪声”（即时间上不独立）。他们证明了即使在非马尔可夫的情况下，许多系统（如2D Navier-Stokes方程）最终也会“忘记”初始状态，收敛到一个唯一的统计平衡态（指数混合性）。
邀请人：薛金鑫
