Stochastic Partial Differential Equations
The first half of the course is devoted to explaining fundamental concepts, terms, facts and tools in probability theory and stochastic analysis. Then, in the second half, we discuss the stochastic partial differential equations as applications of stochastic analysis.
Lecturer
Date
26th September ~ 26th December, 2023
Location
| Weekday | Time | Venue | Online | ID | Password |
|---|---|---|---|---|---|
| Tuesday,Thursday | 09:50 - 11:25 | A3-3-301 | ZOOM 03 | 242 742 6089 | BIMSA |
Prerequisite
It is desirable that the audience is familiar with modern probability theory and some tools in stochastic analysis. But I will try to briefly explain these in my course.
Syllabus
The course consists of the following two parts.
(1) Foundations of Probability Theory and Stochastic Analysis:
Probability space, Convergence of random variables, Independence, Conditional probability, Strong law of large numbers, Kolmogorov's inequality, Convergence in law, Central limit theorem, Discrete and continuous time martingales, Brownian motion, Stochastic integrals, Ito's formula, Stochastic differential equations, Relation to partial differential equations
(2) Stochastic Partial Differential Equations:
Stochastic Allen-Cahn equation, Time-dependent Ginzburg-Landau equation, Singular stochastic partial differential equations, KPZ (Kardar-Parisi-Zhang) equation, coupled KPZ equation
(1) Foundations of Probability Theory and Stochastic Analysis:
Probability space, Convergence of random variables, Independence, Conditional probability, Strong law of large numbers, Kolmogorov's inequality, Convergence in law, Central limit theorem, Discrete and continuous time martingales, Brownian motion, Stochastic integrals, Ito's formula, Stochastic differential equations, Relation to partial differential equations
(2) Stochastic Partial Differential Equations:
Stochastic Allen-Cahn equation, Time-dependent Ginzburg-Landau equation, Singular stochastic partial differential equations, KPZ (Kardar-Parisi-Zhang) equation, coupled KPZ equation
Reference
[1] D. Williams: Probability with Martingales, Cambridge, 1991.
[2] J-F. Le Gall, Brownian Motion, Martingales, and Stochastic Calculus, Springer, 2013.
[3] I. Karatzas and S.E. Shreve: Brownian Motion and Stochastic Calculus, Springer, 1991.
[4] T. Funaki, Lectures on Random Interfaces (Chapters 3, 4, 5), SpringerBriefs, 2016.
[5] Slides and Videos of Funaki's courses at Yau center: 2022, 2020
https://ymsc.tsinghua.edu.cn/en/info/1050/1967.htm
https://ymsc.tsinghua.edu.cn/en/info/1051/1518.htm
[2] J-F. Le Gall, Brownian Motion, Martingales, and Stochastic Calculus, Springer, 2013.
[3] I. Karatzas and S.E. Shreve: Brownian Motion and Stochastic Calculus, Springer, 1991.
[4] T. Funaki, Lectures on Random Interfaces (Chapters 3, 4, 5), SpringerBriefs, 2016.
[5] Slides and Videos of Funaki's courses at Yau center: 2022, 2020
https://ymsc.tsinghua.edu.cn/en/info/1050/1967.htm
https://ymsc.tsinghua.edu.cn/en/info/1051/1518.htm
Audience
Undergraduate
, Graduate
Video Public
No
Notes Public
No
Language
English
Lecturer Intro
Funaki Tadahisa was a professor at University of Tokyo (1995-2017) and at Waseda University (2017-2022) in Japan. His research subject is probability theory mostly related to statistical physics, specifically interacting systems and stochastic PDEs. He was a president of Mathematical Society of Japan (2013-2015), and was an invited sectional lecturer at ICM 2022.