2022年许惟钧教授中科院系列报告
Course Title: PDEs with Singular Randomness
Speaker: Weijun Xu 许惟钧 (BICMR, PKU)
Course Abstract: I will present some classical as well as more recent developments on PDEs with singular randomness, including parabolic stochastic PDEs and dispersive PDEs with random initial data. We will start with rough paths theory, and then go into recent advances, together with interesting questions that remain to be answered.
Time and Venue:
Monday (starting from 2022-09-05) 10:00- 11:30, 中国科学院数学与系统科学研究院 南楼N202
No lectures on 2022-10-03 and 2022-10-10
Videos: Here
For passcode, please write an email to Quan Shi.
Lecture 1 (2022-09-05): Rough Paths I
Abstract: Ito’s theory of stochastic differential equations is probabilistic in nature. It lacks analytic stability, as can been seen from the existence of Ito-Stratonovich correction. We will see how an old idea by Terry Lyons explores the nature of this ill-posedness, and then a theory that gives us a solution as well as a deep understanding.
Lecture 2 (2022-09-19, 南楼N202) Rough Paths II
Abstract: We will introduce the main ingredients of rough paths theory, and see how it helps us understand and resolve the analytic instability in stochastic differential equations. In the end, we will also see how Ito and Stratonovich formulations fit in this framework.
Lecture 3 (2022-09-26, 南楼N202) Rough Paths III
Abstract: We discuss applications of rough paths in classical stochastic analysis. After that, we will switch to para-controlled distributions with 2D parabolic Anderson model as an example, where we will see how insights from rough paths are naturally inherited here.
Lecture 4 (2022-10-17, 南楼N202) From rough paths to paracontrolled distributions
Abstract: We will use 2D PAM and \phi^4_3 as two examples to illustrate the framework of paracontrolled distributions. We will see that how ideas / ingredients of this framework arise naturally from those in rough paths. After that, we will switch to regularity structures.
Lecture 5 (2022-10-24, 南楼N202) Regularity Structures I
Abstract: We start to introduce elements of the theory of regularity structures, and get to the reconstruction theorem. We aim to illustrate that these abstract symbols and rules do arise from very concrete problems, and their abstractions are at a right level that enables them to cover a large class of problems.
Lecture 6 (2022-10-31, 南楼N202) Regularity Structures II
Abstract: We continue our investigations on regularity structures. We will go into some details of the reconstruction theorem, including a sketch of proof and some applications. We will then start to define abstract operations such as kernel convolutions and multiplications.
Lecture 7 (2022-11-14, 南楼N202) Regularity Structures III
Abstract: We continue our investigation on regularity structures, and explain how to solve the enhanced equation in the abstract modelled distribution space.
Lecture 8 (2022-11-14, 南楼N202) Regularity Structures IV
Abstract: We will 2D PAM and \phi^4_3 as two examples to see how to put together all the pieces to give a solution theory to singular SPDEs.
Quan Shi 石权 