19242101 Aufbaumodul: Stochastics IV "Stochastic Partial Differential Equations: Classical and New"
Summer Term 2020
lecture and exercise by Prof. Dr. Nicolas Perkowski
Time and place
Lecture: Video lectures are available online (see below).
Exercise Session: Wednesdays, 10:15 - 11:45, online.
- Final Exam: to be announced in due course
Prerequisits: Stochastics I-II and Analysis I — III. Recommended: Stochastic Analysis and Functional Analysis. Previous knowledge in PDE theory is not required.
To receive credits fo the course you need to
- actively participate in the exercise session
- work on and successfully solve the weekly exercises
- pass the final exam (see above)
If you are an FU student you only need to register for the course via CM (Campus Management).
If you are not an FU student, you are required to register via KVV/Whiteboard.
Problem sets will be put online every Wednesday and can be found under Assignements in the KVV/Whiteboard portal. You do not have to submit your solutions. The solutions will be discussed in the online tutorial.
Course Overview/ Content:
Ito calculus for Gaussian random measures
Semilinear stochastic PDEs in one dimension
Basic rough path theory
Paraproducts and paracontrolled distributions
Local existence and uniqueness for semilinear SPDEs in higher dimensions
Properties of solutions
- Videos for Wednesday, April 22:
- Videos for Wednesday, April 29:
- Videos for Wednesday, May 6:
- Videos for Wednesday, May 13:
- Videos for Wednesday, May 20:
- Videos for Wednesday, May 27:
- Application of the Young theory to fractional Brownian motions (2.10-2.11)
- Motivating examples for rough paths (2.12-2.14)
- Definition of a rough path (2.15-2.18)
- First applications of rough paths (2.19-2.21)
- Videos for Wednesday, June 3:
- Videos for Wednesday, June 10:
- Videos for Wednesday, June 17:
- Applications of the Bernstein-type inequality (3.14-3.15)
- Lemma about functions that are localized in Fourier space (3.16)
- The paraproduct and the resonant product (3.17-3.20)
- Examples for products of distributions (3.21-3.23)
- Videos for Wednesday, June 24:
- Videos for Wednesday, July 1:
- The Phi42 equation (4.10-4.11)
- Hermite polynomials (4.12)
- Wiener-Ito integrals (4.14-4.18)
- Link between Hermite polynomials and Wiener-Ito integrals (4.19)
- Gaussian hypercontractivity (4.20-4.23)
- Videos for Wednesday, July 8:
- Videos for Wednesday, July 15:
- Definition of paracontrolled distribution (5.5-5.8)
- Comparison of modified paraproduct and usual paraproduct (5.9-5.10)
- Operations on paracontrolled distributions (5.11-5.14)
- Paracontrolled Picard iteration (5.15)
- Suggestion of some possible projects for the exam