Summer Term 2020

lecture and exercise by Prof. Dr. Nicolas Perkowski

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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

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Prerequisits: Stochastics I-II and Analysis I — III. Recommended: Stochastic Analysis and Functional Analysis. Previous knowledge in PDE theory is not required.

Assessment

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.    

Exercises 

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

• Schauder estimates

• Gaussian hypercontractivity

• Paraproducts and paracontrolled distributions

• Local existence and uniqueness for semilinear SPDEs in higher dimensions

• Properties of solutions

References

Lecture notes.

Exercises.

Videos

• Videos for Wednesday, April 22:
  • Welcome
  • Introduction (Section 0 in the notes)
  • Covariance measures (1.1-1.2)
  • Gaussian martingale measures (1.3-1.4)
  • Examples of Gaussian martingale measures (1.5-1.8)
• Videos for Wednesday, April 29:
  • Construction of the Ito integral (1.9-1.16)
  • Weak and mild solutions (1.17-1.20)
  • K alpha covariances (1.21-1.23)
• Videos for Wednesday, May 6:
  • Existence and uniqueness of mild solutions (1.24-1.25)
  • Examples of SPDEs (1.26)
  • Towards the regularity of solutions (1.27-1.28)
• Videos for Wednesday, May 13:
  • Regularity of solutions (1.29-1.32)
  • Quartic variation for space-time white noise in 1d (1.33)
  • An example we cannot solve (1.34)
  • The d-dimensional torus (1.35)
  • Energy estimates, a glimpse in the variational approach (1.36)
  • "Stochastic parabolicity", Ito vs Stratonovich (1.37)
• Videos for Wednesday, May 20:
  • Some "singular" SPDEs (1.38-1.41)
  • The sewing lemma (2.1-2.2)
  • The Young integral (2.3-2.5)
  • Some technical results (2.6-2.8)
  • Young integral equations (2.9)
• 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:
  • Controlled paths (2.22)
  • Controlled rough path integral (2.23-2.24)
  • Rough differential equations (2.25-2.29)
  • Brownian motion as a rough path (2.30-2.31)
• Videos for Wednesday, June 10:
  • Linear operations on tempered distributions (3.1-3.3)
  • Fourier transforms and convolutions (3.4-3.8)
  • Smooth dyadic partition of unity (3.9-3.11)
  • Besov spaces and Bernstein-type inequality (3.12-3.13)
• 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:
  • Besov spaces and heat kernel on the torus (3.24)
  • A Kolmogorov type criterion for space-time Hölder-Besov regularity (4.1)
  • Regularity of the stochastic convolution (4.2-4.4)
  • Schauder estimates (4.5-4.6)
  • The Phi41 equation (4.7)
• 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:
  • Application to the Phi42 equation (4.23)
  • Local subcriticality (4.27-4.28)
  • Tree notation
  • Statement of the commutator estimate (5.2-5.3)
  • Proof of the commutator estimate (5.4)
• 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