Stochastic Partial Differential Equations

Graduate Course, period 3 2007-08

The main goal of the course is to understand the first 7 chapters of
G. Da Prato and J. Zabczyk, "Stochastic Equations in Infinite Dimensions", Cambridge University Press, 1992.
This is difficult reading, but fortunately there is a new book
C. Prévôt and M. Röckner, "A Concise Course on Stochastic Partial Differential Equations", Lecture Notes in Mathematics, Vol. 1905, Springer, 2007,
which covers the important parts of Chapters 1-4 in Da Prato-Zabczyk. This book is based on the diploma thesis
C. Knoche and K. Frieler, "Solutions of Stochastic Differential Equations in Infinite Dimensional Hilbert Spaces and Their Dependence on Initial Data".

Literature: Chapter 2 in Prévôt and Röckner. (Perhaps also chapter 3.) Chapters 5, 6, and perhaps 7 in Da Prato and Zabczyk.

Lecture notes: SPDE lecture notes (final version Sept 2008).

Teachers:
Stig Larsson
Mihaly Kovacs

Schedule:
Tuesdays 10.00-11.45, room MV:L14
Fridays 10.0-11.45, room MV:L14, except Feb 22 MV:L15
First time: January 22.

Lectures:

1. Stig: Introduction. See also this seminar talk. Gaussian measures and random variables in Hilbert space. (PR 2.1)

2. Misi: Proof of 2.1.2.

3. Stig: Proof of 2.1.6 and 2.1.10.

4. Feb 1. Misi: Filtration of Wiener process 2.1.11-2.13. Conditional expectation 2.2.1.

5. Feb 5. Stig: Martingales in Banach space.

6. Feb 8. Misi: Remark 2.2.5. Definition 2.3.1.

7. Feb 12. Stig: 2.3.2 - 2.3.4

8. Feb 15. Stig: 2.3.5 - 2.3.6. Misi: strong and uniform sigma-algebras on L(U,H).

9. Feb 19. Misi: pseudo inverse, Cameron-Martin space, characterization of the sigma-algebra P_T.

10. Feb 22. Misi: L_2(U,H) is a separable Hilbert space. 2.3.7 - 2.3.8

11. Feb 26. Stig: 2.3.9 - 2.3.10

12. Feb 29. Misi: 2.4 without proofs. 2.5

13. March 4. Misi: finish 2.5. Stig: DaPrato+Zabczyk 5.1.

14. March 7. Stig: DPZ 5.1. The stochastic heat equation.

15. March 11. Misi: DPZ existence of weak solutions.

16. March 14. Misi: uniqueness of weak solutions.

17. March 18. Stig: The stochastic wave equation. Semilinear stochastic PDE with global lipschitz condition.

/stig