Introduction to stochastic PDEs
Summary
Stochastic PDEs are used to model systems that are spatially extended and include a random component. This course gives an introduction to this topic, including some general measure theory, some Gaussian measure theory and some linear semigroup theory.
Content
Stochastic PDEs form a relatively recent area of mathematics that combines many different fields, including PDE theory, stochastic analysis, ergodic theory, functional analysis, etc. This course is an introduction to the area with the aim of being able to appreciate some 21st century developments towards the end of the course. We will mainly focus on the development of a rather general solution theory for linear and semilinear stochastic PDEs, including stochastically forced heat, Navier-Stokes, and reaction-diffusion equations.
Some of the tools developed in this course, in particular infinite-dimensional measure theory and linear semigroup theory, are of much broader interest.
Keywords
probability, partial differential equations, semigroups, Gaussian measures
Learning Prerequisites
Required courses
Analysis I-IV
Probability
Recommended courses
Measure and integration
Probability theory
Functional Analysis I-II
Important concepts to start the course
Basic concepts in probability theory
Basic properties of Hilbert and Banch spaces
Teaching methods
Weekly lectures (on blackboard) and exercise sessions with assistant
Expected student activities
Attending the lectures and solving the exercises
Assessment methods
Oral exam
Supervision
| Office hours | No |
| Assistants | Yes |
| Forum | No |
Resources
Virtual desktop infrastructure (VDI)
No
Bibliography
G. DA PRATO and J. ZABCZYK. Stochastic equations in infinite dimensions, vol. 44 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge, 1992.
A. LUNARDI. Analytic semigroups and optimal regularity in parabolic problems. Progress in Nonlinear Differential Equations and their Applications, 16. Birkhäuser Verlag, Basel, 1995.
V. I. BOGACHEV. Gaussian measures, vol. 62 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 1998.
P. BILLINGSLEY. Convergence of probability measures. John Wiley & Sons Inc., New York, 1968.
K. YOSIDA. Functional analysis. Classics in Mathematics. Springer-Verlag, Berlin, 1995. Reprint of the sixth (1980) edition.
Ressources en bibliothèque
- Convergence of probability measures / Billingsley
- Analytic semigroups and optimal regularity in parabolic problems / Lunardi
- Gaussian measures / Bogachev
- Functional analysis / Yosida
- Stochastic equations in infinite dimensions / Da Prato
Notes/Handbook
The lecture will mainly follow the notes available at https://www.hairer.org/notes/SPDEs.pdf, but might cover additional material if time permits.
Moodle Link
In the programs
- Semester: Spring
- Exam form: Oral (summer session)
- Subject examined: Introduction to stochastic PDEs
- Courses: 3 Hour(s) per week x 14 weeks
- Exercises: 2 Hour(s) per week x 14 weeks
- Type: optional
- Semester: Spring
- Exam form: Oral (summer session)
- Subject examined: Introduction to stochastic PDEs
- Courses: 3 Hour(s) per week x 14 weeks
- Exercises: 2 Hour(s) per week x 14 weeks
- Type: optional
- Semester: Spring
- Exam form: Oral (summer session)
- Subject examined: Introduction to stochastic PDEs
- Courses: 3 Hour(s) per week x 14 weeks
- Exercises: 2 Hour(s) per week x 14 weeks
- Type: optional
- Semester: Spring
- Exam form: Oral (summer session)
- Subject examined: Introduction to stochastic PDEs
- Courses: 3 Hour(s) per week x 14 weeks
- Exercises: 2 Hour(s) per week x 14 weeks
- Type: optional
- Semester: Spring
- Exam form: Oral (summer session)
- Subject examined: Introduction to stochastic PDEs
- Courses: 3 Hour(s) per week x 14 weeks
- Exercises: 2 Hour(s) per week x 14 weeks
- Type: optional