The course will aim to provide an introduction to stochastic PDEs from the classical perspective, that being a mixture of stochastic analysis and PDE analysis. We will focus in particular on the variational approach to semi-linear parabolic problems, `a la Lions. There will also be comments on other models and approaches.
Suggested Pre-requisites: The course is broadly aimed at graduate students with some knowledge of PDE theory and/or stochastic analysis. Familiarity with measure theory and functional analysis will be useful.
Lecture 1: Introduction and Preliminaries
- Introduction to randomness in PDE
- Stochastic analysis in infinite dimensions
Literature: [DKM+09, Hai09, Par07, PR07, DPZ14]
Lecture 2: Variational Approach to Deterministic PDE
- Variational approach to linear parabolic equations
- Variational approaches to non-linear parabolic equations
Lecture 3: Variational Approach to Parabolic SPDE
- Itˆo’s formula in Hilbert spaces
- Variational approach to monotone, coercive SPDE
- Concrete examples
Lecture 4: Further Topics and Directions (time permitting)
- Regularity of solutions
- Ergodicity
- Pathwise approach to SPDE
Literature: [Hai09, DKM+09, DPZ96, Hai14, GIP15]
References
[DKM+09] Robert Dalang, Davar Khoshnevisan, Carl Mueller, David Nualart, and Yimin Xiao. A minicourse on stochastic partial differential equations, vol- ume 1962 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 2009.
[DPZ96] G. Da Prato and J. Zabczyk. Ergodicity for Infinite Dimensional Systems. London Mathematical Society Lecture Note Series. Cambridge University Press, 1996.
[DPZ14] Giuseppe Da Prato and Jerzy Zabczyk. Stochastic Equations in Infinite Dimensions. Encyclopedia of Mathematics and its Applications. Cambridge University Press, 2 edition, 2014.
[Eva10] Lawrence Craig Evans. Partial Differential Equations. American Mathe- matical Society, 2010.
[GIP15] Massimiliano Gubinelli, Peter Imkeller, and Nicolas Perkowski. Paracon- trolled distributions and singular PDEs. Forum Math. Pi, 3:75, 2015.
[Hai09] Martin Hairer. An Introduction to Stochastic PDEs. Technical report, The University of Warwick / Courant Institute, 2009. Available at: http://hairer.org/notes/SPDEs.pdf
[Hai14] M. Hairer. A theory of regularity structures. Inventiones mathematicae, 198(2):269–504, 2014.
[Par07] Etienne Pardoux. Stochastic partial differential equations. https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.405.4805&rep=rep1&type=pdf 2007.
[PR07] Claudia Pr´evˆot and Michael R¨ockner. A concise course on stochastic partial differential equations. Springer, 2007.
Further Information
Structure: 4 x 1.5hr Lectures
Lecture 1: Introduction and Preliminaries
- Introduction to randomness in PDE
- Stochastic analysis in infinite dimensions