Seminar series
Date
Tue, 20 Apr 2021
Time
10:00 - 11:30
Location
Virtual
Speaker
Dr. Avi Mayorcas
Organisation
Former University of Oxford D. Phil. Student

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

Literature: [Par07, Eva10]

Lecture 3: Variational Approach to Parabolic SPDE

  • Itˆo’s formula in Hilbert spaces
  • Variational approach to monotone, coercive SPDE
  • Concrete examples

Literature: [PR07, Par07]

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
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