Introduction to SPDEs from Probability and PDE - Lecture 1 of 4

20 April 2021
10:00
to
11:30
Introduction to SPDEs from Probability and PDE - Lecture 1 of 4
Dr. Avi Mayorcas

Further Information: 

Structure: 4 x 1.5hr Lectures 

Lecture 1:  Introduction and Preliminaries

  • Introduction to randomness in PDE
  • Stochastic analysis in infinite dimensions
Abstract

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.

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