Meeting times
When: Friday 11:15 - 12:45ish
Where: Kerchof 326
If interested contact Gennady Uraltsev and Juraj Földes.
We can consider remote participation via Zoom should interest arise. Feel free to circulate.
Summary
Martin Hairer received the Fields medal at the ICM in Seoul 2014 for “his outstanding contributions to the theory of stochastic partial differential equations" (quoted from the ICM webpage), in particular for the creation of the theory of regularity structures1. The theory of regularity structures formally subsumes Terry Lyons’ theory of rough paths2 3 and is particularly adapted to solving stochastic parabolic equations4. A more analytic generalization of rough paths has been developed by Gubinelli et. al. in the form of paracontrolled calculus5 and has proven applicable to stochastic PDE with dispersive behavior6.
The goal of this reading seminar is to gain an understanding of the relation of the results panorama of singular stochastic PDEs with particular attention to the results above. We will mainly concentrate on the classic Terry Lyons’ theory and its subsequent development in terms of paracontrolled calculus. In particular, we will follow lecture notes by Gubinelli7 and the intro book8 by Hairer and Friz. This starts mainly with ODE theory. We will then try to progress to SPDEs (parabolic) by following9. Finally, it might be interesting to present how these ideas are applied to dispersive PDE10 11 and also to compare paracontrolled theory with Hairer’s regularity structures8 1.
Gennady has previously partially followed a course and taught on closely related topics so he would gladly provide lectures in the beginning. Initiative to present topics further down the road is appreciated but not required.
Prerequisites
- A solid background in Lebesgue integration theory and classical analysis.
- Exposure to probability and stochastic analysis is suggested.
We plan on developing pathwise stochastic integration in this context but a certain general familiarity with the Brownian motion and Ito’s formula is suggested.
Basics of Fourier analysis is required:
- $L^2$ theory
- theory of distributions
- generalized functions and Sobolev spaces (will be reviewed quickly)
Exposure to basic PDEs (heat, wave) is encouraged (especially starting from the second part of the seminar).
Additional material
In the meantime, to get an informal introduction of the relation of rough path theory to applications outside of math please enjoy the following talk:
What has rough paths got to do with data science - Terry Lyons - YouTube
Other interesting material (that will probably not be covered in the seminar):
We are open to suggestions on closely related topics.
Hairer, Martin. “A theory of regularity structures.” Inventiones mathematicae 198.2 (2014): 269-504. ↩︎ ↩︎
Lyons, T., & Qian, Z. (2002). “System control and rough paths”. Oxford : New York: Clarendon Press ; Oxford University Press. ↩︎
Gubinelli, Massimiliano, Peter Imkeller, and Nicolas Perkowski. “Paracontrolled distributions and singular PDEs.” Forum of Mathematics, Pi. Vol. 3. Cambridge University Press, 2015. ↩︎
Gubinelli, M., et al. “Global dynamics for the two-dimensional stochastic nonlinear wave equations, to appear in Internat.” Math. Res. Not. ↩︎
Unpublished lecture notes ↩︎
Gubinelli, Massimiliano, and Nicolas Perkowski. “Lectures on singular stochastic PDEs.” arXiv preprint arXiv:1502.00157 (2015). ↩︎
Gubinelli, Massimiliano, Herbert Koch, and Tadahiro Oh. “Paracontrolled approach to the three-dimensional stochastic nonlinear wave equation with quadratic nonlinearity.” arXiv preprint arXiv:1811.07808 (2018). ↩︎
Gubinelli, Massimiliano, B. Ugurcan, and Immanuel Zachhuber. “Semilinear evolution equations for the Anderson Hamiltonian in two and three dimensions.” Stochastics and Partial Differential Equations: Analysis and Computations 8.1 (2020): 82-149. ↩︎