Stochastics IV:
Guilherme de Lima Feltes, Nicolas Perkowski
Additional information / Pre-requisites
Prerequisite: Stochastics I, II, III.
Recommended: Functional Analysis.
Comments
Content: We will learn two different methods for solving stochastic partial differential equations. The classical method is based on Ito calculus, and we will use it to solve semilinear SPDEs with space-time white noise in one space dimension. But we will see that in higher dimensions this theory only works for linear equations, and motivated by that we will introduce "paracontrolled distributions“, which we developed in the last years based on ideas from harmonic analysis and rough paths, and which allow us to solve some interesting semilinear equations in higher dimensions. Along the way we will learn about regularity theory for semilinear PDEs, Gaussian Hilbert spaces, and much more.
- Ito calculus for Gaussian random measures;
- semilinear stochastic PDEs in one dimension;
- Schauder estimates;
- Gaussian hypercontractivity;
- paraproducts and paracontrolled distributions;
- local existence and uniqueness for semilinear SPDEs in higher dimensions;
- properties of solutions
Detailed Information can be found on the Homepage of 19246301 SPDEs: Classical and New.
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Literature
There will be lecture notes.
14 Class schedule
Regular appointments