Stochastics IV: Stochastic Partial Differential Equations: Classical and New
Akt: 21.12.2022 16:06
Thu, 10:00 - 12:00
(A6/SR 009 Seminarraum)
Content: "Stochastics IV: Stochastic Partial Differential Equations: Classical and New"
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;
paraproducts and paracontrolled distributions;
local existence and uniqueness for semilinear SPDEs in higher dimensions;
properties of solutions
Pavliotis, Grigoris, and Andrew Stuart: Multiscale methods: averaging and homogenization. Springer Science & Business Media, 2008.
Bensoussan, Alain, Jacques-Louis Lions, and George Papanicolaou: Asymptotic analysis for periodic structures. Vol. 374. American Mathematical Soc., 2011.
Prerequisite: Stochastics I, II.
Recommended: Stochastics III and Functional Analysis.
A6/SR 009 Seminarraum
wöchentlich, ab 20.04.2023, 10:00 - 12:00 (14 Termine)