Introduction to stochastic PDE

In this course we continue the introduction to stochastic partial differential equations, taking for granted the basics on Gaussian measure theory and semigroup theory introduced in the first part of the course.
We will focus on semilinear parabolic problems driven by additive noise, such as stochastic reaction diffusion equations $$du = \Delta u\ dt + f(u)dt + dW_t$$ and stochastic Navier-Stokes equations $$du = \Delta u\ dt - (u\cdot\nabla u)u\ dt-\nabla p\ dt + dW_t,\quad \textrm{div}\ u =0,$$ where \(W\) is an infinite-dimensional Wiener process. Such SPDE can be treated as stochastic evolution equations in some infinite-dimensional Banach space and they already form a rich class of problems with many interesting properties. Furthermore, this class of problems has the advantage of allowing to completely pass under silence many subtle problems arising from stochastic integration in infinite-dimensional spaces.
After having established the basic well-posedness results we will investigate questions on long-time behavior, ergodicity and random dynamics, e.g. the existence of random attractors.
Towards the end of the course an introduction to the recent theory of paracontrolled distributions will be given.

Date and time info
Wednesday 11:00 - 12:30

Keywords
Stochastic Partial Differential Equations, Stochastic Analysis

Prerequisites
basic measure theory, functional analysis and probability theory

Audience
MSc students, PhD students, Postdocs

Language
English