Stochastic partial differential equations

Head:
Benjamin Gess (Email, Homepage)

Phone:
+49 (0) 341 - 9959 - 954

Fax:
+49 (0) 341 - 9959 - 658

Address:
Inselstr. 22
04103 Leipzig

Winterterm 2016/2017

Introduction to stochastic PDE

  • Lecturer: Benjamin Gess
  • Date: Wednesday 11:00 - 12:30
  • Room: MPI MiS, A3 02
  • Language: English
  • Target audience: MSc students, PhD students, Postdocs
  • Keywords: Stochastic Partial Differential Equations, Stochastic Analysis
  • Prerequisites: basic measure theory, functional analysis and probability theory

Abstract

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.

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28.09.2021, 13:34