Introduction to stochastic PDE
Abstract
This course is an attempt to give a reasonably self-contained presentation of the basic theory of stochastic partial differential equations, taking for granted basic measure theory, functional analysis and probability theory, but nothing else.
In view of the available time we shall 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.
The reader who is interested in a more detailed exposition of these more technically subtle parts of the theory might be advised to read the works
- Da Prato, Zabczyk; Stochastic equations in infinite dimensions, 1992
- Da Prato, Zabczyk; Ergodicity for infinite-dimensional systems, 1996
- Prevot, Röckner; A concise course on stochastic partial differential equations, 2007.
Thursday, 16:00 - 18:00
Keywords
Stochastic Partial Differential Equations, Stochastic Analysis
Prerequisites
basic measure theory, functional analysis and probability theory
Audience
MSc students, PhD students, Postdocs
Language
English
