An Introduction to Stochastic PDEs

TL;DR

This paper provides a comprehensive, self-contained overview of the basic theory of stochastic partial differential equations, focusing on semilinear parabolic problems driven by additive noise, highlighting their regularising properties and mathematical structure.

Contribution

It offers a clear, accessible presentation of stochastic PDE theory, emphasizing semilinear parabolic problems and their infinite-dimensional analysis, with simplified treatment of stochastic integration issues.

Findings

Semilinear parabolic SPDEs have regularising properties.

These problems can be modeled as stochastic evolution equations in Banach or Hilbert spaces.

The approach simplifies the treatment of stochastic integration in infinite dimensions.

Abstract

These notes are based on a series of lectures given first at the University of Warwick in spring 2008 and then at the Courant Institute, Imperial College London, and EPFL. It 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. The approach taken in these notes is to focus on semilinear parabolic problems driven by additive noise. These can be treated as stochastic evolution equations in some infinite-dimensional Banach or Hilbert space that usually have nice regularising properties and they already form a very 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.