An Introduction to Stochastic PDEs
Martin Hairer
TL;DR
The notes lay a rigorous, self-contained foundation for stochastic PDEs by focusing on semilinear parabolic equations with additive noise and treating them as stochastic evolution equations in infinite-dimensional spaces. A core thread is the interplay between Gaussian measures, Cameron–Martin spaces, and semigroup theory, which together provide mild solution frameworks and regularity insights for SPDEs driven by space–time white noise. The text develops essential probability tools on Polish spaces (tightness, weak convergence, Wasserstein metrics) alongside Gaussian measure theory, culminating in a robust semigroup-based machinery for linear and perturbative analysis of SPDEs. Overall, the work establishes a cohesive framework linking probabilistic Gaussian structure with analytic semigroup methods to study stochastic evolution in infinite dimensions and to model regularity phenomena such as the Gaussian free field and cylindrical Wiener integration.
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.