• Corpus ID: 115155988

An Introduction to Stochastic PDEs

  title={An Introduction to Stochastic PDEs},
  author={Martin Hairer},
  journal={arXiv: Probability},
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 in spring 2009. 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… 

Stochastic PDEs, Regularity Structures, and Interacting Particle Systems

These lecture notes grew out of a series of lectures given by the second named author in short courses in Toulouse, Matsumoto, and Darmstadt. The main aim is to explain some aspects of the theory of

The Navier-Stokes Equations on T with Stochastic Forcing

These notes results from a class project in MATH 595, a topics course on the analysis of the Navier-Stokes equations, at McGill University during the Fall semester 2014. They are heavily based on a

An introduction to singular stochastic PDEs: Allen-Cahn equations, metastability and regularity structures

These notes have been prepared for a series of lectures given at the Sarajevo Stochastic Analysis Winter School, from January 28 to February 1, 2019. There already exist several excellent lecture

A solution theory for a general class of SPDEs

This article presents a way of treating stochastic partial differential equations with multiplicative noise by rewriting them as stochastically perturbed evolutionary equations in the sense of Picard and McGhee (Partial differential equations: a unified Hilbert space approach, DeGruyter, Berlin, 2011), which allows for an effective treatment of coupled systems of SPDEs.

A Theory of Hypoellipticity and Unique Ergodicity for Semilinear Stochastic PDEs

We present a theory of hypoellipticity and unique ergodicity for semilinear parabolic stochastic PDEs with "polynomial" nonlinearities and additive noise, considered as abstract evolution equations

Discretisations of rough stochastic partial differential equations

This thesis consists of two parts, in both of which we consider approximations of rough stochastic PDEs and investigate convergence properties of the approximate solutions. In the first part we use

Hörmander’s theorem for semilinear SPDEs

We consider a broad class of semilinear SPDEs with multiplicative noise driven by a finite-dimensional Wiener process. We show that, provided that an infinite-dimensional analogue of Hormander's

Malliavin calculus and densities for singular stochastic partial differential equations

We study Malliavin differentiability of solutions to sub-critical singular parabolic stochastic partial differential equations (SPDEs) and we prove the existence of densities for a class of singular

Sampling conditioned hypoelliptic diffusions

A series of recent articles introduced a method to construct stochastic partial differential equations (SPDEs) which are invariant with respect to the distribution of a given conditioned diffusion.

Diffusions, Markov processes, and martingales

This celebrated book has been prepared with readers' needs in mind, remaining a systematic treatment of the subject whilst retaining its vitality. The second volume follows on from the first,

Singular perturbations to semilinear stochastic heat equations

  • Martin Hairer
  • Mathematics
    Probability Theory and Related Fields
  • 2010
We consider a class of singular perturbations to the stochastic heat equation or semilinear variations thereof. The interesting feature of these perturbations is that, as the small parameter ε tends

Spectral gaps in Wasserstein distances and the 2D stochastic Navier–Stokes equations

We develop a general method to prove the existence of spectral gaps for Markov semigroups on Banach spaces. Unlike most previous work, the type of norm we consider for this analysis is neither a

Rough stochastic PDEs

  • Martin Hairer
  • Mathematics
    Communications on Pure and Applied Mathematics
  • 2011
In this article, we show how the theory of rough paths can be used to provide a notion of solution to a class of nonlinear stochastic PDEs of Burgers type that exhibit too‐high spatial roughness for

Convergence of probability measures

The author's preface gives an outline: "This book is about weakconvergence methods in metric spaces, with applications sufficient to show their power and utility. The Introduction motivates the

Measure Theory

Uniqueness of solutions of stochastic differential equations

It follows from a theorem of Veretennikov [4] that (1) has a unique strong solution, i.e. there is a unique process x(t), adapted to the filtration of the Brownian motion, satisfying (1).

Approximations to the Stochastic Burgers Equation

This article is devoted to the numerical study of various finite-difference approximations to the stochastic Burgers equation, and forms a number of conjectures for more general classes of equations, supported by numerical evidence.

On nonseparable Banach spaces

Combining combinatorial methods from set theory with the functional structure of certain Banach spaces we get some results on the isomorphic structure of nonseparable Banach spaces. The conclusions