An Introduction to Stochastic PDEs

By Martin Hairer. 2009

289 citations recorded by the source corpus.

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 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.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Finance

View Paper: https://www.semanticscholar.org/paper/78cc77b964ec51a48c8b7251b6dad4a7b31c2f77

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Citations in the corpus

The Bayesian Approach to Inverse Problems (2013, background): “22 in [40] to the non-Gaussian setting and to Hölder, rather than Lipschitz, functions {ψk}).”
Inverse problems: A Bayesian perspective (2010, background): “In particular, Theorem 6.4 is proved in Lifshits (1995), and Hairer (2009) has a nice exposition of the Fernique Theorem.”
A Computational Framework for Infinite-Dimensional Bayesian Inverse Problems Part I: The Linearized Case, with Application to Global Seismic Inversion (2013, unknown): “We recall that the Cameron-Martin space E equipped with the inner product (· , ·)E := (C 0 ·, C −1/2 0 ·) associated with C0 is the range of C 1/2 0 [27], and hence coincides with D (A).”
A mild Itô formula for SPDEs (2010, background): “chastic partial diﬀerential equation (SPDE) (1) (SPDEs have been extremely intensively studied in the last decades; see, e.g., the books [95, 21, 35, 22, 12, 19, 96, 14, 89, 61, 47] and lecture notes [105, 57, 67, 90, 1, 24, 40] and the references therein). A simple example of this framework is the ”
Spectral gaps for a Metropolis–Hastings algorithm in infinite dimensions (2011, methodology): “In this paper we substantiate these ideas by using spectral gaps derived by an application of the weak Harris theorem [Hairer, Mattingly and Scheutzow (2011)].”
Sparse polynomial approximation of parametric elliptic PDEs. Part II: lognormal coefficients (2015, unknown): “A strategy to establish that b with representation (10) is almost surely in a Hölder space C κ for some κ > 0 is discussed in [18, 11].”