Abstract
This survey is devoted to Martin Hairer’s Reconstruction Theorem, which is one of the cornerstones of his theory of Regularity Structures \[6]. Our aim is to give a new self-contained and elementary proof of this theorem, together with some applications, including a characterization, based on a single arbitrary test function, of negative Hölder spaces. We present the Reconstruction Theorem as a general result in the theory of distributions that can be understood without any knowledge of Regularity Structures themselves, which we do not even need to define.
Highlights
Consider the following problem: if at each point x P Rd we are given a distribution Fx on Rd, is there a distribution f on Rd which is well approximated by Fx around each point x P Rd?A classical example is when f : Rd Ñ R is a smooth function and Fx is the Taylor polynomial of f based at x, of some fixed order r P N; we know that f pyqFxpyq is of order |yx|r1 for y P Rd close to x
We present some applications of independent interest, including a characterization of negative Hölder spaces based on a single arbitrary test function
With his theory of Rough Paths [Lyo98], Terry Lyons introduced the idea of a local description of the solution to a stochastic differential equation as a generalized Taylor expansion, where classical monomials are replaced by iterated integrals of the driving Brownian motion. This idea led Massimiliano Gubinelli to introduce his Sewing Lemma [Gub04], which is a version of the Reconstruction Theorem in R1. With his theory of regularity structures [Hai14], Martin Hairer translated these techniques in the context of stochastic partial differential equations (SPDEs), whose solutions are defined on Rd with d ą 1
Summary
Consider the following problem: if at each point x P Rd we are given a distribution (generalized function) Fx on Rd, is there a distribution f on Rd which is well approximated by Fx around each point x P Rd?. With his theory of Rough Paths [Lyo98], Terry Lyons introduced the idea of a local description of the solution to a stochastic differential equation as a generalized Taylor expansion, where classical monomials are replaced by iterated integrals of the driving Brownian motion This idea led Massimiliano Gubinelli to introduce his Sewing Lemma [Gub04], which is a version of the Reconstruction Theorem in R1 (the name “Sewing Lemma” is due to Feyel and de La Pradelle [FdLP06], who gave the proof which is commonly used). With his theory of regularity structures [Hai14], Martin Hairer translated these techniques in the context of stochastic partial differential equations (SPDEs), whose solutions are defined on Rd with d ą 1 (see [Zam21] for a history of SPDEs). In Section we construct a suitable product between distributions and non smooth functions, see Theorem 14.1, which is a multi-dimensional analogue of Young integration
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