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A tourist's guide to regularity structures and singular stochastic PDEs

I. Bailleul, M. Hoshino

Abstract

We give an essentially self-contained treatment of the fundamental analytic and algebraic features of regularity structures and its applications to the study of singular stochastic PDEs.

A tourist's guide to regularity structures and singular stochastic PDEs

Abstract

We give an essentially self-contained treatment of the fundamental analytic and algebraic features of regularity structures and its applications to the study of singular stochastic PDEs.

Paper Structure

This paper contains 41 sections, 11 theorems, 362 equations.

Table of Contents

  1. Introduction
  2. Basics on regularity structures
  3. Algebra as the mechanics of local expansion devices
  4. Regularity structures
  5. Models and modelled distributions
  6. Products and derivatives
  7. Regularity structures built from integration operators
  8. Operators on $\mathbb{R}\times\mathbb{R}^d$
  9. Regularity structures with some abstract integration operators
  10. Admissible models
  11. Lifting ${\bf K}$ as a continuous map from $\mathcal{D}^\gamma(T,\sf g)$ into $\mathcal{D}^{\gamma+2}(T,\sf g)$
  12. Building admissible smooth models
  13. Solving singular PDEs within regularity structures
  14. Spatially periodic models
  15. Modelled distributions with singularity at $x_0=0$
  16. ...and 26 more sections

Key Result

Theorem 4

(Reconstruction theorem) Let $\mathscr{T}$ be a concrete regularity structure and ${\sf M}=(\sf g,\Pi)$ be a model over $\mathscr{T}$. Fix a regularity exponent $\gamma\in\mathbb{R}\backslash\{0\}$. There exists a linear continuous operator satisfying the property uniformly in $\boldsymbol{f}\in\mathcal{D}^\gamma(T, {\sf g}), x\in\mathbb{R}\times\mathbb{R}^d$ and $0<t\leq 1$. Such an operator is

Theorems & Definitions (38)

  • Definition
  • Definition
  • Definition
  • Definition
  • Remark
  • Theorem 4
  • Definition
  • Definition
  • Definition
  • Theorem 17
  • ...and 28 more