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Discretisation of regularity structures

Dirk Erhard, Martin Hairer

Abstract

We introduce a general framework allowing to apply the theory of regularity structures to discretisations of stochastic PDEs. The approach pursued in this article is that we do not focus on any one specific discretisation procedure. Instead, we assume that we are given a scale $\varepsilon > 0$ and a "black box" describing the behaviour of our discretised objects at scales below $\varepsilon $.

Discretisation of regularity structures

Abstract

We introduce a general framework allowing to apply the theory of regularity structures to discretisations of stochastic PDEs. The approach pursued in this article is that we do not focus on any one specific discretisation procedure. Instead, we assume that we are given a scale and a "black box" describing the behaviour of our discretised objects at scales below .

Paper Structure

This paper contains 17 sections, 11 theorems, 159 equations.

Table of Contents

  1. Introduction
  2. Discussion
  3. Strategy
  4. Structure of the article
  5. Notation
  6. A framework for discretisations
  7. The reconstruction theorem
  8. The reconstruction operator in weighted $\CD_\eps^\gamma$-spaces
  9. Convolution operators
  10. Comparison between continuous and discrete convolution operators
  11. Convolution operators in weigthed $\CD^\gamma_\eps$-spaces
  12. Local operations
  13. Multiplication
  14. Composition with a smooth function
  15. Differentiation
  16. ...and 2 more sections

Key Result

theorem \oldthetheorem

Let $\gamma>0$, and fix a compact set $\mathfrak{K}$. Fix a discrete model $(\Pi^\eps,\Gamma^\eps)$ such that Assumption a:rec is satisfied. We then have the estimate uniformly over all test function $\eta\in \Phi$, all $\delta\in(\eps,1]$, all $f\in \CD_\eps^\gamma$, all $z\in\mathfrak{K}$, and all $\eps\in (0,1]$. Given a second discrete model $(\bar{\Pi}^{\eps},\bar{\Gamma}^{\eps})$ such that

Theorems & Definitions (73)

  • remark 1
  • remark 2
  • definition 1
  • remark 3
  • remark 4
  • remark 5
  • remark 6
  • remark 7
  • definition 2
  • remark 8
  • ...and 63 more