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Locality for singular stochastic PDEs

I. Bailleul, Y. Bruned

Abstract

This work deals with singular stochastic PDEs driven by non-translation invariant differential operators. We describe the renormalized equation for a very large class of spacetime dependent renormalization schemes. Our approach bypasses in particular the use of decorated trees with extended decorations.

Locality for singular stochastic PDEs

Abstract

This work deals with singular stochastic PDEs driven by non-translation invariant differential operators. We describe the renormalized equation for a very large class of spacetime dependent renormalization schemes. Our approach bypasses in particular the use of decorated trees with extended decorations.

Paper Structure

This paper contains 11 sections, 13 theorems, 176 equations.

Table of Contents

  1. Introduction
  2. Decorated trees
  3. Preparation maps and their associated models
  4. Strong preparation maps
  5. Models associated with a preparation map.
  6. A short proof for the renormalized equation for translation invariant operators
  7. Renormalization schemes and renormalized equation in a non-translation invariant setting
  8. The analytic setting for non-translation invariant operators
  9. State space dependent preparation maps and their associated models
  10. BHZ renormalization
  11. Renormalization maps for branched rough paths

Key Result

Theorem \oldthetheorem

Let $\xi$ be a continuous noise. Let $R : (\mathbb{R}\times\textbf{T})\times T\rightarrow T$ be a strong preparation map such that $R\tau=\tau$ for all polynomials and planted trees $\tau$ in $\bigoplus_{a\in\frak{T}^+\times\{0\}}\mathcal{I}_a(T)$. Denote by ${\sf M}^{\!R}$ the admissible model asso is a solution of the renormalized system for some explicit functions $\Upsilon_p(\tau)(u,\partial_

Theorems & Definitions (26)

  • Theorem \oldthetheorem
  • Proposition 2.1
  • Lemma \oldthetheorem
  • proof
  • Proposition 2.2
  • proof
  • Definition \oldthetheorem
  • Proposition 3.1
  • proof
  • Definition \oldthetheorem
  • ...and 16 more