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Random models for singular SPDEs

I. Bailleul, Y. Bruned

Abstract

We give a proof of the convergence of the BHZ renormalized model associated with the generalized (KPZ) equation that does not require the full strength of the BPHZ renormalisation. Our approach is based on a convenient form of chaos decomposition. The other key ingredient is a generalisation of the Hairer-Quastel convergence theorem for Feynman diagrams with certain decorations encoding Taylor remainders. With these ideas we are able to construct the model for the generalised KPZ equation.

Random models for singular SPDEs

Abstract

We give a proof of the convergence of the BHZ renormalized model associated with the generalized (KPZ) equation that does not require the full strength of the BPHZ renormalisation. Our approach is based on a convenient form of chaos decomposition. The other key ingredient is a generalisation of the Hairer-Quastel convergence theorem for Feynman diagrams with certain decorations encoding Taylor remainders. With these ideas we are able to construct the model for the generalised KPZ equation.
Paper Structure (8 sections, 1 theorem, 82 equations)

This paper contains 8 sections, 1 theorem, 82 equations.

Table of Contents

  1. Introduction
  2. The generalized (KPZ) equation
  3. The (gKPZ) regularity structure and the BHZ renormalized model
  4. Tree-like graphs and their mirror graphs
  5. Malliavin derivatives of trees and their expectation
  6. Convergence of the BHZ model
  7. A workhorse
  8. Proof of convergence

Key Result

Theorem 2

If our graph $\bf G$ satisfies Condition (C) then there exists a constant $c\in(0,\infty)$ depending only on $\bf G$ such that

Theorems & Definitions (1)

  • Theorem 2