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Itô perspective on variance renormalisation

Konstantinos Dareiotis, Máté Gerencsér

Abstract

We show that the Itô solutions of the nonlinear stochastic heat equation $$ \partial_t u^\varepsilon- Δu^\varepsilon =\varepsilon^{3/4} g (u^\varepsilon) \nabla ξ_\varepsilon, $$ where $ ξ_\varepsilon$ denotes the mollification in space at scale $\varepsilon>0$ of a space-time white noise $ξ$, converge in law, as $\varepsilon\to 0$, to the solution of the stochastic heat equation with right-hand side $cg'g(u)ξ$ with a constant $c>0$. Since the noise $\nablaξ$ is supercritical, the small prefactor is not unexpected to obtain a limit, but the exponent $3/4$ is not predicted by naive scaling arguments. The case $g(u)=u$, modulo a Cole-Hopf transform, corresponds to the result of [Hai25] for the KPZ equation. Our argument is relatively short and relies solely on stochastic analytic techniques.

Itô perspective on variance renormalisation

Abstract

We show that the Itô solutions of the nonlinear stochastic heat equation where denotes the mollification in space at scale of a space-time white noise , converge in law, as , to the solution of the stochastic heat equation with right-hand side with a constant . Since the noise is supercritical, the small prefactor is not unexpected to obtain a limit, but the exponent is not predicted by naive scaling arguments. The case , modulo a Cole-Hopf transform, corresponds to the result of [Hai25] for the KPZ equation. Our argument is relatively short and relies solely on stochastic analytic techniques.
Paper Structure (12 sections, 22 theorems, 3 equations)

This paper contains 12 sections, 22 theorems, 3 equations.

Table of Contents

  1. Introduction
  2. Setup and notation
  3. A priori estimates
  4. Estimate for the solutions
  5. Tightness
  6. Identification of the limit
  7. The limiting noise
  8. Sketch of the strategy
  9. Convergence of the main term
  10. Estimating the remainders
  11. Proof of the main theorem
  12. Auxiliary Lemmata

Key Result

Theorem 1.1

Let $\psi:\mathbb{T}\to{\mathbb{R}}$ be $1/4$-Hölder continuous and let $g:{\mathbb{R}}\to{\mathbb{R}}$ have bounded derivatives of order $1$, $2$, and $3$. For $\varepsilon>0$, let $u^\varepsilon$ denote the Itô solution of on $[0,1]\times\mathbb{T}$, and let $u$ denote the Itô solution of on $[0,1]\times\mathbb{T}$. Then, as $\varepsilon \to 0$, $u^\varepsilon$ converge weakly to $u$, as rando

Theorems & Definitions (47)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Definition 1.7
  • Remark 1.8
  • Theorem 2.1
  • Lemma 2.2
  • proof
  • ...and 37 more