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Renormalization of a 1d quadratic Schr{ö}dinger model with additive noise

Aurélien Deya, Reika Fukuizumi, Laurent Thomann

TL;DR

This work analyzes a 1D quadratic Schrödinger equation driven by additive space-time fractional noise, showing that the equation is ill-posed in rough noise regimes and enabling a rigorous renormalization for a critical subrange. By constructing an explicit renormalization via a Da Prato–Debussche-type decomposition and carefully treating the stochastic products as random operators, the authors establish local well-posedness of the renormalized remainder equation and prove convergence of renormalized strong solutions as the noise approximation is refined. The approach separates stochastic and deterministic components pathwise, providing a transparent renormalization scheme with explicitly computable terms $\Lambda^{(n)}$ and $\sigma^{(n)}$, and yielding a limit solution $u$ with a precise regularity description. The results extend the pathwise analysis of stochastic NLS to fractional noise on the torus, highlighting intricate regularity and renormalization phenomena unique to dispersive equations with rough forcing and opening avenues for further study of higher-order expansions and optimal noise-regularity thresholds.

Abstract

The study is devoted to the interpretation and wellposedness of the stochastic NLS model \begin{equation*} (\imath \partial_t-Δ)u=|u|^2+\dot{B}, \quad u_0=0,\quad \quad t\in \mathbb{R}, \ x\in \mathbb{T}, \end{equation*} where $\dot{B}$ stands for a space-time fractional noise with index $H=(H_0,H_1)$ in a subset of $(0,1)^{2}$. We first establish that in the situation where $0<2H_0+H_1\leq 2$, the equation cannot be interpreted in a (classical) functional sense.\\ \indent Our investigations then focus on the rough regime corresponding to the condition $\frac74<2H_0+H_1\leq 2$. In this specific case, we exhibit an \textit{explicit} renormalization procedure allowing to restore the (local) convergence of the approximated solutions. We follow a pathwise-type approach emphasizing the distinction between the stochastic objects at the core of the dynamics and the general deterministic machinery.

Renormalization of a 1d quadratic Schr{ö}dinger model with additive noise

TL;DR

This work analyzes a 1D quadratic Schrödinger equation driven by additive space-time fractional noise, showing that the equation is ill-posed in rough noise regimes and enabling a rigorous renormalization for a critical subrange. By constructing an explicit renormalization via a Da Prato–Debussche-type decomposition and carefully treating the stochastic products as random operators, the authors establish local well-posedness of the renormalized remainder equation and prove convergence of renormalized strong solutions as the noise approximation is refined. The approach separates stochastic and deterministic components pathwise, providing a transparent renormalization scheme with explicitly computable terms and , and yielding a limit solution with a precise regularity description. The results extend the pathwise analysis of stochastic NLS to fractional noise on the torus, highlighting intricate regularity and renormalization phenomena unique to dispersive equations with rough forcing and opening avenues for further study of higher-order expansions and optimal noise-regularity thresholds.

Abstract

The study is devoted to the interpretation and wellposedness of the stochastic NLS model \begin{equation*} (\imath \partial_t-Δ)u=|u|^2+\dot{B}, \quad u_0=0,\quad \quad t\in \mathbb{R}, \ x\in \mathbb{T}, \end{equation*} where stands for a space-time fractional noise with index in a subset of . We first establish that in the situation where , the equation cannot be interpreted in a (classical) functional sense.\\ \indent Our investigations then focus on the rough regime corresponding to the condition . In this specific case, we exhibit an \textit{explicit} renormalization procedure allowing to restore the (local) convergence of the approximated solutions. We follow a pathwise-type approach emphasizing the distinction between the stochastic objects at the core of the dynamics and the general deterministic machinery.
Paper Structure (36 sections, 26 theorems, 322 equations)

This paper contains 36 sections, 26 theorems, 322 equations.

Table of Contents

  1. Introduction
  2. General considerations
  3. Reformulation of the equation and main steps of the strategy
  4. Main results
  5. Fractional noise on $\mathbb R\times \mathbb T$
  6. Approximation of the noise
  7. Stochastic tree-elements
  8. Stochastic product as a random operator
  9. Wellposedness of the (renormalized) remainder equation
  10. Convergence of the (renormalized) strong solutions
  11. Preliminary estimates and fully-deterministic controls
  12. Basic controls on time integration and time localization
  13. Singular integrals
  14. Fully deterministic controls on the product
  15. Study of the tree-elements I: first-order
  16. ...and 21 more sections

Key Result

Theorem 1.2

Assume that the indexes $H_0,H_1$ satisfy the three conditions Then for a natural smooth approximation $(\dot{B}^{(n)})_{n\geq 1}$ of $\dot{B}$, there exist two sequences such that: $(i)$ For every $n\geq 1$, $\Lambda^{(n)}:\Omega \times \mathbb R \to \mathbb C$ is a time process, while $\sigma^{(n)}:\mathbb R \to \mathbb C$ is a deterministic time function. $(ii)$ For every $n\geq 1$, both $\La

Theorems & Definitions (56)

  • Remark 1.1
  • Theorem 1.2
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Lemma 1.7
  • Remark 2.1
  • Proposition 2.2
  • Remark 2.3
  • Proposition 2.5
  • ...and 46 more