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On the 1d stochastic Schr{ö}dinger product

Aurélien Deya

TL;DR

The paper analyzes the 1D stochastic Schrödinger product under space-white, time-fractional noise (H>1/2) and demonstrates precise limits of a first-order Da Prato–Debussche-type expansion. By decomposing the solution u=Ψ+z and renormalizing the Schrödinger product, it derives sharp thresholds in Bourgain-type spaces Z^{c,b} for the convergence of the renormalized product tree and the resonant/non-resonant product operators. It shows that, for H∈(1/2,3/4), the first-order approach is insufficient beyond certain regimes: the renormalized tree remains controlled only in restricted (c,b) ranges, while both the resonant L^{∘,(n)} and the non-resonant L^{♯,(n)} components impose strict conditions (c≥3/2−2H and c<b−1/4, respectively). Consequently, the combined obstructions imply that, for H up to 9/16, the first-order method fails to extend, and even at white noise (H=1/2) the method breaks, pointing to the need for higher-order or paracontrolled techniques to address rough stochastic NLS in this setting. The results delineate rather concrete barriers to well-posedness via first-order expansions and highlight the role of renormalization in controlling stochastic Schrödinger products.

Abstract

We exhibit various restrictions about the wellposedness of the Schr{\''o}dinger product $$\cl:z \longmapsto -\imath \int\_0^t e^{\imath s {\cop \partial^2\_x}}\big( z\_s\cdot Ψ\_s\big) ds $$ where $Ψ$ refers to the so-called linear solution of the stochastic Schr{\''o}dinger problem. We focus more specifically on the case where $Ψ$ satisfies \begin{equation}\label{starting-equation-abstract} (\imath \partial\_t-\partial^2\_x)Ψ=\dot{B}, \quad Ψ\_0=0,\quad \quad t\in \R, \ x\in \mathbb{T}, \end{equation} where $\dot{B}$ is a white noise in space with fractional time covariance of index $H>\frac12$. \smallskip As an consequence of our analysis, we obtain that if $H$ is close to $\frac12$ (that is $\dot{B}$ is close to a space-time white noise), then it is essentially impossible to treat the stochastic NLS problem \begin{equation*} (\imath \partial\_t-\partial^2\_x)u= |u|^2+\dot{B}, \quad u\_0=0,\quad \quad t\in \R, \ x\in \mathbb{T}, \end{equation*} using only a first-order expansion of the solution (\enquote{$u=Ψ+z$}).

On the 1d stochastic Schr{ö}dinger product

TL;DR

The paper analyzes the 1D stochastic Schrödinger product under space-white, time-fractional noise (H>1/2) and demonstrates precise limits of a first-order Da Prato–Debussche-type expansion. By decomposing the solution u=Ψ+z and renormalizing the Schrödinger product, it derives sharp thresholds in Bourgain-type spaces Z^{c,b} for the convergence of the renormalized product tree and the resonant/non-resonant product operators. It shows that, for H∈(1/2,3/4), the first-order approach is insufficient beyond certain regimes: the renormalized tree remains controlled only in restricted (c,b) ranges, while both the resonant L^{∘,(n)} and the non-resonant L^{♯,(n)} components impose strict conditions (c≥3/2−2H and c<b−1/4, respectively). Consequently, the combined obstructions imply that, for H up to 9/16, the first-order method fails to extend, and even at white noise (H=1/2) the method breaks, pointing to the need for higher-order or paracontrolled techniques to address rough stochastic NLS in this setting. The results delineate rather concrete barriers to well-posedness via first-order expansions and highlight the role of renormalization in controlling stochastic Schrödinger products.

Abstract

We exhibit various restrictions about the wellposedness of the Schr{\''o}dinger product where refers to the so-called linear solution of the stochastic Schr{\''o}dinger problem. We focus more specifically on the case where satisfies \begin{equation}\label{starting-equation-abstract} (\imath \partial\_t-\partial^2\_x)Ψ=\dot{B}, \quad Ψ\_0=0,\quad \quad t\in \R, \ x\in \mathbb{T}, \end{equation} where is a white noise in space with fractional time covariance of index . \smallskip As an consequence of our analysis, we obtain that if is close to (that is is close to a space-time white noise), then it is essentially impossible to treat the stochastic NLS problem \begin{equation*} (\imath \partial\_t-\partial^2\_x)u= |u|^2+\dot{B}, \quad u\_0=0,\quad \quad t\in \R, \ x\in \mathbb{T}, \end{equation*} using only a first-order expansion of the solution (\enquote{}).
Paper Structure (30 sections, 18 theorems, 332 equations)

This paper contains 30 sections, 18 theorems, 332 equations.

Table of Contents

  1. Presentation of the problem
  2. The first-order strategy and related questions
  3. About renormalization
  4. Stochastic Schrödinger product as a random operator
  5. Objective of the study
  6. Main results
  7. Problem (P1'): control of the renormalized Schrödinger product tree
  8. Problem (P2'): about the Schrödinger product operator $\mathcal{L}^{\circ,(n)}$
  9. Problem $\mathbf{(P2"')}$: about the Schrödinger product operator $\mathcal{L}^{\sharp,(n)}$
  10. Convergence of the Schrödinger product tree
  11. Covariance of the linear solution
  12. Proof of Proposition \ref{['prop:ice-cream-frac-intro']}
  13. Proof of Proposition \ref{['prop:ice-cream-frac-intro']}, item $(i)$
  14. Proof of Proposition \ref{['prop:ice-cream-frac-intro']}, item $(ii)$
  15. Proof of Proposition \ref{['prop:ice-cream-frac-intro']}, item $(iii)$
  16. ...and 15 more sections

Key Result

Lemma 1.1

In the above setting, the following assertions hold true. $(i)$ If $H>\frac{3}{4}$, then for every $T>0$, one has $\Psi\in L^2([0,T]\times \mathbb T)$ almost surely. $(ii)$ If $H\leq \frac{3}{4}$, then for every $T>0$, one has $\mathbb{E}[\|\Psi\|_{L^2([0,T]\times \mathbb{T})}^2] = \infty$.

Theorems & Definitions (37)

  • Lemma 1.1
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Remark 1.6
  • Remark 2.1
  • Proposition 2.2
  • Proposition 2.3
  • Remark 2.4
  • Corollary 2.5
  • ...and 27 more