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Nonlinear Schrödinger equations with spatial white noise potential on full space for $d\le 3$

This paper establishes the existence, uniqueness, and local well-posedness of energy solutions for nonlinear Schrödinger equations with multiplicative spatial white noise on $\mathbb{R}^d$ for dimensions $d \le 3$ , utilizing exponential transformations, conserved quantities, and paracontrolled calculus to achieve propagation without loss of regularity or localization.

Original authors: Antoine Mouzard, Immanuel Zachhuber

Published 2026-04-17

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CC BY 4.0

Original authors: Antoine Mouzard, Immanuel Zachhuber

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to predict how a ripple moves across a pond. In a perfect, calm world, this is easy: the water is smooth, the wind is steady, and the math works out beautifully. This is the standard Nonlinear Schrödinger Equation (NLS), a famous formula used to describe waves in physics, from light in fiber optics to atoms in a Bose-Einstein condensate.

But now, imagine that the pond isn't calm. Imagine that the water is being hit by a chaotic, unpredictable storm of raindrops that never stop, hitting every single point on the surface at random. This is what mathematicians call "spatial white noise." It's a mathematical representation of pure, unstructured chaos.

The problem is that this "rain" is so violent and irregular that the water surface becomes jagged and broken. In fact, it's so broken that the standard math tools we use to predict the waves simply break down. If you try to calculate the wave's path, you get infinite numbers or nonsense results. This is the challenge the authors, Antoine Mouzard and Immanuel Zachhuber, set out to solve.

Here is a breakdown of their breakthrough, using some everyday analogies:

1. The Problem: A Broken Compass in a Storm

The authors are studying these waves in full space (an infinite pond), not just a small, contained bathtub (a torus).

The Noise: The "white noise" is like a static-filled radio signal that gets louder the further you go from the center. It's not just messy; it's growing wilder at the edges of the universe.
The Difficulty: In dimensions 2 and 3 (a flat sheet or a 3D volume), this noise is so rough that the wave equation literally doesn't make sense. It's like trying to drive a car on a road made of jagged glass; the wheels (the math) can't grip.

2. The Old Solution: The "Exponential Mask" (and its flaws)

Previous researchers tried to solve this by putting a "mask" over the chaos. They used a trick called an exponential transform.

The Analogy: Imagine the wave is a person walking through a storm. The old method said, "Let's wrap the person in a giant, magical raincoat ( $e^X$ ) that absorbs all the rain."
The Flaw: While this kept the person dry, the raincoat was so heavy and weirdly shaped that it distorted the person's movement.
1. Loss of Smoothness: The person started walking clumsily. The solution became "rougher" than the starting point.
2. Loss of Location: Because the raincoat grew infinitely large at the edges of the storm, the person eventually got lost. The math couldn't tell you where the wave was anymore because the "coat" stretched to infinity.

3. The New Solution: A Smart, Localized Raincoat

Mouzard and Zachhuber invented a better way. They realized they didn't need a raincoat that covered the entire universe. They needed a smart, localized shield.

The Trick: They split the storm into two parts:
1. The Local Chaos: The jagged, dangerous raindrops right next to the wave.
2. The Global Growth: The fact that the storm gets louder far away.
The Innovation: They built a shield that only handles the Local Chaos. It smooths out the jagged glass right under the wave's feet so the math works. Crucially, they designed this shield so it doesn't grow as you move away from the center.
The Result: The wave can still feel the storm, but it can walk smoothly without getting distorted or lost. They proved that the wave stays exactly where it should be and keeps its shape, even in the chaos.

4. The "Magic Glasses" (Paracontrolled Calculus)

To prove that their solution is unique (that there is only one correct path for the wave), they needed to see the wave's "dispersion"—how it spreads out over time.

The Analogy: Imagine trying to watch a fast-moving car in a foggy, chaotic street. You can't see it clearly.
The Tool: They used a mathematical technique called Paracontrolled Calculus. Think of this as putting on a pair of high-tech "Magic Glasses."
- These glasses don't just filter the fog; they reorganize the chaos. They separate the "noise" from the "signal" so clearly that the wave's path becomes visible.
- Using these glasses, they proved that the wave spreads out exactly as physics predicts, even with the noise. This allowed them to prove that their solution is the only possible solution.

Why This Matters

Before this paper, we could only solve these equations in small, contained boxes (like a torus) or with solutions that got messy and lost their way over time.

Global Success: They proved that in 2D and 3D, you can predict these chaotic waves forever without losing track of them or their smoothness.
Firsts: This is the first time anyone has successfully solved this specific type of "singular" wave equation on an infinite 3D space without the solution falling apart.
Real World: While this is pure math, it helps us understand how waves behave in extremely turbulent environments, which could eventually help with better models for quantum physics, fluid dynamics, or signal processing in noisy environments.

In short: The authors took a chaotic, infinite storm that broke all previous math tools, built a custom "smart shield" to handle the local mess without getting lost in the global chaos, and used "magic glasses" to prove that the waves move exactly as they should. They finally tamed the wild, infinite pond.

1. Problem Statement

The paper addresses the global well-posedness and uniqueness of energy solutions for singular stochastic dispersive partial differential equations (SPDEs) on the full Euclidean space $\mathbb{R}^d$ with $d \in \{2, 3\}$ . Specifically, the authors study:

The Nonlinear Schrödinger Equation (NLS):
$i\partial_t u = -\Delta u + u\xi + |u|^{m-1}u$
The Hartree Equation:
$i\partial_t u = -\Delta u + u\xi + u \cdot (V_\beta * |u|^2)$

Key Challenges:

Singular Noise: The potential $\xi$ is a spatial Gaussian white noise, which belongs to the Hölder-Besov space $C^{-d/2-\kappa}_{-\delta}$ . In dimensions $d \ge 2$ , the noise is too irregular for the product $u\xi$ to be defined in the classical sense, requiring probabilistic renormalization.
Full Space vs. Compact Domains: Previous results on tori ( $\mathbb{T}^d$ ) relied on the spectral properties of the Anderson Hamiltonian ( $H = -\Delta + \xi$ ), which has a compact resolvent. On $\mathbb{R}^d$ , the spectrum of $H$ is the entire real line $\mathbb{R}$ , and the operator lacks compactness, making spectral methods inapplicable.
Loss of Regularity and Localization: Previous approaches using an exponential transform ( $u = e^X v$ ) to handle the singularity often resulted in a loss of regularity (solutions becoming rougher than initial data) and a loss of spatial localization (solutions growing at infinity due to the unbounded nature of the random field $X$ ).

2. Methodology

The authors employ a combination of exponential transforms, paracontrolled calculus, and renormalization techniques adapted to weighted spaces.

A. Exponential Transform and Renormalization

To handle the singularity, the authors introduce a random field $X$ and define a new variable $v$ such that $u = e^X v$ .

Renormalization: They construct $X$ such that the term $\xi - \Delta X - |\nabla X|^2$ converges to a well-defined random distribution $Y$ (after subtracting diverging constants $c_\epsilon$ ).
Spatial Localization (Crucial Innovation): Unlike previous works on $\mathbb{R}^d$ $R^{d}$ where $X$ $X$ grows at infinity, the authors decompose the noise $\xi = \xi_\le + \xi_>$ $ξ = ξ_{\leq} + ξ_{>}$ using a frequency/space localization operator (based on Gubinelli and Hofmanovà).
- $X$ is constructed only from the high-frequency, decaying part of the noise ( $\xi_>$ ).
- This ensures that $e^X$ and $e^{-X}$ are bounded ( $L^\infty$ ), preserving the equivalence of weighted $L^2$ norms and preventing the loss of spatial localization.

B. Definition of Energy Solutions

The authors define energy solutions in the dual space of $e^X(H^1 \cap L^2_\mu)$ .

Instead of the standard weak formulation which leads to ill-defined products like $\nabla X \cdot \nabla v$ , they multiply the equation by $e^X$ to obtain a formulation involving the operator $e^X H e^X$ .
The equation is tested against functions in the form domain, allowing for a weak formulation that makes sense for $v \in H^1$ .

C. Paracontrolled Calculus and Strichartz Inequalities

To prove uniqueness and low-regularity well-posedness, the authors derive Strichartz inequalities for the linear flow associated with the singular operator.

Perturbative Argument: They cannot directly use the Anderson Hamiltonian's spectral properties. Instead, they use a paracontrolled transformation $\Gamma$ .
Conjugated Operator: They define a transformed operator $H^\sharp = (\Gamma e^X)^{-1} H (e^X \Gamma)$ .
Regularity Gain: The map $\Gamma$ is constructed via a paracontrolled ansatz ( $v = \Gamma v^\sharp$ ) such that $H^\sharp$ is a "better behaved" perturbation of the Laplacian. Specifically, $H^\sharp + \Delta$ maps into a space of positive regularity, unlike the original singular operator.
This allows the use of the standard Schrödinger group $e^{it\Delta}$ to derive Strichartz estimates for the singular flow.

3. Key Contributions

First Results on $\mathbb{R}^3$ : This is the first proof of well-posedness for singular dispersive SPDEs on $\mathbb{R}^3$ . Previous results were limited to the torus $\mathbb{T}^3$ .
No Loss of Regularity or Localization: The authors prove global existence and uniqueness of energy solutions without the loss of regularity or spatial decay that plagued previous exponential transform methods on full space.
Strichartz Inequalities for Singular Flows: They establish Strichartz estimates for the linear equation with white noise potential on $\mathbb{R}^d$ ( $d=2,3$ ) using paracontrolled calculus, a significant technical hurdle previously unsolved in this context.
Deterministic Initial Data: The results allow for deterministic initial data in two dimensions ( $d=2$ ) with polynomial decay, a setting where previous results required stochastic initial data or suffered from regularity loss.

4. Main Results

Existence of Energy Solutions

Theorem: For initial data $u_0 \in e^{-X}(L^2_\mu \cap H^1)$ $u_{0} \in e^{- X} (L_{μ}^{2} \cap H^{1})$ , there exists a unique global energy solution $u \in C(\mathbb{R}, e^{-X}(L^2_\mu \cap H^1))$ $u \in C (R, e^{- X} (L_{μ}^{2} \cap H^{1}))$ .
- NLS on $\mathbb{R}^2$ : Valid for all $m \ge 1$ .
- NLS on $\mathbb{R}^3$ : Valid for $1 \le m < 6$ .
- Hartree on $\mathbb{R}^3$ : Valid for $\beta > 0$ (existence) and $\beta > 1/2$ (uniqueness).
Method: Existence is proven via a compactness argument using uniform bounds on the renormalized energy of regularized solutions.

Uniqueness and Low Regularity Well-Posedness

Uniqueness: Uniqueness of energy solutions is established using the derived Strichartz inequalities and Gronwall's lemma.
Local Well-Posedness:
- $\mathbb{R}^2$ (NLS): Local well-posedness for initial data in $L^2_\sigma \cap H^\sigma$ with $\sigma \in (\sigma_m, 1)$ , where $\sigma_m = 1 - \frac{1}{m-1}$ .
- $\mathbb{R}^3$ (Hartree): Local well-posedness for $\beta > 1/2$ with $\sigma \in (\sigma_\beta, 1)$ , where $\sigma_\beta = 5/4 - \beta/2$ .
Significance: These results improve upon previous conditions (e.g., $\beta > 17/20$ on the torus) by leveraging the better dispersive properties of the free Schrödinger flow on $\mathbb{R}^3$ compared to $\mathbb{T}^3$ .

5. Significance

This work represents a major advancement in the theory of singular SPDEs. By successfully extending the theory from compact manifolds to the full Euclidean space $\mathbb{R}^d$ without sacrificing regularity or localization, the authors bridge a critical gap between parabolic singular SPDEs (where such results are more common) and dispersive ones.

The introduction of a localized random field to handle the growth at infinity, combined with paracontrolled calculus to recover dispersive estimates, provides a robust framework for studying singular dispersive equations in unbounded domains. This opens the door to further studies on long-time dynamics, scattering, and other singular dispersive models in non-compact settings.