Qualitative quasi-invariance of low regularity Gaussian measures for the 1d quintic nonlinear Schrödinger equation

Problem Statement
This paper addresses the qualitative quasi-invariance of Gaussian measures under the flow of the one-dimensional quintic nonlinear Schrödinger equation (NLS) on the torus $\mathbb{T} := \mathbb{R}/2\pi\mathbb{Z}$. Specifically, it considers initial data distributed according to Gaussian measures $\mu_s$ with covariance operator $(1-\Delta)^{-s}$. The central question is whether the law of the solution at any time $t$, denoted $\Phi(t)\#\mu_s$, is absolutely continuous with respect to the initial measure $\mu_s$.

The study focuses on the low regularity regime. While global well-posedness for the quintic NLS is currently known for initial data in $H^\sigma(\mathbb{T})$ with $\sigma > 2/5$ (established by Li-Wu-Xu [26]), the Gaussian measure $\mu_s$ is supported on $H^\sigma(\mathbb{T})$ only if $\sigma < s - 1/2$. Consequently, to apply the global flow to $\mu_s$-almost every initial datum, one requires $s - 1/2 > 2/5$, which implies the threshold $s > 9/10$. Prior results on quasi-invariance for this equation had been established for $s > 3/2$ (extending earlier work on even integers $s=2k$). This work aims to extend the quasi-invariance result down to the critical threshold $s > 9/10$.

Methodology
The approach combines probabilistic tools with deterministic energy estimates, following the framework initiated by Sun and Tzvetkov [36] and extended in [23]. The methodology proceeds through several key steps:

1. Truncation and Approximation: The authors work with a truncated version of the NLS, $\Phi_N(t)$, where high frequencies are projected out. They utilize the factorization of the truncated flow and the approximation properties of $\Phi_N(t)$ converging to the full flow $\Phi(t)$ to transfer quasi-invariance from the finite-dimensional approximation to the infinite-dimensional system.
2. Poincaré-Dulac Normal Form Reduction: To handle the non-conservation of the $H^s$ norm, the authors employ a normal form reduction. This involves defining a modified energy functional $E_{s,N}(u) = \frac{1}{2}\|\pi_N u\|_{H^s}^2 + R_{s,N}(u)$, where $R_{s,N}$ is an energy correction term derived from the resonant interactions of the nonlinearity. This transforms the problem into analyzing the transport of a weighted Gaussian measure $\rho_{s,N} = e^{-R_{s,N}} d\mu_s$.
3. Probabilistic Estimates via Boué-Dupuis Formula: A crucial innovation in this work is the use of the Boué-Dupuis variational formula (Lemma 2.11) to establish exponential integrability of the energy correction $R_{s,N}$ and its time derivative $Q_{s,N}$. This replaces the use of Wiener Chaos expansions, which the author notes is insufficient or overly complicated for reaching the $s > 9/10$ threshold.
4. Deterministic and Counting Estimates: The proof relies heavily on refined deterministic estimates for the nonlinearity $|u|^4u$ localized in dyadic frequency blocks. Key to this is the analysis of the resonant function $\Omega(\vec{k})$ and the symmetrized derivative $\psi_{2s}(\vec{k})$. The authors establish lower bounds on the resonant function away from the diagonal and utilize counting bounds (Lemmas 6.3 and 6.4) to control the number of resonant interactions. These estimates are combined with Strichartz estimates and $L^2$ bounds on the nonlinearity (Lemma 7.1) to derive the necessary dyadic bounds.

Key Contributions and Results
The primary result is Theorem 1.1, which states that for every $s > 9/10$, the Gaussian measure $\mu_s$ is quasi-invariant along the flow of the 1D quintic NLS. In other words, for any time $t \in \mathbb{R}$, the push-forward measure $\Phi(t)\#\mu_s$ is absolutely continuous with respect to $\mu_s$.

The paper achieves this by:

• Extending the range of quasi-invariance from $s > 3/2$ (the result of [23]) to $s > 9/10$, which coincides with the current threshold for global well-posedness of the equation in the relevant Sobolev spaces.
• Proving quantitative $L^p$ estimates for the energy functionals $R_{s,N}$ and $Q_{s,N}$ (Propositions 3.8 and 3.9) using the Boué-Dupuis formula rather than Wiener Chaos.
• Establishing a rigorous link between the quasi-invariance of the weighted measures $\rho_{s,N}$ and the original Gaussian measures $\mu_s$ via uniform estimates and approximation arguments (Section 5).

Significance and Limitations
The authors position this work as a partial extension of [23], pushing the regularity threshold for quasi-invariance to the limit of current global well-posedness knowledge. The significance lies in demonstrating that the quasi-invariance property holds even in low regularity regimes where the Hamiltonian is not conserved and standard energy methods fail.

However, the paper explicitly notes its limitations regarding the strength of the result:

• Qualitative vs. Quantitative: Unlike the result in [23] for $s > 3/2$, this work does not provide quantitative bounds on the Radon-Nikodym derivatives (i.e., it does not establish $L^p$ bounds for the density itself, only the absolute continuity). The authors attribute this to the inability to use the conservation of mass and Hamiltonian in the low regularity regime ($s \leq 3/2$).
• Sharpness: The paper does not claim $s > 9/10$ is the sharp threshold for quasi-invariance. It acknowledges that determining the exact threshold where singular behavior (mutual singularity) begins remains an open problem, similar to progress made for the cubic Szegő equation in [9].
• Methodological Constraint: The reliance on the Boué-Dupuis formula is highlighted as the enabling factor for this extension, whereas the previously used Wiener Chaos method appears insufficient for this specific range of $s$.

In summary, the paper provides a rigorous proof of the quasi-invariance of Gaussian measures for the 1D quintic NLS down to the regularity $s > 9/10$, utilizing a novel combination of normal form reductions and variational probabilistic estimates, while acknowledging that quantitative control of the resulting densities remains out of reach with current techniques.