Unconditional well-posedness of the stochastic Korteweg-de Vries equation on the real line

Technical Summary: Unconditional Well-Posedness of the Stochastic Korteweg-de Vries Equation on the Real Line

Problem Statement
The paper investigates the well-posedness of the stochastic Korteweg-de Vries equation (SKdV) with additive noise on the real line $\mathbb{R}$:
$$\partial_t u + \partial_x^3 u + \partial_x(u^2) = \phi \xi, \quad u|_{t=0} = u_0,$$
where $\xi$ is a space-time white noise and $\phi$ is a linear operator mapping $L^2(\mathbb{R})$ to $L^2(\mathbb{R})$. The primary focus is on establishing global well-posedness and unconditional uniqueness in the space $L^2(\mathbb{R})$.

Prior work by de Bouard, Debussche, and Tsutsumi (1999) established global well-posedness in $L^2(\mathbb{R})$ but required an artificial regularity assumption on the noise operator $\phi$, specifically $\phi \in \mathcal{H}S(L^2; \dot{H}^{-3/8})$, in addition to the natural assumption $\phi \in \mathcal{H}S(L^2; L^2)$. Furthermore, while unconditional uniqueness (uniqueness within the entire class $C([0,T]; L^2)$ without reference to auxiliary function spaces) is well-understood for the deterministic KdV, no such result existed for the stochastic case with low-regularity initial data.

Methodology
The authors employ the Fourier restriction norm method adapted to Fourier-Lebesgue spaces in time to address the regularity of the stochastic convolution.

1. Adaptation of Function Spaces:
   Standard $X^{s,b}$ spaces (Bourgain spaces) typically require $b > 1/2$ to ensure continuity in time, but the stochastic convolution $\Psi$ (driven by Brownian motion) only possesses temporal regularity corresponding to $b < 1/2$. To resolve this, the authors utilize spaces $X^{s,b}_{q}$ defined via the norm:
   $$\|u\|_{X^{s,b}_{q}} = \|\langle \xi \rangle^s \langle \tau - \xi^3 \rangle^b \hat{u}(\tau, \xi)\|_{L^2_\xi L^q_\tau}.$$
   By choosing $q > 2$ and $b > 1/2$ such that $(b-1)q < -1$, these spaces capture the local-in-time regularity of the stochastic convolution while allowing for $b > 1/2$. This choice is crucial for handling the "high $\times$ low $\to$ high" frequency interactions in the bilinear estimates without the artificial noise regularity assumption.

2. Local Well-Posedness:
   The authors prove local well-posedness in $X^{s,b}_{q}$ spaces. The key innovation is establishing a bilinear estimate (Proposition 3.1) in these spaces for $b > 1/2$, which was previously obstructed by the need to handle the stochastic convolution's low regularity. Once local well-posedness is established under the natural assumption $\phi \in \mathcal{H}S(L^2; L^2)$, global well-posedness follows from standard a priori $L^2$ bounds derived via Itô's lemma.

3. Unconditional Uniqueness:
   To prove unconditional uniqueness, the authors adapt an argument by Zhou (1997) for the deterministic KdV to the stochastic setting.

   • Decomposition: They decompose the solution $u$ into a linear part $\Phi$ (containing the initial data and stochastic convolution) and a remainder $v = u - \Phi$.
   • Estimates: They establish bounds for $v$ in specific low-regularity spaces ($X^{-3/2-\epsilon, 1}$) by exploiting the equation satisfied by $v$. Unlike the deterministic case, they cannot rely on pointwise time estimates of the equation due to the noise; instead, they use a first-order expansion and careful time-localization arguments to avoid scaling issues.
   • Bilinear Estimates: They prove modified bilinear estimates (Lemmas 4.2 and 4.3) involving the $X^\epsilon$ and $Y^\epsilon$ norms, which allow them to control the difference of two solutions $w = u_1 - u_2$ in a contraction argument.

Key Contributions and Results

• Theorem 1.1 (Global Well-Posedness): The authors prove that the SKdV is globally well-posed in $H^s(\mathbb{R})$ for $s \ge 0$ under the minimal assumption $\phi \in \mathcal{H}S(L^2; H^s)$. This result removes the necessity of the condition $\phi \in \mathcal{H}S(L^2; \dot{H}^{-3/8})$ imposed in previous literature, showing that the condition was artificial for $s \ge 0$.
• Theorem 1.3 (Pathwise Unconditional Uniqueness): The paper establishes the first pathwise unconditional uniqueness result for stochastic dispersive PDEs in a low-regularity setting. Specifically, given two global solutions $u_1, u_2 \in C(\mathbb{R}; L^2(\mathbb{R}))$ to the SKdV with the same initial data and noise realization, they are identical almost surely. This uniqueness holds in the entire class $C([0,T]; L^2)$ without intersecting with auxiliary spaces like $X^{s,b}$.
• Appendix: The paper includes a technical argument proving the boundedness of multiplication by a sharp cutoff function in Fourier-Lebesgue and Sobolev spaces, a tool of independent interest.

Significance and Claims
The paper claims to resolve a question open for over two decades regarding the necessity of the $\dot{H}^{-3/8}$ regularity assumption on the noise operator for global well-posedness in $L^2$. By utilizing Fourier-Lebesgue spaces in time, the authors demonstrate that the standard regularity assumption on $\phi$ is sufficient.

Furthermore, the work provides the first unconditional uniqueness result for stochastic dispersive equations with low-regularity initial data. The authors note that $L^2$ regularity is optimal for this result, as it is the minimal space required to make sense of the quadratic nonlinearity $\partial_x(u^2)$ as a spatial distribution without auxiliary function spaces. The proof strategy adapts Zhou's deterministic argument to the stochastic setting, overcoming the lack of pointwise time regularity in the noise term through a decomposition and careful estimation of the remainder term.

The authors explicitly state that while normal form approaches (as used in recent works on the circle) could provide alternative proofs, their method is more direct and avoids the lengthier iterations required by normal form reductions. They also note that extending these results to negative Sobolev spaces remains an open problem for future work.