Nonlinear PDEs with modulated dispersion III: multiplicative noises

Technical Summary: Nonlinear PDEs with Modulated Dispersion III: Multiplicative Noises

Problem Statement
This paper investigates the pathwise well-posedness of the stochastic modulated Korteweg-de Vries (KdV) equation on the circle $\mathbb{T}$ with a multiplicative noise. The equation is given by:
$$\partial_t u + \partial_x^3 u \cdot \partial_t w = \partial_x (u^2) + u \phi \zeta$$
where $w: \mathbb{R}_+ \to \mathbb{R}$ is a time-non-homogeneous modulation acting on the linear dispersion term, and $\zeta$ represents a noise that is either fractional-in-time (Young case, Hurst parameter $\beta \in (1/2, 1)$) or white-in-time (rough case, $\beta = 1/2$). The noise term is multiplicative, taking the form $u \phi \zeta$, where $\phi$ is a Hilbert-Schmidt operator and $\zeta$ is a space-time white noise or fractional noise.

The primary challenge addressed is the lack of temporal regularity in the modulation $w$ and the noise $\zeta$, which prevents the application of standard stochastic calculus. Furthermore, the paper seeks to determine whether the irregularity of the modulation $w$ can induce a "regularization by noise" phenomenon, specifically smoothing the stochastic convolution term in a pathwise manner.

Methodology
The authors employ a pathwise approach, avoiding probabilistic averaging in favor of deterministic analysis for almost every realization of the noise and modulation. The methodology relies on two main pillars:

1. Nonlinear Young and Rough Integration Theory: The equation is reformulated using the modulated interaction representation $\mathbf{u}(t) = U_w(t)^{-1} u(t)$, where $U_w(t) = e^{-w(t)\partial_x^3}$. This transforms the equation into a nonlinear Young differential equation (YDE) with a perturbation:
   $$\mathbf{u} = \mathbf{u}_0 + I_X(\mathbf{u}) + I_{\text{noise}}(\mathbf{u})$$
   Here, $I_X$ is a nonlinear Young integral associated with the bilinear nonlinearity, and $I_{\text{noise}}$ is the stochastic term.

   • In the Young case ($\beta > 1/2$), the stochastic term is constructed as a linear Young integral.
   • In the rough case ($\beta = 1/2$), the stochastic term is constructed as a rough integral using the theory of controlled rough paths (Gubinelli derivative).

2. Random Tensor Estimates and Sewing Lemma: To establish the regularity of the stochastic drivers (the operators defining the integrals), the authors utilize:

   • The sewing lemma (introduced by Gubinelli) to construct the integrals as limits of Riemann-Stieltjes sums.
   • Random tensor estimates (developed in prior works by the authors and others) to bound the operator norms of the stochastic drivers. These estimates are crucial for proving that the drivers possess sufficient spatial regularity despite the roughness of the noise.
   • Kolmogorov's continuity criterion for operator-valued processes to upgrade $L^p$ bounds to almost sure pathwise regularity.

Key Contributions and Results

1. The Young Case (Fractional-in-Time Noise, $1/2 < \beta < 1$):

• Regularization by Noise: The paper establishes a novel regularization-by-noise phenomenon for the stochastic convolution $\Psi(u)$. It proves that if the modulation function $w$ is sufficiently irregular (specifically, $(\rho, \gamma)$-irregular with large $\rho$), the stochastic convolution gains arbitrary spatial regularity.
  • Specifically, for any initial regularity $s \in \mathbb{R}$ and any noise regularity $\sigma \in \mathbb{R}$, if the modulation is sufficiently irregular ($\rho \gg 1$), the solution exists in $H^s(\mathbb{T})$.
  • Moreover, the nonlinear part of the solution exhibits "nonlinear smoothing," belonging to a space $H^{s_0}(\mathbb{T})$ where $s_0$ can be made arbitrarily large by increasing $\rho$.
• Pathwise Well-Posedness: The authors prove local well-posedness in $H^s_0(\mathbb{T})$ (mean-zero subspace) for any $s \in \mathbb{R}$, provided the modulation is sufficiently irregular. This extends previous results on deterministic modulated KdV to the stochastic multiplicative setting.
• Necessity of Mean-Zero Noise: The regularization effect relies critically on the condition that the noise has zero spatial mean ($\phi_0 = 0$). If the noise has a non-zero mean component, the smoothing effect on the stochastic convolution fails.

2. The Rough Case (White-in-Time Noise, $\beta = 1/2$):

• Loss of Regularity: In contrast to the Young case, the authors find that irregularity of the modulation $w$ does not induce smoothing on the stochastic convolution when the noise is white in time. Instead, there is a slight loss of spatial regularity in the mapping properties of the driver.
• Regularized Noise Requirement: To achieve pathwise local well-posedness in the rough case, the authors must slightly regularize the noise term by applying a smoothing operator $\langle \partial_x \rangle^{-\epsilon_0}$.
• Well-Posedness: Under this slight regularization, pathwise local well-posedness is established in $H^s_0(\mathbb{T})$ for any $s \in \mathbb{R}$, provided the noise is sufficiently smooth in space. The condition $\phi_0 = 0$ is not required in this case.

3. General Framework:

• The paper develops a general framework for stochastic modulated dispersive PDEs with multiplicative noises. The techniques are shown to be applicable to other equations, such as the stochastic modulated Benjamin-Ono (BO) and Intermediate Long Wave (ILW) equations, as well as the stochastic modulated Schrödinger equation (discussed in the Appendix).

Significance and Claims
The paper claims to provide the first pathwise local well-posedness results for stochastic modulated dispersive PDEs with multiplicative noises. Its significance lies in:

• Unifying Framework: It successfully combines the nonlinear Young integration theory (originally applied to deterministic modulated equations) with the pathwise construction of stochastic convolutions for multiplicative noise.
• Novel Regularization Mechanism: It identifies a new mechanism where the irregularity of the modulation (not just the noise) acts as a regularizer for the stochastic convolution in the Young regime. This contrasts with classical regularization-by-noise results which typically rely on the noise itself or random initial data.
• Pathwise vs. Probabilistic: By working pathwise, the results hold almost surely for specific realizations of the noise and modulation, rather than relying on probabilistic averaging, which is particularly relevant for equations where the noise is multiplicative and the solution map is not smooth.
• Limitations: The authors explicitly note that the regularization by noise phenomenon is specific to the Young case ($\beta > 1/2$). In the rough case ($\beta = 1/2$), the modulation's irregularity does not compensate for the roughness of the noise, necessitating spatial regularization of the noise term.

The paper does not claim to solve the global well-posedness problem for all parameters but establishes robust local existence and uniqueness results that depend on the interplay between the modulation's irregularity and the noise's temporal regularity.