Well-posedness and existence of an invariant measure for the linearly-damped KdV equations driven by a jump noise

1. Problem Statement

The paper investigates the linearly damped Korteweg-de Vries (KdV) equation on a one-dimensional periodic domain (the torus $\mathbb{T}$) subject to multiplicative Lévy noise. The governing equation is:
$$du(t) + (\partial_x^3 u(t) + u(t)\partial_x u(t) + \gamma u(t))dt = G(u(t-))dW(t) + \int_{E_0} K(u(t-), \xi) d\tilde{\pi}(t, \xi) + \int_{E \setminus E_0} K(u(t-), \xi) d\pi(t, \xi)$$
where:

• $u$ represents the wave elongation.
• $\gamma > 0$ is the linear damping coefficient (friction).
• $W$ is a Wiener process (Gaussian noise).
• $\pi$ is a Poisson random measure representing jumps, and $\tilde{\pi}$ is the compensated measure.
• The noise is split into small jumps (integrated over $E_0$, where the Lévy measure is finite) and large jumps (integrated over $E \setminus E_0$).

The study addresses two primary objectives:

1. Well-posedness: Establish the existence and uniqueness of pathwise weak solutions in the space $H^2(\mathbb{T})$ for general Lévy noise.
2. Long-time Behavior: Prove the existence of an invariant measure for the system, specifically when the noise is square-integrable (i.e., neglecting large jumps, $K=0$ on $E \setminus E_0$), provided the damping $\gamma$ is sufficiently large.

2. Methodology

The authors employ a multi-layered analytical approach combining stochastic analysis, functional analysis, and approximation techniques.

A. Mathematical Framework

• Spaces: Solutions are sought in Sobolev spaces $H^s(\mathbb{T})$, specifically focusing on $H^2(\mathbb{T})$. The space of càdlàg (right-continuous with left limits) functions is used to handle the jump discontinuities inherent in Lévy processes.
• Noise Assumptions: The noise coefficients $G$ (diffusion) and $K$ (jump) satisfy linear growth and Lipschitz conditions in $L^2, H^1,$ and $H^2$. The Lévy measure $\nu$ is assumed to be finite on bounded subsets of the jump space.

B. Well-posedness Strategy (Theorem 3.1)

To prove the existence and uniqueness of a pathwise solution, the authors use a three-step approximation scheme:

1. Regularization: A fourth-order artificial viscosity term $\epsilon \partial_x^4 u$ is added to the equation to smooth the solution and ensure higher regularity.
2. Galerkin Approximation: The system is projected onto finite-dimensional subspaces $H_N$ (spanned by Fourier modes $|k| \le N$).
3. Truncation: Nonlinear terms ($u \partial_x u$) are truncated using a smooth cutoff function $\theta_N$ to ensure global Lipschitz continuity, allowing for the existence of unique solutions to the approximate ODEs/SDEs.

Key Steps in the Proof:

• A Priori Estimates: The authors derive uniform bounds (independent of $N$ and $\epsilon$) for the approximate solutions. They utilize specific conserved functionals of the deterministic KdV equation (specifically $I_0 = \|u\|_{L^2}^2$ and $I_2$, a higher-order energy functional) combined with Itô's formula.
  • $L^p$ estimates are derived for $p \in [2, 8]$ to control the $L^2$ norm.
  • Higher-order estimates in $H^2$ are derived using the functional $I_2$.
• Compactness: The tightness of the laws of the approximate solutions is established in the space $L^2(0, T; H^1) \cap D([0, T]; H^{-2})$. This relies on:
  • Aubin-Lions Lemma: For spatial compactness ($H^2 \hookrightarrow \hookrightarrow H^1$).
  • Aldous' Tightness Criterion: To handle the temporal regularity and the càdlàg nature of the solutions in the Skorokhod topology.
• Convergence: Using the Jakubowski-Skorokhod Representation Theorem, the authors construct a new probability space where the approximate solutions converge almost surely. They pass to the limit $N \to \infty$ and $\epsilon \to 0$ to obtain a global martingale solution.
• Pathwise Uniqueness: Uniqueness is proven by analyzing the difference between two solutions and applying the Gyöngy-Krylov argument (an infinite-dimensional extension of the Yamada-Watanabe theorem). This relies on the pathwise uniqueness of the martingale solution to conclude the existence of a unique pathwise solution.

C. Invariant Measure Strategy (Theorem 3.2)

For the long-time behavior, the authors consider the case where large jumps are absent ($K=0$ on $E \setminus E_0$), ensuring the noise is square-integrable.

• Method: They apply the Maslowski-Seidler Theorem, which generalizes the Krylov-Bogoliubov theorem to spaces with weak topologies.
• Conditions Verified:
  1. Sequential Weak Feller Property: The transition semigroup $P_t$ maps continuous bounded functions on the weak topology of $H^2$ to sequentially continuous bounded functions. This is proven using the pathwise uniqueness and tightness arguments.
  2. Tightness of Time-Averages: They prove that the time-averaged probability of the solution leaving a large ball in $H^2$ vanishes as the radius increases. This requires establishing linear-in-time bounds for the $H^2$ norm.
• Energy Estimates: The proof of the linear bounds relies on the damping term $\gamma u$. The authors show that if $\gamma$ is sufficiently large (specifically $\gamma > \frac{50}{197}(\kappa_1 + \kappa_2)$), the dissipation overcomes the growth induced by the nonlinearity and noise, ensuring the system does not explode and admits an invariant measure.

3. Key Contributions

1. First Study of Damped KdV with Lévy Noise: This is the first work to establish well-posedness and invariant measures for the KdV equation driven by general Lévy noise (including jumps), extending previous results limited to Gaussian (Wiener) noise.
2. Pathwise Well-posedness in $H^2$: The authors prove the existence of a unique pathwise weak solution in $H^2(\mathbb{T})$ for the damped KdV equation with multiplicative Lévy noise. This is a significant regularity result, as many stochastic KdV results are limited to $L^2$ or $H^1$.
3. Handling Jump Discontinuities: The paper rigorously handles the technical difficulties arising from the jump terms (both small and large) using the "piecing out" argument and compensated Poisson integrals, ensuring the solution remains càdlàg.
4. Existence of Invariant Measure: The paper proves the existence of an invariant measure in $H^2(\mathbb{T})$ for the system driven by square-integrable Lévy noise, provided the damping is strong enough. This characterizes the statistical steady state of the system.

4. Main Results

• Theorem 3.1 (Well-posedness): For any damping coefficient $\gamma > 0$ and initial data $u_0 \in H^2(\mathbb{T})$ with finite moments, there exists a unique pathwise weak solution $u \in L^8(\Omega; L^\infty(0, T; L^2)) \cap L^2(\Omega; L^\infty(0, T; H^2))$ to the damped KdV equation with general Lévy noise.
• Theorem 3.2 (Invariant Measure): If the noise is square-integrable (no large jumps) and the damping coefficient $\gamma$ is sufficiently large, the transition semigroup associated with the equation admits an invariant measure $\mu$ on $H^2(\mathbb{T})$.

5. Significance

• Physical Relevance: The inclusion of Lévy noise allows for modeling intermittent and impulsive disturbances in the medium (e.g., sudden shocks or non-Gaussian fluctuations in shallow water waves), which is more realistic than purely Gaussian models.
• Mathematical Rigor: The work bridges a gap in the literature by combining the analysis of jump processes with the specific nonlinear structure of the KdV equation. The derivation of high-order ($H^2$) estimates in the presence of jumps is technically demanding and novel.
• Long-term Dynamics: By establishing the existence of an invariant measure, the paper provides a foundation for studying the statistical properties (e.g., ergodicity, mixing) of stochastic KdV systems under non-Gaussian forcing, which is crucial for understanding the long-term predictability and stability of such wave systems.

In summary, this paper provides a comprehensive mathematical framework for analyzing the linearly damped KdV equation under realistic, non-Gaussian stochastic forcing, proving both the short-term well-posedness and the long-term statistical stability of the system.