Stochastic Hydrodynamics with Lévy Noise

• Stochastic hydrodynamics with Lévy noise is a study of fluid systems influenced by jump processes that induce non-Gaussian, intense fluctuations.
• The framework employs rigorous SPDE formulations, Galerkin approximations, and energy estimates to secure well-posedness and precise scaling laws.
• Advanced probabilistic techniques establish invariant measures and ergodicity, ensuring exponential convergence and insights into long-term behavior.

Stochastic hydrodynamics with Lévy noise examines the probabilistic dynamics of fluid systems under the influence of highly irregular, discontinuous stochastic forcing. Unlike classical Gaussian or Wiener noise, Lévy processes incorporate jumps—random discontinuities—that encode rare but intense fluctuations, giving rise to fundamentally non-Gaussian behavior. The rigorous theory integrates methods from stochastic analysis on infinite-dimensional spaces, nonlinear PDE theory, variational analysis, and large deviation theory, enabling the study of well-posedness, path regularity, invariant measures, and scaling laws for a wide class of hydrodynamic SPDEs subject to jump noise.

1. Mathematical Formulation: Lévy-Driven Hydrodynamic SPDEs

The canonical framework involves an abstract nonlinear evolution equation on a Hilbert space $H$, posed as

$$du(t) + [A u(t) + B(u(t), u(t))]\,dt = f(t)\,dt + \sigma(t,u(t))\,dW(t) + \int_Z G(t,u(t^-),z) \,\tilde{N}(dt,dz), \quad u(0) = u_0,$$

where:

• $A$ is a linear, typically coercive (elliptic) operator, such as the Stokes or Laplace operator;
• $B$ encodes the (possibly nonlinear, often skew-symmetric) transport/nonlinearity of the fluid; e.g., the Navier–Stokes convective term $B(u,v) = P(u \cdot \nabla v)$;
• $W(t)$ is a cylindrical Wiener process, while $\tilde{N}(dt,dz) = N(dt,dz) - dt\,\nu(dz)$ is the compensated Poisson random measure for jumps;
• $\sigma$ and $G$ determine the structure of the Gaussian and jump noise, respectively.

Paradigmatic systems include the 2D/3D Navier–Stokes equations, MHD, Boussinesq, shell turbulence models, viscous Burgers, and non-Newtonian fluids, all extended to include either additive or (multiplicative) Lévy noise (Mohan, 2021, Peng et al., 2020, Motyl, 2013, Shang et al., 2017).

2. Well-posedness and Structural Assumptions

Existence and uniqueness of solutions rely on sharp analytic properties of the drift and noise terms:

• Drift: Skew-symmetry and local monotonicity of $B$ ensure energy balance and allow elimination of the worst nonlinear terms in energy estimates (Peng et al., 2020). For non-Newtonian or power-law fluids, additional coercivity is provided by monotone damping (e.g., Forchheimer or second-grade terms) (Mohan, 2021, Shang et al., 2017).
• Noise: The crucial advances of (Peng et al., 2020) demonstrate that boundedness of the $V$-components of Lipschitz/growth constants for $\sigma,G$ (specifically, $L_2,L_5 < 2$) is both necessary and sufficient for global well-posedness. No control on higher moments or small-jump measures is required, in contrast to previous literature.
• Galerkin approximations: The solution theory is built on finite-dimensional approximations, a priori uniform energy estimates (using Itô–Lévy calculus), tightness arguments in non-metrizable spaces, and Skorokhod embeddings (Mohan, 2021, Motyl, 2013, Karczewska et al., 2017).

Typical results guarantee, for $u_0 \in H$ and $f \in L^2(0,T;V')$, a unique $H$-valued càdlàg solution $u \in D([0,T];H) \cap L^2(0,T;V)$, with uniform energy bounds

$$\sup_{t \in [0,T]} \mathbb{E} |u(t)|^2 + \mathbb{E} \int_0^T \|u(t)\|^2\,dt < \infty \quad [2005.04476].$$

3. Long-Time Behavior: Ergodicity and Invariant Measures

For a broad class of locally monotone, coercive hydrodynamic SPDEs with pure-jump or mixed noise, ergodicity results establish the existence and uniqueness of an invariant probability measure $\mu$ for the associated Markov–Feller semigroup. This measure is characterized by finite moments of all orders and exponential convergence to equilibrium in Wasserstein or Lipschitz (e-)norms (Barrera et al., 2 Dec 2024).

In the context of the incompressible 2D Navier–Stokes equations with additive Lévy noise, the corresponding semigroup $(P_t)_{t\geq0}$ is Feller, and possesses a unique invariant measure $\mu \in M_1(H)$, with

$\int_H \|u\|_H^4\, \mu(du) < \infty,$

and exponential convergence

$$|P_t F(x) - \int_H F\, d\mu| \leq e^{-\nu t}\|x\|_H, \quad \forall F \in {\rm Lip}_b(H),\ {\rm Lip}(F) \leq 1.$$

Key technical novelties include the absence of compactness constraints on the embedding $V \hookrightarrow H$ and the proof of the exponential equicontinuity (“e-property”) for the Markov semigroup (Barrera et al., 2 Dec 2024).

4. Scaling Laws, Turbulent Quantities, and Moderate Deviations

In 1D models, such as the stochastic viscous Burgers equation with cylindrical Lévy forcing, a complete description of moment growth, scaling of structure functions, and spectral laws is achievable (Yuan et al., 2021):

• Moments: For Sobolev norms, $E[\|u(t)\|_{H^n}^2] \sim \nu^{-(2n-1)}$ as viscosity $\nu \to 0$.
• Structure functions: For $\ell$ in the inertial range,

$$S_p(\ell) = E[|u(x+\ell) - u(x)|^p] \sim \begin{cases} \ell^p\, \nu^{-(p-1)}, & \ell \ll \nu, \ p \geq 1, \ \ell^p, & p < 1, \ \ell \ll \nu, \ \ell^{\min(1,p)}, & \nu \ll \ell \ll 1. \end{cases}$$

• Energy spectrum: Inertial range scaling $E_n \sim n^{-2}$ is obtained rigorously.

Moderate deviation principles (MDP) for Lévy-driven SPDEs describe the probabilities of intermediate fluctuations between the law of large numbers and large deviation regimes. The rate function is given explicitly in terms of control solutions to a associated “skeleton” deterministic SPDE. Results cover stochastic 2D hydrodynamical models without the need for compact embedding in the functional setting (Li et al., 11 Feb 2025).

5. Applications: Classical and Extended Hydrodynamic Systems

The abstract frameworks accommodate a wide family of models:

• 2D/3D Navier–Stokes: Classical viscous incompressible flows, with rigorous theory of weak/strong martingale or pathwise strong solutions driven by both Gaussian and jump noise (Mohan, 2021, Motyl, 2013).
• MHD and Boussinesq: Multicomponent systems with coupled velocity, magnetic, and temperature fields. Existence of martingale solutions established in unbounded domains (Motyl, 2013, Peng et al., 2020).
• Shell models (GOY/Sabra): Discrete turbulence models with multiplicative Lévy forcing, treated within the same functional analytic and probabilistic frameworks (Peng et al., 2020, Li et al., 11 Feb 2025).
• Non-Newtonian fluids (second grade): Unique global strong solutions with uniform moment bounds for stochastic models incorporating nonlocal diffusive effects (Shang et al., 2017).
• Dispersive equations: Martingale solutions for Korteweg–de Vries equations with Lévy noise constructed via stochastic compactness (Karczewska et al., 2017).

6. Analytical Techniques and Probabilistic Tools

Central methodology includes:

• Faedo–Galerkin approximation for finite-dimensional truncations;
• Energy and moment estimates via Itô–Lévy calculus;
• Compactness in Fréchet/nonmetrizable spaces to overcome lack of compact embedding (crucial in 3D/unbounded domains) (Motyl, 2013, Li et al., 11 Feb 2025);
• Aldous’ tightness criterion, Skorokhod embedding theorems (standard and generalized for nonmetric spaces);
• Cut-off/localization and contraction techniques for global strong solutions in the presence of only sharp growth/Lipschitz constants (Peng et al., 2020);
• Martingale problem and Yamada–Watanabe type arguments for strong uniqueness and law uniqueness (Mohan, 2021, Shang et al., 2017).

7. Open Directions and Research Frontier

Major current and emerging lines of investigation include:

• Strong solutions and pathwise uniqueness for multi-dimensional and non-Newtonian flows with jump noise, especially in critical/supercritical regimes (Mohan, 2021).
• Fluctuation analysis: Moderate and large deviation theory for a broader class of Lévy-driven infinite-dimensional SPDEs, particularly in settings lacking compactness (Li et al., 11 Feb 2025).
• Turbulent regimes and scaling laws: Inviscid limits, anomalous dissipation, and non-Gaussian scaling for hydrodynamic SPDEs with heavy-tailed noise (Yuan et al., 2021).
• Long-time asymptotics and ergodicity under degenerate or purely discontinuous forcing (Barrera et al., 2 Dec 2024).
• Extension to unbounded domains, passive/active scalar coupling, and further singular or higher-order models (Motyl, 2013, Li et al., 11 Feb 2025).

The current theory demonstrates a robust interplay between nonlinear PDE analysis and infinite-dimensional stochastic processes, fundamentally enriching the understanding of randomness and intermittency in continuum fluid models under realistic, jump-driven stochastic dynamics.