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Kening Lu

Publications and source records attributed to Kening Lu.

At least 19 recordsLinked to original sources
arXiv2026

Polynomial mixing for the 3D damped cubic nonlinear Schr\"odinger equation with degenerate noise

We prove polynomial mixing for the defocusing damped cubic stochastic nonlinear Schr\"odinger equation on the three-dimensional torus under saturating smooth finite rank Brownian forcing. The mixing rate is measured in the $p$-Wasserstein metric induced by the $H^1$ distance for every $1\le p<\infty$. We also obtain sharp geometric characterizations of saturation. The proof is based on a polynomial mixing criterion built on a stable--compact decomposition of the exact solution differences with polynomial moment control of the logarithmic path amplification. Dense Malliavin range allows the compact defect to be compensated by finite-dimensional Cameron--Martin shifts, producing a block multiplier with negative mean logarithm. A logarithmic transportation gauge, combined with a renewal--reset coupling scheme, then yields mixing at every prescribed polynomial order in the weaker $L^2$ distance. A stationary regularity gain to $H^{2-}$ then enables us to upgrade the convergence to $H^1$.

math.PR↗
arXiv2026

Spectral gap for the three-dimensional damped cubic wave equation with degenerate noise

We establish a weighted Wasserstein spectral gap for the three-dimensional damped cubic wave equation with genuinely finite rank Brownian forcing. Under a saturation condition, the gap holds with respect to the negative phase topology $\mathcal E_s=H^{-s}\times H^{-1-s}$ for every $0<s<1/2$, from which we deduce unique ergodicity and exponential mixing in the energy topology. Sharp geometric characterizations of the saturation condition are also obtained. The method we develop is a stable--compact asymptotic coupling mechanism for hypoelliptic dissipative SPDEs beyond the parabolic setting. Instead of relying on positive time smoothing or asymptotic gradient estimates, it reduces the infinite-dimensional obstruction to contraction to a compact defect in a weaker coupling topology. Dense Malliavin range then permits this defect to be compensated by a finite dimensional Cameron--Martin shift, producing a finite distance contraction on bounded Lyapunov cores. Together with a separate high energy contraction from dissipation, this yields a global weighted Wasserstein spectral gap.

math.AP↗
arXiv2026

Unique Ergodicity for the Projective Process of the 2D Navier--Stokes Equation with Nondegenerate Noise

We prove unique ergodicity of the projective process associated with the two dimensional Navier--Stokes equation in vorticity form, with additive diagonal noise acting on every nonzero real Fourier phase and satisfying two sided power law bounds. Consequently, the exact Furstenberg--Khasminskii formula for the top Lyapunov exponent holds. The main new ingredient is a compact dense mechanism for asymptotic generalized coupling in the absence of a Foia\c{s}--Prodi type high-low mode decomposition for the projective dynamics. Using the dense range of the Malliavin derivative and compactness of the state derivative, we construct finite rank perturbations of the Wiener path that compensate, to first order, for perturbations of the initial condition, leaving a residual whose logarithmic growth has negative stationary mean and hence contracts locally at an exponential rate, with a cost controlled by a triangular scheme of blockwise Ramer transformations.

math.PR↗
arXiv2026

Exponential mixing and Freidlin--Wentzell large deviation principle for Markov cocycles

This paper studies the long time statistics and small noise asymptotics of Markov cocycles associated with Markov processes in random environments modeled by measure preserving dynamical systems on a standard Borel probability space. Our first result provides an abstract criterion for exponential mixing of stationary measures for such cocycles, formulated toward SPDE applications with assumptions that can be verified directly from a priori estimates. To overcome the nonuniformity from the environment, we combine generalized coupling arguments with ergodic theoretic methods. This allows us to convert nonuniform estimates along the environment into contraction on a positive density set of times, and then upgrade this to all time contraction by introducing a block gap-counting argument. Our second result establishes a Freidlin--Wentzell large deviation principle(LDP) for the unique stationary measure in the small noise limit with a good rate function. For the upper bound, the noise is allowed to be degenerate, while the deterministic pullback attractor may have nontrivial dynamics. The abstract theory applies to nonautonomous SPDEs. We illustrate it with two examples: the two-dimensional Navier--Stokes equations on bounded domains and damped Sine--Gordon equations, where both the deterministic forcing and the degenerate additive noise depend on the random environment.

math.PR↗
arXiv2026

Mixed Third-Order Flux Laws for Dual Cascade in the Stochastic SQG Equation

We study dual-cascade flux laws for the stochastic forced--dissipative surface quasi-geostrophic (SQG) equation on a large periodic box. For statistically stationary solutions, under a weak anomalous dissipation assumption, we derive rigorous mixed third-order structure-function laws for the dual cascade: a Yaglom-type law for the direct cascade of surface potential energy (SPE) and an antisymmetrized mixed flux law for the inverse cascade of the Hamiltonian. In particular, the inverse Hamiltonian law appears to be new even as an explicit third-order structure-function relation. We also prove Onsager-type obstruction results showing that sufficiently regular stationary families cannot sustain the corresponding non-zero fluxes: $B^s_{3,\infty}$-regularity above the Onsager threshold $1/3$ rules out the direct SPE flux, while sufficient low-frequency Besov regularity rules out the inverse Hamiltonian flux. These results provide a rigorous formulation of the SQG dual-cascade phenomenology in a stochastic stationary setting.

math.AP↗
arXiv2024

Pullback measure attractors and limiting behaviors of McKean-Vlasov stochastic delay lattice systems

We study the long-term behavior of the distribution of the solution process to the non-autonomous McKean-Vlasov stochastic delay lattice system defined on the integer set $\mathbb{Z}$. Specifically, we first establish the well-posedness of solutions for this non-autonomous, distribution-dependent stochastic delay lattice system. Then, we prove the existence and uniqueness of pullback measure attractors for the non-autonomous dynamical system generated by the solution operators, defined in the space of probability measures. Furthermore, as an application of the pullback measure attractor, we prove the ergodicity and exponentially mixing of invariant measures for the system under appropriate conditions. Finally, we establish the upper semi-continuity of these attractors as the distribution-dependent stochastic delay lattice system converges to a distribution-independent system.

math.DS↗
arXiv2024

Invariant measures, periodic measures and pullback measure attractors of McKean-Vlasov stochastic reaction-diffusion equations on unbounded domains

This paper deals with the long term dynamics of the non-autonomous McKean-Vlasov stochastic reaction-diffusion equations on R^n. We first prove the existence and uniqueness of pullback measure attractors of the non-autonomous dynamical system generated by the solution operators defined in the space of probability measures. We then prove the existence and uniqueness of invariant measures and periodic measures of the equation under further conditions. We finally establish the upper semi-continuity of pullback measure attractors as well as the convergence of invariant measures and periodic measures when the distribution dependent stochastic equations converge to a distribution independent system.

math.PR↗
arXiv2023

Exponential mixing and limit theorems of quasi-periodically forced 2D stochastic Navier-Stokes Equations in the hypoelliptic setting

We consider the incompressible 2D Navier-Stokes equations on the torus driven by a deterministic time quasi-periodic force and a noise that is white in time and degenerate in Fourier space. We show that the asymptotic statistical behavior is characterized by a quasi-periodic invariant measure that exponentially attracts the law of all solutions. The result is true for any value of the viscosity $ν>0$ and does not depend on the strength of the external forces. By utilizing this quasi-periodic invariant measure, we establish a quantitative version of the strong law of large numbers and central limit theorem for the continuous time inhomogeneous solution processes with explicit convergence rates. It turns out that the convergence rate in the central limit theorem depends on the time inhomogeneity through the Diophantine approximation property on the quasi-periodic frequency of the quasi-periodic force.

math.PR↗
arXiv2023

Smooth invariant foliations without a bunching condition and Belitskii's $C^{1}$ linearization for random dynamical systems

Smooth linearization is one of the central themes in the study of dynamical systems. The classical Belitskii's $C^1$ linearization theorem has been widely used in the investigation of dynamical behaviors such as bifurcations, mixing, and chaotic behaviors due to its minimal requirement of partial second order non-resonances and low regularity of systems. In this article, we revisit Belitskii's $C^1$ linearization theorem by taking an approach based on smooth invariant foliations and study this problem for a larger class of dynamical systems ({\it random dynamical systems}). We assumed that the linearized system satisfies the condition of Multiplicative Ergodic Theorem and the associated Lyapunov exponents satisfy Belitskii's partial second order non-resonant conditions. We first establish the existence of $C^{1,β}$ stable and unstable foliations without assuming the bunching condition for Lyapunov exponents, then prove a $C^{1,β}$ linearization theorem of Belitskii type for random dynamical systems. As a result, we show that the classical Belitskii's $C^1$ linearization theorem for a $C^{2}$ diffeomorphism $F$ indeed holds without assuming all eigenspaces of the linear system $DF(0)$ are invariant under the nonlinear system $F$, a requirement previously imposed by Belitskii in his proof.

math.DS↗
arXiv2023

Large deviations for 2D stochastic Navier-Stokes Equations driven by a periodic force and a degenerate noise

We consider the incompressible 2D Navier-Stokes equations on the torus, driven by a deterministic time periodic force and a noise that is white in time and degenerate in Fourier space. The main result is twofold. Firstly, we establish a Ruelle-Perron-Frobenius type theorem for the time inhomogeneous Feynman-Kac evolution operators with regular potentials associated with the stochastic Navier-Stokes system. The theorem characterizes asymptotic behaviors of the Feynman-Kac operators in terms of the periodic family of principal eigenvalues and corresponding unique eigenvectors. The proof involves a time inhomogeneous version of Ruelle's lower bound technique. Secondly, utilizing this Ruelle-Perron-Frobenius type theorem and a Kifer's criterion, we establish a Donsker-Varadhan type large deviation principle with a nontrivial good rate function for the occupation measures of the time inhomogeneous solution processes.

math.PR↗
arXiv2021

Statistical Properties of 2D Stochastic Navier-Stokes Equations with Time-Periodic Forcing and Degenerate Stochastic Forcing

We consider the incompressible 2D Navier-Stokes equations with periodic boundary conditions driven by a deterministic time periodic forcing and a degenerate stochastic forcing. We show that the system possesses a unique ergodic periodic invariant measure which is exponentially mixing under a Wasserstein metric. We also prove the weak law of large numbers for the continuous time inhomogeneous solution process. In addition, we obtain the weak law of large numbers and central limit theorem by restricting the inhomogeneous solution process to periodic times. The results are independent of the strength of the noise and hold true for any value of viscosity with a lower bound $ν_1$ characterized by the Grashof number $G_1$ associated with the deterministic forcing. In the laminar case, there is a larger lower bound $ν_2$ of the viscosity characterized by the Grashof number $G_2$ associated with both the deterministic and random forcing. We prove that in this laminar case, the system has trivial dynamics for any viscosity larger than $ν_2$ by demonstrating the existence of a unique globally exponentially stable random periodic solution that supports the unique periodic invariant measure.

math.DS↗
arXiv2020

Rough Path Theory to approximate Random Dynamical Systems

We consider the rough differential equation $dY=f(Y)d\bm \om$ where $\bm \om=(ω,\bbomega)$ is a rough path defined by a Brownian motion $ω$ on $\RR^m$. Under the usual regularity assumption on $f$, namely $f\in C^3_b (\RR^d, \RR^{d\times m})$, the rough differential equation has a unique solution that defines a random dynamical system $ϕ_0$. On the other hand, we also consider an ordinary random differential equation $dY_δ=f(Y_δ)dω_\de$, where $ω_\de$ is a random process with stationary increments and continuously differentiable paths that approximates $ω$. The latter differential equation generates a random dynamical system $ϕ_δ$ as well. We show the convergence of the random dynamical system $ϕ_δ$ to $ϕ_0$ for $δ\to 0$ in Hölder norm.

math.PR↗
arXiv2019

Ergodic theory of Random Anosov systems mixing on fibers

In this paper, we study the complicated dynamics of Anosov systems driven by an external force in the context of geometric theory (an abundance of random periodic points and random horseshoes) and smooth ergodic theory (random periodic measures and random Liv\v sic Theorem).

math.DS↗
arXiv2015

SRB Measures for A Class of Partially Hyperbolic Attractors in Hilbert spaces

In this paper, we study the existence of SRB measures and their properties for infinite dimensional dynamical systems in a Hilbert space. We show several results including (i) if the system has a partially hyperbolic attractor with nontrivial finite dimensional unstable directions, then it has at least one SRB measure; (ii) if the attractor is uniformly hyperbolic and the system is topological mixing and the splitting is Hölder continuous, then there exists a unique SRB measure which is mixing; (iii) if the attractor is uniformly hyperbolic and the system is non-wondering and and the splitting is Hölder continuous, then there exists at most finitely many SRB measures; (iv) for a given hyperbolic measure, there exist at most countably many ergodic components whose basin contains an observable set.

math.DS↗
arXiv2015

Entropy, chaos and weak horseshoe for infinite dimensional random dynamical systems

In this paper, we study the complicated dynamics of infinite dimensional random dynamical systems which include deterministic dynamical systems as their special cases in a Polish space. Without assuming any hyperbolicity, we proved if a continuous random map has a positive topological entropy, then it contains a topological horseshoe. We also show that the positive topological entropy implies the chaos in the sense of Li-Yorke. The complicated behavior exhibiting here is induced by the positive entropy but not the randomness of the system.

math.DS↗
arXiv2015

Random dynamical systems for stochastic evolution equations driven by multiplicative fractional Brownian noise with Hurst parameters $H\in (1/3,1/2]$

We consider the stochastic evolution equation $ du=Audt+G(u)dω,\quad u(0)=u_0 $ in a separable Hilbert--space $V$. Here $G$ is supposed to be three times Fréchet--differentiable and $ω$ is a trace class fractional Brownian--motion with Hurst parameter $H\in (1/3,1/2]$. We prove the existence of a global solution where exceptional sets are independent of the initial state $u_0\in V$. In addition, we show that the above equation generates a random dynamical system.

math.DS↗
arXiv2014

Local pathwise solutions to stochastic evolution equations driven by fractional Brownian motions with Hurst parameters $H\in (1/3,1/2]$

In this article we are concerned with the study of the existence and uniqueness of pathwise mild solutions to evolutions equations driven by a Hölder continuous function with Hölder exponent in $(1/3,1/2)$. Our stochastic integral is a generalization of the well-known Young integral. To be more precise, the integral is defined by using a fractional integration by parts formula and it involves a tensor for which we need to formulate a new equation. From this it turns out that we have to solve a system consisting in a path and an area equations. In this paper we prove the existence of a unique local solution of the system of equations. The results can be applied to stochastic evolution equations with a non-linear diffusion coefficient driven by a fractional Brownian motion with Hurst parameter in $(1/3,1/2]$, which is particular includes white noise.

math.AP↗