[論文レビュー] Exponential mixing for a class of dissipative PDEs with bounded degenerate noise

Sergei Kuksin, Vahagn Nersesyan+1 more|arXiv (Cornell University)|Feb 9, 2018

Stochastic processes and financial applications参考文献 64被引用数 43

ひとこと要約

tldr: 本論文は、コンパクトな位相空間上の離散時間マルコフ過程の一部のクラスに対して、固有の定常測度と指数的混合性を証明する。対象には、いくつかのフーリエモードに作用する有界で退化ノイズにより摂動を受ける非線形散逸性PDEが含まれ、線形化が取り扱いやすく、ノイズ構造が分解可能である。

ABSTRACT

We study a class of discrete-time random dynamical systems with compact phase space. Assuming that the deterministic counterpart of the system in question possesses a dissipation property, its linearisation is approximately controllable, and the driving noise is bounded and has a decomposable structure, we prove that the corresponding family of Markov processes has a unique stationary measure, which is exponentially mixing in the dual-Lipschitz metric. The abstract result is applicable to nonlinear dissipative PDEs perturbed by a bounded random force which affects only a few Fourier modes. We assume that the nonlinear PDE in question is well posed, its nonlinearity is non-degenerate in the sense of the control theory, and the random force is a regular and bounded function of time which satisfies some decomposability and observability hypotheses. This class of forces includes random Haar series, where the coefficients for high Haar modes decay sufficiently fast. In particular, the result applies to the 2D Navier-Stokes system and the nonlinear complex Ginzburg-Landau equations. The proof of the abstract theorem uses the coupling method, enhanced by the Newton-Kantorovich-Kolmogorov fast convergence.

研究の動機と目的

• Motivate and study uniqueness and exponential mixing for discretized random dynamical systems with bounded degenerate noise.
• Develop an abstract framework that links controllability of linearised dynamics to ergodic properties of the Markov chain.
• Apply the abstract result to PDEs such as 2D Navier–Stokes and nonlinear complex Ginzburg–Landau equations perturbed by Haar-type noise.

提案手法

• Formulate a four-hypothesis framework (regularity, dissipativity, approximate controllability of the linearisation, and decomposability of the noise).
• Use a Markovian discrete-time setting with a compact phase space and a contraction/Doeblin-type coupling via a Newton–Kantorovich–Kolmogorov fast convergence scheme.
• Construct a measurable coupling in infinite dimensions to achieve exponential convergence in the dual-Lipschitz metric.
• Exploit a homological-analogy inspired approach to approximately invert the linearised noise map to enable coupling even under bounded noise.

実験結果

リサーチクエスチョン

• RQ1 Under what conditions on the nonlinear PDE and the bounded, decomposable noise does the discrete-time Markov process admit a unique stationary measure?
• RQ2 Can one obtain exponential mixing in the dual-Lipschitz metric for systems with degenerate, bounded noise that does not act on all determining modes?
• RQ3 How do approximate controllability of the linearisation and decomposability of the noise contribute to constructing effective couplings in infinite dimensions?
• RQ4 To what PDEs (e.g., Navier–Stokes, complex Ginzburg–Landau) can the abstract result be applied, and what observables or noise structures ensure mixing?
• RQ5 What is the role of Haar-type noise and observability assumptions in establishing the required non-degeneracy?

主な発見

• There exists a unique stationary measure for the Markov process and exponential convergence in the dual-Lipschitz metric for sufficiently large N and positive viscosity ν.
• The result applies to nonlinear dissipative PDEs perturbed by bounded random forces that affect only a finite or decomposed set of Fourier modes, including 2D Navier–Stokes and complex Ginzburg–Landau equations.
• Mixing is established under hypotheses of regularity, dissipativity, approximate linearised controllability, and decomposable noise with Lipschitz densities.
• The forcing can be represented as Haar series with coefficients decaying sufficiently fast, enabling observable and decomposable noise.
• The coupling approach is aided by a Newton–Kantorovich–Kolmogorov fast convergence scheme, avoiding Malliavin calculus requirements for bounded noise.
• If the initial law is stationary, the solution becomes statistically periodic in time, and convergence to the stationary regime is exponential in time for arbitrary initial data.