[Paper Review] On Limiting Behavior of Stationary Measures for Stochastic Evolution Systems with Small Noise Intensity

Lifeng Chen, Zhao Dong+2 more|arXiv (Cornell University)|Nov 22, 2016

Stochastic processes and financial applications3 citations

TL;DR

This paper establishes that as noise intensity ε → 0, all weak limits of stationary measures μ^ε for stochastic evolution systems converge to invariant measures supported within the Birkhoff center of the corresponding deterministic semiflow X⁰. The result holds under tightness of {μ^ε} and applies broadly to SDEs, SPDEs, and SFDEs with Brownian or Lévy noise, offering a precise characterization of limiting support beyond global attractors, including cases with unstable equilibria or saddle points.

ABSTRACT

The limiting behavior of stochastic evolution processes with small noise intensity $ε$ is investigated in distribution-based approach. Let $μ^ε$ be stationary measure for stochastic process $X^ε$ with small $ε$ and $X^{0}$ be a semiflow on a Polish space. Assume that $\{μ^ε: 0

Motivation & Objective

• To identify the limiting support of stationary measures μ^ε for stochastic evolution systems as noise intensity ε → 0.
• To extend existing results on zero-noise limits beyond global attractors to the more refined Birkhoff center.
• To establish that all weak limits of μ^ε are invariant under the deterministic semiflow X⁰.
• To provide a general framework applicable to SDEs, SPDEs, and SFDEs driven by Brownian or Lévy noise.
• To resolve the open question of where limiting measures concentrate when the global attractor contains no stable periodic orbits or equilibria.

Proposed method

• Uses a distribution-based approach to analyze the limiting behavior of stationary measures μ^ε as ε → 0.
• Imposes tightness of {μ^ε : 0 < ε ≤ ε₀} as a key assumption to ensure weak convergence subsequences.
• Applies the Birkhoff center theory to characterize the support of limit measures under the deterministic semiflow X⁰.
• Employs weak convergence techniques and properties of ω-limit sets to prove invariance of limit measures under X⁰.
• Leverages the Poincaré recurrence theorem and support invariance to show that the support of any limit measure is contained in the Birkhoff center of X⁰.
• Constructs explicit examples (e.g., Bernoulli lemniscate, May-Leonard system) to demonstrate that limit measures can concentrate at saddle points, not just stable equilibria or attractors.

Experimental results

Research questions

• RQ1 Where do the limiting measures of stationary distributions concentrate as noise intensity ε → 0, especially when the global attractor contains no stable periodic orbits?
• RQ2 Can the support of zero-noise limits be precisely characterized beyond the global attractor of the drift vector field?
• RQ3 Is it possible for a limiting measure to concentrate at an unstable equilibrium or saddle point under small noise perturbations?
• RQ4 How does the Birkhoff center of the deterministic semiflow relate to the support of zero-noise limits of stationary measures?
• RQ5 Can the framework be extended to systems with degenerate or Lévy noise, not just non-degenerate diffusion?

Key findings

• All weak limits of stationary measures μ^ε are invariant under the deterministic semiflow X⁰ as ε → 0.
• The support of any such limit measure is contained within the Birkhoff center of the semiflow X⁰.
• For dissipative systems, the Birkhoff center has zero Lebesgue measure, even when the global attractor has positive measure.
• In the absence of stable periodic orbits, limiting measures may concentrate at saddle points—demonstrated via the Bernoulli lemniscate example.
• In the May-Leonard system with one-dimensional white noise, the limiting measures concentrate at three saddle points, not the global attractor.
• The result is optimal in the sense that for any invariant measure of X⁰, one can construct a noise structure such that μ^ε weakly converges to it (Proposition 3.2).