[Paper Review] Concentration and limit behaviors of stationary measures

Wen Huang, Min Ji+2 more|arXiv (Cornell University)|Jul 21, 2016

Stability and Controllability of Differential Equations19 references3 citations

TL;DR

This paper investigates the concentration and limit behaviors of stationary measures for Fokker-Planck equations driven by multiplicative white noise perturbations of ordinary differential equations. Using Lyapunov functions and weak∗-compactness, it establishes that as noise vanishes, stationary measures converge weakly to invariant measures supported on the global attractor of the unperturbed system; further, under strong local attractor/repeller structures, limit measures concentrate on or avoid these sets depending on noise admissibility, with applications to stochastic Hopf bifurcation.

ABSTRACT

In this paper, we study limit behaviors of stationary measures of the Fokker-Planck equations associated with a system of ordinary differential equations perturbed by a class of multiplicative including additive white noises. As the noises are vanishing, various results on the invariance and concentration of the limit measures are obtained. In particular, we show that if the noise perturbed systems admit a uniform Lyapunov function, then the stationary measures form a relatively sequentially compact set whose weak$^*$-limits are invariant measures of the unperturbed system concentrated on its global attractor. In the case that the global attractor contains a strong local attractor, we further show that there exists a family of admissible multiplicative noises with respect to which all limit measures are actually concentrated on the local attractor; and on the contrary, in the presence of a strong local repeller in the global attractor, there exists a family of admissible multiplicative noises with respect to which no limit measure can be concentrated on the local repeller. Moreover, we show that if there is a strongly repelling equilibrium in the global attractor, then limit measures with respect to typical families of multiplicative noises are always concentrated away from the equilibrium. As applications of these results, an example of stochastic Hopf bifurcation is provided. Our study is closely related to the problem of noise stability of compact invariant sets and invariant measures of the unperturbed system.

Motivation & Objective

• To analyze the limit behavior of stationary measures for stochastic perturbations of ODEs with multiplicative white noise.
• To determine under what conditions these limit measures concentrate on global or local attractors/repellers of the unperturbed system.
• To establish conditions under which limit measures avoid strongly repelling equilibria.
• To connect the stochastic stability of invariant sets with the existence of admissible noise families.
• To provide a distribution-based framework for understanding noise stability in deterministic dynamical systems.

Proposed method

• Analyzes the Fokker-Planck equation associated with Itô SDEs of the form dx = V(x)dt + G(x)dW.
• Uses a uniform Lyapunov function for the unperturbed system to prove relative sequential compactness of stationary measures.
• Applies weak∗-topology convergence to identify limit measures as invariant measures of the unperturbed system.
• Introduces the concept of admissible multiplicative noise families to control concentration behavior of limit measures.
• Employs isolating neighborhoods and C2 boundary conditions to characterize strong local attractors/repellers.
• Uses Lyapunov/anti-Lyapunov functions to link dynamical structure to stochastic stability.

Experimental results

Research questions

• RQ1 Under what conditions do stationary measures of noisy perturbations converge weakly to invariant measures of the unperturbed system as noise vanishes?
• RQ2 When are the limit measures concentrated on the global attractor of the unperturbed system?
• RQ3 Can the choice of admissible multiplicative noise families force limit measures to concentrate on a strong local attractor or avoid a strong local repeller?
• RQ4 How does the presence of a strongly repelling equilibrium affect the concentration of limit measures?
• RQ5 What is the role of the Fokker-Planck equation in characterizing the long-term distributional behavior of noisy dynamical systems?

Key findings

• If the unperturbed system admits a uniform Lyapunov function, the family of stationary measures is relatively sequentially compact in the weak∗-topology, and all weak∗-limit measures are invariant for the unperturbed flow.
• When the global attractor contains a strong local attractor, there exists a family of admissible multiplicative noises such that all limit measures are concentrated on that local attractor.
• When the global attractor contains a strong local repeller, there exists a family of admissible multiplicative noises such that no limit measure can be concentrated on the local repeller.
• If there is a strongly repelling equilibrium in the global attractor, then for typical families of admissible multiplicative noises, the limit measures are always concentrated away from the equilibrium.
• The results are applied to a stochastic Hopf bifurcation example, illustrating the concentration behavior near bifurcation-induced attractors.
• The paper establishes a rigorous connection between stochastic stability of invariant sets and the existence of noise families that preserve or avoid them.