[Paper Review] Quasi-stationary distributions of multi-dimensional diffusion processes

Alexandru Hening, Weiwei Qi+2 more|arXiv (Cornell University)|Feb 11, 2021

Mathematical and Theoretical Epidemiology and Ecology Models44 references4 citations

TL;DR

This paper establishes the existence, uniqueness, and exponential convergence to quasi-stationary distributions (QSDs) for multi-dimensional diffusion processes that almost surely absorb in finite time. By analyzing a singular, uniformly elliptic operator derived from the Fokker-Planck generator, the authors prove sharp exponential convergence for compactly supported initial laws and uniqueness with exponential convergence for all initial distributions under stronger Lyapunov conditions, generalizing prior one-dimensional results and extending spectral theory to singular coefficients.

ABSTRACT

The present paper is devoted to the investigation of the long term behavior of a class of singular multi-dimensional diffusion processes that get absorbed in finite time with probability one. Our focus is on the analysis of quasi-stationary distributions (QSDs), which describe the long term behavior of the system conditioned on not being absorbed. Under natural Lyapunov conditions, we construct a QSD and prove the sharp exponential convergence to this QSD for compactly supported initial distributions. Under stronger Lyapunov conditions ensuring that the diffusion process comes down from infinity, we show the uniqueness of a QSD and the exponential convergence to the QSD for all initial distributions. Our results can be seen as the multi-dimensional generalization of Cattiaux et al (Ann. Prob. 2009) as well as the complement to Hening and Nguyen (Ann. Appl. Prob. 2018) which looks at the long term behavior of multi-dimensional diffusions that can only become extinct asymptotically. The centerpiece of our approach concerns a uniformly elliptic operator that we relate to the generator, or the Fokker-Planck operator, associated to the diffusion process. This operator only has singular coefficients in its zeroth-order terms and can be handled more easily than the generator. For this operator, we establish the discreteness of its spectrum, its principal spectral theory, the stochastic representation of the semigroup generated by it, and the global regularity for the associated parabolic equation. We show how our results can be applied to most ecological models, among which cooperative, competitive, and predator-prey Lotka-Volterra systems.

Motivation & Objective

• To analyze the long-term behavior of multi-dimensional diffusion processes that almost surely absorb in finite time.
• To establish the existence and uniqueness of quasi-stationary distributions (QSDs) for such processes.
• To prove sharp exponential convergence to the QSD for compactly supported initial distributions.
• To extend spectral theory to a class of uniformly elliptic operators with singular zeroth-order coefficients.
• To apply the results to ecological models, including competitive, cooperative, and predator-prey Lotka-Volterra systems.

Proposed method

• Introduce a uniformly elliptic operator derived from the Fokker-Planck generator, which has singular coefficients in its zeroth-order term.
• Establish the discreteness of the spectrum and principal spectral theory for this singular operator.
• Develop a stochastic representation for the semigroup generated by the operator using time-changed processes.
• Prove global regularity for solutions to the associated parabolic equation via approximation and density arguments.
• Use Lyapunov-type conditions to control the behavior near the boundary and ensure coming down from infinity.
• Apply the spectral theory to derive convergence rates and uniqueness of the QSD.

Experimental results

Research questions

• RQ1 Under what conditions does a quasi-stationary distribution exist for a multi-dimensional diffusion process that absorbs almost surely in finite time?
• RQ2 Can exponential convergence to the QSD be established for all initial distributions, not just compactly supported ones?
• RQ3 What role does the spectral theory of a singular, uniformly elliptic operator play in characterizing the QSD and its convergence properties?
• RQ4 How do the Lyapunov conditions ensure that the process comes down from infinity, and how does this relate to uniqueness of the QSD?
• RQ5 To what extent can these results be applied to classical ecological models such as Lotka-Volterra systems?

Key findings

• Under natural Lyapunov conditions, a quasi-stationary distribution exists and exponential convergence to it holds for compactly supported initial distributions.
• With stronger Lyapunov conditions ensuring the process comes down from infinity, the QSD is unique and exponential convergence holds for all initial distributions.
• The spectrum of the associated uniformly elliptic operator is discrete, and its principal eigenvalue governs the exponential decay rate.
• The semigroup generated by the operator admits a stochastic representation via time-changed diffusion processes.
• The principal eigenfunction of the operator corresponds to the density of the QSD.
• The results generalize Cattiaux et al. (2009) to multi-dimensional settings and complement Hening and Nguyen (2018) by covering finite-time absorption.