The present paper is devoted to the investigation of the long term behavior of a class of higher-dimensional singular diffusion processes that get absorbed by the extinction set in finite time with probability one. Our primary 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 higher-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 higher-dimensional diffusions that can only become extinct asymptotically. As applications, we show how our results can be applied to many ecological models, among which cooperative, competitive, and predator-prey Lotka-Volterra systems. The cornerstone of our approach revolves around a uniformly elliptic operator that we relate through a two-step transform to the Fokker-Planck operator associated with the diffusion process. This operator only has singular coefficients in its zeroth-order terms and can be handled more easily than the Fokker-Planck operator, which is defined on an unbounded domain and exhibits degeneracy in the extinction set. 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. These results extend beyond the study of QSDs and are of independent interest, especially in the context of spectral theory for degenerate elliptic operators on unbounded domains. As direct consequences, we show that the extinction rate associated with the QSD and the sharp exponential convergence rate are respectively given by the absolute value of the principal eigenvalue and the spectral gap, between the principal eigenvalue and the rest of the spectrum, of this operator. Such characterizations of the QSD and exponential convergence rate were previously unknown in the context of irreversible singular diffusion processes.
Publication:
JOURNAL OF DYNAMICS AND DIFFERENTIAL EQUATIONS
http://dx.doi.org/10.1007/s10884-026-10494-2
Author:
Alexandru Hening
Department of Mathematics, Texas A&M University, College Station 77843-3368, US
Weiwei Qi
State Key Laboratory of Mathematical Sciences (SKLMS), Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China
wwqi@amss.ac.cn
Zhongwei Shen
Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton AB T6G 2G1, Canada
Yingfei Yi
Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton AB T6G 2G1, Canada
yingfei@ualberta.ca
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