Stationary distributions of systems with Discreteness Induced Transitions

非对称自催化反应及其平稳分布

Asymmetric autocatalytic reactions and their stationary distribution

本文研究非对称自催化反应，给出其近似平稳分布公式，并证明该分布在三个参数区域中呈现不同性质，尤其在离散性诱导转变区域可刻画各稳定点处的时间占比。

Constrained Langevin approximation for the Togashi-Kaneko model of autocatalytic reactions

The Togashi Kaneko model (TK model), introduced by Togashi and Kaneko in 2001, is a simple stochastic reaction network that displays discreteness-induced transitions between meta-stable patterns. Here we study a constrained Langevin approximation (CLA) of this model. The CLA, obtained by Anderson et al. in 2019, is an obliquely reflected diffusion process on the positive orthant and hence it respects the constrain that chemical concentrations are never negative. We show that the CLA is a Feller process, is positive Harris recurrent, and converges exponentially fast to the unique stationary dis…

Spectral Conditions for Equivalence of Gaussian Random Fields with Stationary Increments

This paper studies the problem of equivalence of Gaussian measures induced by Gaussian random fields (GRFs) with stationary increments and proves a sufficient condition for the equivalence in terms of the behavior of the spectral measures at infinity. The main results extend those of Stein (2004), Van Zanten (2007, 2008) and are applicable to a rich family of nonstationary space-time models with possible anisotropy behavior.

Quasi-stationary distributions of multi-dimensional diffusion processes

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 infini…

A stochastic approximation approach to quasi-stationary distributions on finite spaces

This work is concerned with the analysis of a stochastic approximation algorithm for the simulation of quasi-stationary distributions on finite state spaces. This is a generalization of a method introduced by Aldous, Flannery and Palacios. It is shown that the asymptotic behavior of the empirical occupation measure of this process is precisely related to the asymptotic behavior of some deterministic dynamical system induced by a vector field on the unit simplex. This approach provides new proof of convergence as well as precise rates for this type of algorithm. We then compare this algorithm w…

局部平稳Hawkes过程的若干极限定理

Some limit theorems for locally stationary Hawkes processes

本文针对具有时变繁殖率的多元Hawkes过程，结合经典鞅方法与新思想解决了非卷积Volterra过程的困难，证明了大数定律和泛函中心极限定理，并应用于金融统计推导出流动性约束下价格扭曲的闭式表达式。