AI 中文总结
研究构建分层黎曼空间上的几何状态切换扩散，通过鞅问题刻画，结合泊松稀疏机制构建过程并证明唯一性，应用于协方差矩阵锥，得到 \(V\)-均匀几何遍历过程。
AI 中文摘要
我们构建了几何状态切换扩散，这是一类在局部紧致分层黎曼状态空间上的马尔可夫过程。与经典状态切换和随机混合扩散不同，状态不是外部标签，而是奇异几何状态空间的内在分层。状态变化可能改变维度、秩、几何或组合类型，同时状态空间保持其周围拓扑。每层上的运动是保守费勒扩散，层间转换由沿有向图的状态依赖跳跃率和着陆核指定。我们通过自然分层核心上的鞅问题来刻画该过程。在总跳跃率的均匀界下，通过将每层扩散与泊松稀疏机制相结合构建保守的 càdlàg 强马尔可夫过程。利用辅助不相交并拓扑和有界扰动论证证明了唯一性。扩展生成器的标准福斯特 - 李雅普诺夫条件给出正哈里斯常返性、不变概率测度的唯一性以及在非周期性下的 \(V\)-均匀几何遍历性。该框架应用于按秩分层的半正定协方差矩阵锥。所得过程将固定秩协方差扩散与随机秩变化相结合，并且是 \(V\)-均匀几何遍历的。
英文摘要
We construct geometric regime-switching diffusions, a class of Markov processes on locally compact stratified Riemannian state spaces. In contrast with classical regime-switching and stochastic hybrid diffusions, the regimes are not external labels, but intrinsic strata of a singular geometric state space. Changes of regime may therefore change dimension, rank, geometry or combinatorial type while the state space maintains its ambient topology. On each stratum the motion is a conservative Feller diffusion, while inter-stratum transitions are specified by state-dependent jump rates and landing kernels along a directed graph. We characterize the process through a martingale problem on a natural stratified core. Under a uniform bound on the total jump rate, we construct a conservative càdlàg strong Markov process by combining the stratumwise diffusions with a Poisson thinning mechanism. Uniqueness is proved using an auxiliary disjoint-union topology and a bounded perturbation argument. Standard Foster--Lyapunov conditions for the extended generator give positive Harris recurrence, uniqueness of the invariant probability measure and, under aperiodicity, \(V\)-uniform geometric ergodicity. The framework is applied to the cone of positive semidefinite covariance matrices, stratified by rank. The resulting process combines fixed-rank covariance diffusions with stochastic rank changes and is \(V\)-uniformly geometrically ergodic.
