AI 中文总结
该研究针对状态依赖反射切换扩散过程，提出模态加权传输证书方法，通过耦合推导生成元不等式，保证马尔可夫半群的全局成对指数收缩，并通过示例验证了方法的有效性。
AI 中文摘要
即使某些模态单独具有扩张性，状态依赖切换扩散过程在分布上也可能具有收缩性。我们针对这类过程在$\boldsymbol{\text{R}}^n$上以及带法向反射的紧致凸域上的收缩性，提出了一种基于传输的可计算条件。传输代价结合了模态依赖的空间权重与离散模态惩罚，同时保留了跨模态对的空间分离性。通过布朗运动的同步耦合与状态依赖跳跃时钟的极大耦合，我们推导了同模态与跨模态构型各自的生成元不等式。凸法向反射贡献了一个非正的有限变差项，因此相同条件适用于关联的无通量Fokker-Planck-Kolmogorov系统。这些条件的可行性保证了马尔可夫半群与弱测度解的全局成对指数收缩，具有单位前置因子和显式速率。对于固定的模态权重排序与规定的衰减速率，这些条件在空间权重与图代价上是仿射的，构成一个半无限线性可行性问题。带有Lipschitz余量的有限网格条件可证明不等式在整个域上成立。一个一维反射示例验证了分布计算，一个平面三模态示例展示了依赖于两个状态坐标的转移率的合成过程。
英文摘要
State-dependent switching diffusions can be contractive in distribution even when some modes are individually expansive. We develop a computable transport-based condition for such contraction on $\mathbb{R}^n$ and on compact convex domains with normal reflection. The transport cost combines mode-dependent spatial weights with a discrete mode penalty while preserving spatial separation for cross-mode pairs. Using synchronous coupling of the Brownian motions and maximal coupling of the state-dependent jump clocks, we derive separate generator inequalities for same-mode and cross-mode configurations. Convex normal reflection contributes a nonpositive finite-variation term, so the same conditions apply to the associated no-flux Fokker-Planck-Kolmogorov system. Their feasibility guarantees global pairwise exponential contraction of the Markov semigroup and weak measure solutions, with unit prefactor and an explicit rate. For a fixed ordering of the mode weights and a prescribed decay rate, the conditions are affine in the spatial weights and graph costs and form a semi-infinite linear feasibility problem. A finite-mesh condition with a Lipschitz margin certifies the inequalities over the full domain. A reflected one-dimensional example validates the distributional computation, and a planar three-mode example demonstrates the synthesis procedure for transition rates depending on both state coordinates.
