AI 中文总结
本文推导了光滑黎曼开卷上加权Kirchhoff扩散作为状态切换布朗运动的标度极限，证明在快速切换下收敛到加权Kirchhoff界面条件，且曲率仅通过体算子保留。
AI 中文摘要
我们在光滑黎曼开卷上推导出一个加权Kirchhoff扩散，作为无边界流形上状态切换布朗运动的标度极限。页面是紧致黎曼流形，共享一个公共边界流形，并在装订线上诱导相同的度量，而它们的第二基本形式和内部几何可能不同。对于每个 $\varepsilon>0$，我们在一个 $o(\varepsilon)$ 领口中正则化度量，通过页面的光滑加倍，在每个加倍上运行布朗运动，并允许页面标签在宽度为 $\varepsilon$ 的领口中以速率 \\(q_\varepsilon(r)Q_{ij}\\) 切换，其中 $Q$ 是任意不可约有限状态马尔可夫生成器，具有不变测度 $\pi$，而 \\(q_\eps\\) 有界且支撑在 $\eps$ 领口上。记 \\(\Theta_\eps\\) 为 \\(q_\eps\\) 的总质量，\\(\Xi_\eps\\) 为其第二原始矩，我们证明，如果 \\(\Theta_\varepsilon\to\infty\\) 且 \\(\Xi_\varepsilon\to0\\)，那么从每个确定性起始点序列出发，粘合过程收敛到具有加权Kirchhoff界面条件的布朗运动。不需要 $Q$ 的可逆性、$q_\varepsilon$ 的逐点标度假设或一阶矩条件。此外，我们表明页面的第二基本形式不贡献额外的界面项：曲率仅通过体Laplace--Beltrami算子保留。
英文摘要
We derive a weighted Kirchhoff diffusion on a smooth Riemannian open book as a scaling limit of regime-switching Brownian motions on boundaryless manifolds. The pages are compact Riemannian manifolds sharing a common boundary manifold and inducing the same metric on the binding, while their second fundamental forms and interior geometries may differ. For each $\varepsilon>0$, we regularize the metric in an $o(\varepsilon)$ collar, pass to smooth doubles of the pages, run Brownian motion on each double, and allow the page label to switch in a collar of width $\varepsilon$ at rates \(q_\varepsilon(r)Q_{ij},\) where $Q$ is an arbitrary irreducible finite-state Markov generator with invariant law $π$, while \(q_\eps\) is bounded and supported on an \(\eps\) collar. Denoting \(Θ_\eps\) the total mass of \(q_\eps\) and \(Ξ_\eps\) its second raw moment, we prove that, if \(Θ_\varepsilon\to\infty\) and \(Ξ_\varepsilon\to0\), then the glued processes converge, from every deterministic sequence of starting points, to Brownian motion with weigthed Kirchhoff interface conditions. No reversibility of $Q$, pointwise scaling ansatz for $q_\varepsilon$, or first-moment condition is required. Furthermore, we show that the second fundamental form of the pages contributes no additional interface term: curvature remains only through the bulk Laplace--Beltrami operators.
