Fast Switching Near a Smooth Interface: A Curvature-Universal Kirchhoff Limit

Source abstract

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.