Fast Switching Near a Smooth Interface: A Curvature-Universal Kirchhoff Limit
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 ε>0, we regularize the metric in an o(ε) 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 ε at rates qε(r)Qij, where Q is an arbitrary irreducible finite-state Markov generator with invariant law π, while qeps is bounded and supported on an eps collar. Denoting Θeps the total mass of qeps and Ξeps its second raw moment, we prove that, if Θε→∞ and Ξε→0, 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 qε, 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.
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BibTeX
@misc{marconi2026fast,
title = {{Fast Switching Near a Smooth Interface: A Curvature-Universal Kirchhoff Limit}},
author = {Leonardo Marconi},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.02570},
eprint = {2610.02570},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.02570},
note = {CC BY-SA 4.0},
}
RIS
TY - UNPB
TI - Fast Switching Near a Smooth Interface: A Curvature-Universal Kirchhoff Limit
AU - Leonardo Marconi
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.02570
UR - https://arxiv.org/abs/2610.02570
AB - 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 ε>0, we regularize the metric in an o(ε) 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 ε at rates q_ε(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_ε→∞ and Ξ_ε→0, 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_ε, 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.
ER -