AI 中文总结
本文在度量图上定义多调和距离，统一扩展电阻与双调和距离，并构造各向同性高斯过程，实现任意有限阶均方可微且满足基尔霍夫条件。
AI 中文摘要
度量图是线性网络的推广，为定义连续索引的高斯过程提供了自然框架。我们在这些拓扑上定义了一类新的距离，称为多调和距离，它统一并扩展了有效电阻距离和双调和距离背后的谱构造。我们给出了谱和变分两种刻画。此外，我们展示了一类显式的随机过程，其变异函数与平方多调和距离一致。最后，我们展示了如何将这些度量与合适的完全单调函数复合，以定义各向同性过程，这些过程沿边具有任意指定的有限阶均方可微性，并在顶点处满足直至一阶的基尔霍夫条件。
英文摘要
Metric graphs are generalisations of linear networks and provide a natural framework for the definition of continuously-indexed Gaussian processes. We define a new class of distances on these topologies, termed polyharmonic distances, which unify and extend the spectral construction underlying the effective resistance distance and the biharmonic one. We give both a spectral and a variational characterisation. Furthermore, we show an explicit class of stochastic processes whose variograms coincide with the squared polyharmonic distances. Finally, we show how these metrics can be composed with suitable completely monotonic functions to define isotropic processes having any prescribed finite-order mean-square differentiability along the edges and satisfying the Kirchhoff conditions up to order one at the vertices.
Comments64 pages, of which 21 of main article
