度量图上扩散过程不变测度的组合表示
A combinatorial representation for the invariant measure of diffusion processes on metric graphs
摘要：
本文将经典的马尔可夫链树定理推广到一个连续集上。特别地，我们考虑度量图上的不可约扩散过程。唯一的不变测度在顶点上有一个原子分量，在边上有一个绝对连续的部分。我们表明，相应的密度在$x$可以表示为一个归一化的叠加的权重与面向点$x$的度量树形。每个定向度量树形图的权重通过沿定向边的$\int\frac{b}{σ^2}$形式的积分的指数乘以每个节点的权重来获得，该权重由节点周围树形图的局部定向乘以$x$处的扩散系数的倒数来确定。度量树状图是将原度量图沿若干条边切割而得到的。
英文摘要：
We give a generalization to a continuous setting of the classic Markov chain tree Theorem. In particular, we consider an irreducible diffusion process on a metric graph. The unique invariant measure has an atomic component on the vertices and an absolutely continuous part on the edges. We show that the corresponding density at $x$ can be represented by a normalized superposition of the weights associated to metric arborescences oriented toward the point $x$. The weight of each oriented metric arborescence is obtained by the exponential of integrals of the form $\int\frac{b}{σ^2}$ along the oriented edges time a weight for each node determined by the local orientation of the arborescence around the node time the inverse of the diffusion coefficient at $x$. The metric arborescences are obtained cutting the original metric graph along some edges.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
