论文标题
公制图上扩散过程不变度度量的组合表示
A combinatorial representation for the invariant measure of diffusion processes on metric graphs
论文作者
论文摘要
我们对经典马尔可夫链树定理的连续设置进行了概括。特别是，我们考虑了公制图上的不可减少扩散过程。独特的不变测度在顶点上具有一个原子成分，边缘上具有绝对连续的部分。我们表明，$ x $时的相应密度可以用与公制轴承相关的权重的归一化叠加来表示。每个方向的公式轴承的重量是通过形式的积分的指数获得的，$ \ int \ frac {b} {σ^2} $沿着方向的边缘时间沿着定向的边缘时间为每个节点确定的重量，这是由于节点时间左右的局部取向的局部取向。获得公制的轴心沿一些边缘切割原始度量图。
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.
