扩散决定递归图
Diffusion determines the recurrent graph
摘要：
我们考虑离散测度空间上的扩散问题，它是由加权图中的马尔可夫半群编码的。本文研究了当扩散满足序同构时，图是否唯一确定。如果图是递归的，则确定完全图结构和测度空间（直到整体缩放）。通过反例证明了这一结果是最优的。在没有递归假设的情况下，在具有标准权重的图上的归一化扩散的情况下，以及在每个点具有相同质量的空间上的任意图的情况下，图仍然是确定的。这些调查提供了离散对应的研究扩散欧几里得整环和流形发起的阿伦特和继续阿伦特/Biegert/特尔埃尔斯特和阿伦特/特尔埃尔斯特。我们考虑的一个关键步骤表明，在我们的上下文中，序同构实际上是酉映射（直到缩放）。
英文摘要：
We consider diffusion on discrete measure spaces as encoded by Markovian semigroups arising from weighted graphs. We study whether the graph is uniquely determined if the diffusion is given up to order isomorphism. If the graph is recurrent then the complete graph structure and the measure space are determined (up to an overall scaling). As shown by counterexamples this result is optimal. Without the recurrence assumption, the graph still turns out to be determined in the case of normalized diffusion on graphs with standard weights and in the case of arbitrary graphs over spaces in which each point has the same mass. These investigations provide discrete counterparts to studies of diffusion on Euclidean domains and manifolds initiated by Arendt and continued by Arendt/Biegert/ter Elst and Arendt/ter Elst. A crucial step in our considerations shows that order isomorphisms are actually unitary maps (up to a scaling) in our context.
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