量化图结构对随机偏微分方程强Feller性质的影响
Quantifying the effect of graph structure on strong Feller property of SPDEs
AI总结：
本文提出了一种基于图的空分解方法，用于分析树图上SPDEs的强Feller性质，并证明了在特定条件下唯一不变测度的存在性。
摘要：
本文研究了底层图结构如何影响在有限树图上随机偏微分方程（SPDEs）的行为，其中每条边由空间-时间白噪声驱动。我们首先引入了一种基于图的空分解方法，用于分析由SPDEs生成的马尔可夫半群的强Feller性质。通过考察图拉普拉斯算子特征函数中零元素的位置，我们建立了确保强Feller性质和不可约性的噪声自由边数的精确上界。有趣的是，我们发现对链图中的任一条边添加噪声就足以满足条件，而对于星图，最多只有一条边可以保持噪声自由而不影响系统的性质。此外，在耗散条件下，我们证明了唯一不变测度的存在性和指数 ergodicity。
英文摘要：
This paper investigates how the structure of the underlying graph influences the behavior of stochastic partial differential equations (SPDEs) on finite tree graphs, where each edge is driven by space-time white noise. We first introduce a novel graph-based null decomposition approach to analyzing the strong Feller property of the Markov semigroup generated by SPDEs on tree graphs. By examining the positions of zero entries in eigenfunctions of the graph Laplacian operator, we establish a sharp upper bound on the number of noise-free edges that ensures both the strong Feller property and irreducibility. Interestingly, we find that the addition of noise to any single edge is sufficient for chain graphs, whereas for star graphs, at most one edge can remain noise-free without compromising the system's properties. Furthermore, under a dissipative condition, we prove the existence and exponential ergodicity of a unique invariant measure.
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。
