AI 中文总结
研究图中区域在狄利克雷边界条件下的刘维尔定理，通过格林算子、随机完备性、格林公式及图的常返性等进行刻画，并应用于欧几里得空间子集和弱球对称图。
AI 中文摘要
我们展示了具有狄利克雷边界条件的图中区域的一个刘维尔定理。具体而言，我们刻画了非零有界调和函数不存在的情况。由于狄利克雷边界条件会产生带有正杀伤项的拉普拉斯算子，我们首先可以通过应用于杀伤项的格林算子等于1，或者换句话说，常数函数1是一个势，来刻画刘维尔定理的有效性。其次，我们根据无穷远处的随机完备性和热的完全损失推导出一个刻画。第三，我们根据超调和势的格林公式给出一个刻画。最后，我们根据无杀伤项图的常返性和暂留性来研究刘维尔性质。作为应用，我们考虑欧几里得空间的子集，如锥体和渗流簇，以及弱球对称图。
英文摘要
We show a Liouville theorem for domains in graphs with Dirichlet boundary conditions. More specifically, we characterize the non-existence of non-zero bounded harmonic functions. Since Dirichlet boundary conditions give rise to Laplacians with a positive killing term, we can first characterize the validity of the Liouville theorem by the fact that the Green operator applied to the killing terms is equal to 1, or in other words, that the constant function $1$ is a potential. Secondly, we derive a characterization in terms of stochastic completeness at infinity and and total loss of heat. Thirdly, we give a characterization in terms of a Green formula for superharmonic potentials. Finally, we investigate the Liouville property in terms of recurrence and transience of the graph without killing term. As an application, we consider subsets of the Euclidean space such as cones and percolation clusters, and weakly spherically symmetric graphs.
Comments37 pages, comments are welcome
