图和度量图上的随机过程、泛函不等式及其应用

For Markov chains on weighted graphs, sharp volume growth criteria for stochastic completeness and upper escape rate have been obtained in full analogue with the diffusion case. However, examples indicate that these criteria are not sharp for tree-type graphs.We develop a new technique to compare the upper escape rate of a Markov chain and that of the diffusion on the corresponding metric graph. We plan to study the escape rate for tree-type weighted graphs via this tool..In contrary to the weighted graph case, for general jump processes the current known volume growth criteria are much weaker than those for diffusions. The main difficulty comes from nonlocality which is an obstruction to the classical heat equation estimation techniques. We plan to overcome this difficulty by developing a “metric graph” model for a jump process. .Heat kernel estimation on manifolds is one of the central problems in geometric analysis. Functional inequalities on weighted graphs are connected with heat kernel estimates via the discretization technique. There has been deep work on two-sided Gaussian type heat kernel estimates under unbounded weight perturbation. However, this strong stability for Gaussian upper bound is still open. We investigate the prolem by combining tools from discretization and harmonic analysis.

对带权重图上的Markov链， 随机完备性与上逃逸速度有与扩散过程情形相同的最优体积增长刻画。 但对树型图这一特殊情形，例子表明该刻画应不是最优的。我们发展了比较Markov链与相应度量图上扩散过程的上逃逸速度的新技术，计划以此为工具研究树型带权重图的逃逸速度。.与带权重图情形不同，对一般的跳过程已知的体积增长刻画比扩散情形要弱很多。主要的技术困难是非局部性，给应用经典的热方程估计技术带来了障碍。我们期望通过发展跳过程的“度量图”模型来克服这一困难。.热核估计是几何分析是的中心问题之一。通过离散化技术，带权重图上的泛函不等式与热核估计紧密联系。热核的Gauss型双边估计在加无界权重下的稳定性问题已有深入研究，但Gauss型上界估计的强稳定性仍是开问题。我们结合离散化技术与调和分析的工具来研究该问题。

带权重图和度量图是研究扩散过程和跳过程、泛函不等式的重要对象，通过离散化方法又可以反过来应用到流形等复杂空间的相关研究上。以度量图为工具，我们改进了带权重图上Markov链上逃逸速度的体积增长刻画。我们建立了度量图和带权重图上的Sobolev不等式与等容度不等式的关系，研究了带权重图的离散Ricci曲率与热半群梯度估计之间的关系。同时我们在泛函不等式稳定性问题上取得了部分进展。