度量测度空间上奇异的非局部狄氏型的热核估计

Heat kernel estimation is one of the important problems in Probability theory, PDE theory and Geometric Analysis. Since Professor S.T. Yau and Professor P. Li published their seminal work on 《Acta Math.》 in 1986, many experts have made great contributions in this research area. Recently there has been intense interest in studying heat kernels and potential theory of non-local Dirichlet forms or of discontinuous Markov processes. However, almost all literatures concern only non-degenerate cases. Due to their importance both in theory and in applications, degenerate non-local Dirichlet forms or discontinuous Markov processes receive more and more interest. In fact, many physical and economic systems have been successfully modeled by non-Gaussian jump processes. However, there are few results on heat kernels and potential theory of degenerate non-local Dirichlet forms. In this project, we are concerned with degenerate non-local Dirichlet forms on abstract metric measure spaces, and try to establish general conclusions on the existence and optimal estimates of heat kernels. Meanwhile, we study the related potential theory: weak Harnack inequality of harmonic functions, Hölder regularity, etc. We expect that the results of the project will help to expand our knowledge about analysis and stochastic analysis on abstract metric measure spaces.

热核估计是概率论、偏微分方程以及几何分析的重要研究内容。自从1986年丘成桐教授和Peter Li教授在《Acta Math.》上发表了关于热核估计的重要结果以来，越来越多的专家学者在此领域开展了大量的研究工作。近年来，非局部狄氏型及其相应的跳跃过程的热核估计、位势理论等更是备受关注，目前已有研究结果考虑的基本都是非退化的情况，而退化的情况不仅在理论方面有重要作用，而且在实际中也有着许多应用，例如可以模拟很多物理现象和经济学现象等，因此有着重要的研究意义，但是目前关于此方面的研究结果非常少。本项目将在抽象度量测度空间上考虑退化的非局部狄氏型，研究其热核的存在性，给出热核满足一定双边估计的等价刻画等一般性的结论，同时研究相关调和函数的弱Harnack不等式、Hölder估计等位势理论。项目的研究方法和研究结果将有助于拓展我们对度量测度空间上方程理论、随机过程理论的认识和理解。

热核估计建立了多个学科之间的联系，而且是这些学科中的重要研究内容。目前关于热核估计的结果基本都是非退化的情况，而关于退化狄氏型的热核估计的结果非常少。本项目旨在度量空间上研究一大类退化的非局部狄氏型及相应的热核估计。我们首先在超距空间上研究了非局部正则狄氏型的等价刻画条件，然后研究了一大类退化的非局部狄氏型的热核估计，我们首次提出一种新类型的热核估计并给出了等价刻画条件，同时我们还研究了相关的位势理论，例如（上、下）调和函数的弱Harnack不等式、调和函数及热核的Hölder估计及连续性等等。然后，我们进一步把超距空间上的一部分结果推广到了抽象度量空间上，为此我们还引进了一种新方法证明全局热核的存在性，即利用半群的局部化超压缩不等式来构造热核。项目的结果对我们理解退化狄氏型的热核的行为有着重要意义。