非时齐流形上扩散半群的性质及其应用

Many problems in several branches of mathematics, such as modern analysis and partial differential equations, could be attributed to study behaviors of some special operators, and using stochastic analysis method can help us to well understand the microscopic properties of these operators. Since pursuing the degree for PhD, the applicant established derivative formula and integration by parts formula, and studied functional inequalities for semigroups on time-inhomogeneous manifolds, including Harnack inequalities, transportation-cost inequalities, and gradient estimates. As a continuation and depth of these topics, this subject will intend to further study the existence and uniqueness of evolution system for measures on time-inhomogeneous manifolds, and then taking evolution system for measures for reference measures, it will then study the asymptotic behavior, hypercontractivity, supercontractivity and ultracontractivity of the diffusion semigroup. Moreover, this subject will aim to study the short time behavior of diffusion processes, which will be further applied to estimate heat kernel in short time.

现代分析学与偏微分方程等数学分支中的众多问题均可归结为研究某些特殊算子的性质, 而用随机分析的方法可以更好地理解这些算子的微观意义. 自攻读博士学位以来, 申请人建立了在度量随时间变化的流形上关于扩散半群的导数公式和分部积分公式, 并研究了关于非时齐扩散半群的泛函不等式, 其中包括 Harnack 不等式, 运费不等式, 梯度不等式等重要不等式. 作为这些课题的继续和深入, 本课题拟进一步研究关于扩散半群的测度发展系统存在唯一性, 进而以测度发展系统为参考测度族研究扩散半群的 H 超压缩性, S 超压缩性和 U 超压缩性以及渐进性质. 同时本课题还将研究扩散过程的短时间行为并给出热核短时间行为的估计.

现代分析学与偏微分方程等数学分支中的众多问题均可归结为研究某些算子的性质,而用随机分析的方法可以更好地理解这些算子的微观意义. 本课题研究了关于扩散半群的测度发展系统存在唯一性, 进而以测度发展系统为参考测度族研究扩散半群的 H 超压缩性, S 超压缩性和 U 超压缩性以及渐进性质.同时本课题还调整增加研究曲率上下界的泛函刻画和一般曲率流的泛函刻画，O-U半群的谱系估计，函数导数的上界估计以及带边流形上的泛函不等式。新增内容同样为流形上随机分析的重要课题，同时为进一步的研究提供基础。