Dirichlet space and analysis of harmonic map over the space of Gromov-Hausdorff limit spaces

(1)度量空间上的变分收敛研究：这一结果是与东北大学数学研究所研究生院副教授Takashi Shioya共同研究的。我们引入了度量空间直和上的渐近关系的概念，其中包括Gromov-Hausdorff收敛的概念。我们讨论了泛函的几个概念，如Gamma收敛、Mosco收敛和紧收敛等。给出了泛函紧收敛的一个充分条件。我们还证明了CAT(0)-空间上能量泛函的预解子收敛的一个充分条件。(2)非对称扰动下半群的随机表示的研究：本研究是与P.J.Fitzsimmons，Z.Q.Chen和T.S.Zhang教授共同完成的。考虑对称正则Dirichlet形式和允许其样本路径跳跃的相关Hunt过程。我们利用局部平方可积可积的可加泛函和有限变差的连续可加泛函考虑一类非对称摄动。证明了相应的半群在样本路径上具有关于时间逆算子的随机表示。(3)Calabis型强极大值原理的研究：在与局部正则半Dirichlet形式相关的强Feller扩散过程的框架下，给出了E.Calabi强极大值原理的一个推广的随机证明。我们的结果可以应用于一族具有一致Ricci曲率下界和一致直径下界的紧致黎曼流形上的Gromov-Hausedorff极限空间。

(1)The study of variational convergence over metric measured space : This result is a joint work with Professor Takashi Shioya, who is an associate professor of Graduate School of Mathematical Institute, Tohoku University. We introduce a notion called asymptotic relation over a direct sum of metric spaces, which includes the notion of Gromov-Hausdorff convergence as an example. We discuss several notions of functionals over it, for example, Gamma convergence, Mosco convergence and compact convergence and so on. We give a sufficient condition for the compact convergence of functionals. We also prove a sufficient condition for the convergence of resolvents of energy functionals over CAT(0)-spaces.(2)The study on the stochastic representation of semigroups obtained from a non-symmetric perturbation : This study is a joint work with Professors P.J.Fitzsimmons, Z.Q.Chen and T.S.Zhang. Consider a symmetric regular Dirichlet form and the associated Hunt process admitting jumps of its sample paths. We consider a non-symmetric perturbation by use of locally square integrable martingale additive functionals and a continuous additive functional of finite variation. We prove that the corresponding semigroup has a stochastic representation in terms of time reverse operator on sample paths.(3)The study of Calabis type strong maximum principle : We give a stochastic proof of an extension of E.Calabi's strong maximum principle in the framework of strong Feller diffusion processes associated with local regular semi-Dirichlet forms. Our results can be applicable to the Gromov-Hausedorff limit space over a family of compact Riemannian manifolds with uniform lower bounds of Ricci curvature and uniform bounds of diameter.