Analysts on harmonic maps over geometric singular spaces via Dirichlet forms

建立了如下结果：1)度量度量空间的变分收敛：我们引入了映射的Lp-收敛的自然定义。$PGE 1$，在域是关于测量的Gromov-Hausdorff拓扑的测量度量空间的收敛序列并且目标是Gromov-Hausdorff收敛序列的情况下。利用Lp-收敛，我们建立了一个变分收敛理论。证明了带附加条件的Poincare不等式蕴含渐近紧性。渐近紧性等价于能量次水平集的Gromov-Hausdorff紧性。假设目标是CAT(0)-空间，我们研究预解的收敛问题。作为应用，我们研究了度量空间上的近似能量泛函和具有Ricci曲率下界的能量泛函的收敛。这项工作是与T.Shiya教授一起完成的。2)对称马尔可夫过程的摄动及其相关的Stocha…更一般的演算：我们给出了一般对称马尔可夫过程的L2-无穷小生成元L的低阶扰动所得到的与二次型有关的半群的路径空间积分表示。利用时间反转，我们引入了对称马尔可夫过程零能可加泛函的随机积分，推广了S.Nakao的早期工作。讨论了这类随机积分的各种性质，得到了Dirichlet过程的It^o公式。这项工作是与陈振全教授共同完成的。P.J.Fitzsimmons和T.S.Zhang.3)对称马尔可夫过程上的Kato类测度：我们证明了$fin L^p(X；m)$蕴含$|f|dmin S_K^1$对于具有$D&gt；0$的$p&gt；D；D$，其中$S_K^1$是相对于$L^2(X；m)$上的半群核$p_t(x，y)$的Kato类测度的子族.我们只假设$p_t(x，y)$满足在$D上防御的小时间的Nash型估计。结果不需要$p_t(X，V)$的具体表达式。4)对称马尔可夫过程关于(n，p)-容量的例外集的精化：我们在(n，p)-意义下的一类光滑度量和一类允许(n，p)-例外集的正连续可加泛函之间建立了一一对应。这项工作是用A.Sato.5)关于马氏链上的调和映射到凸空间的Liouville定理完成的：我们给出了从具有保守马氏链的函数的调和空间到含有重心的凸空间的调和映射的一个Liouville类型的定理。不假定域和目标的结构是可区分的。这项工作是与他一起完成的。K.Th.6)Alexandrov空间上的拉普拉斯比较定理：对于Alexandrov空间，我们把Bishop-Gromov相对体积比较的一个方向受限形式看作是广义Ricci曲率的下有界概念。在此条件下，我们证明了Alexandrov空间的拉普拉斯比较定理。作为应用，我们证明了一个拓扑分裂定理。这项工作是与T.Shioya教授一起完成的。较少

We establish the following result :1) Variational convergence of metric measure spaces:We introduce a natural definition of Lp-convergence of maps. $pge 1$, in the case where the domain is a convergent sequence of measured metric space with respect to the measured Gromov-Hausdorff topology and the target is a Gromov-Hausdorff convergent sequence. With the Lp-convergence, we establish a theory of variational convergences. We prove that the Poincare inequality with some additional condition implies the asymptotic compactness. The asymptotic compactness is equivalent to the Gromov-Hausdorff compactness of the energy-sublevel sets. Supposing that the targets are CAT(0)-spaces, we study convergence of resolvents. As applications, we investigate the approximating energy functional over a measured metric space and convergence of energy functionals with a lower bound of Ricci curvature. This work was done with Prof. T. Shioya.2) Perturbation of symmetric Markov processes and its related stocha … More stic calculus:We present a path-space integral representation of the semigroup associated with the quadratic form obtained by a lower orderperturbation of the L2-infinitesimal generator L of a general symmetric Markov process. Using time-reversal, we introduce a stochastic integral for zero-energy additive functionals of symmetric Markov processes, extending earlier work of S. Nakao. Various properties of such stochastic integrals are discussed and an It^o formula for Dirichlet processes is obtained. This work was done with Professors Z.Q. Chen. P.J. Fitzsimmons and T.S. Zhang.3) Kato class measures over symmetric Markov processes :We show that $fin L^p(X ; m)$ implies $|f|dmin S_K^1$ for $p>D$ with $D>0$, where $S_K^1$ is a subfamily of Kato class measures relative to a semigroup kennel $p_t(x, y)$ of a Markov process associated with a (non-symmetric) Dirichlet form on $L^2(X ; m)$. We only assume that $p_t(x, y)$ satisfies the Nash type estimate of small time defending on $D$. No concrete expression of $p_t(X, V)$ is needed for the result. This wonk was done with M. Takahashi.4) Refinements of exceptional sets with respect to (n, p)-capacity oven symmetric Markov processes:We establish a one to one correspondence between a class of smooth measures in the (n, p)-sense and a class of positive continuous additive functionals admitting (n, p)-exceptional sets. This work was done with A. Sato.5) Liouville theorems for harmonic maps to convex spaces over Markov chains:We give a Liouville type theorem for harmonic maps from the space equipped with the harmonicity of functions in terms of conservative Markov chains to convex spaces admitting barycenters. No differentiable structures for the domain and the target are assumed. This work was done with prof. k.Th. Sturm.6) Laplacian comparison theorem on Alexandrov spaces :We consider a directionally restricted version of the Bishop-Gromov relative volume comparison as generalized notion of Ricci curvature bounded below for Alexandrov spaces. We prove a Laplacian comparison theorem for Alexandrov spaces under the condition. As an application we prove a topological splitting theorem. This work was done with Prof. T.Shioya. Less