Analysts on harmonic maps over geometric singular spaces via Dirichlet forms

我们得到了以下结果：1)度量测度空间的变分收敛性：引入映射的lp收敛性的一个自然定义。$ page 1$，在定域是测量度量空间相对于测量的Gromov-Hausdorff拓扑的收敛序列，目标是Gromov-Hausdorff收敛序列的情况下。利用lp收敛性，我们建立了变分收敛理论。在附加条件下证明了庞加莱不等式的渐近紧性。渐近紧性等价于能量子水平集的Gromov-Hausdorff紧性。假设目标是CAT(0)-空间，我们研究了解的收敛性。作为应用，我们研究了测量度量空间上的近似能量泛函以及具有里奇曲率下界的能量泛函的收敛性。2)对称马尔可夫过程的摄动及其相关的stocha . More stic calculus：我们给出了由一般对称马尔可夫过程的l2 -无穷小发生器L的低阶摄动得到的二次型半群的路径空间积分表示。利用时间反转，我们引入了对称马尔可夫过程的零能量加性泛函的随机积分，扩展了S. Nakao的早期工作。讨论了这类随机积分的各种性质，得到了狄利克雷过程的It^o公式。这项工作是与陈志强教授共同完成的。3)对称马尔可夫过程上的Kato类测度：我们证明了$fin L^p(X; m)$暗示$|f|dmin S_K^1$对于$p>D$与$D>0$，其中$S_K^1$是与$L^2(X; m)$上的（非对称）Dirichlet形式相关的马尔可夫过程的半群窝$p_t(X, y)$相关的Kato类测度的子族。我们只假设$p_t(x, y)$满足$D$上小时间防御的纳什型估计。结果不需要$p_t(X， V)$的具体表达式。4)关于(n, p)-容量炉对称马尔可夫过程的例外集的改进：我们建立了一类(n, p)-意义上的光滑测度与一类允许(n, p)-例外集的正连续加性泛函之间的一一对应关系。5)关于马尔可夫链上凸空间调和映射的Liouville定理：我们给出了从具有保守马尔可夫链上函数调和性的空间到允许质心的凸空间调和映射的Liouville型定理。假设域和目标均无可微结构。这项工作是与k.Th教授共同完成的。6) Alexandrov空间上的Laplacian比较定理：我们将Bishop-Gromov相对体积比较的方向限制形式看作是Alexandrov空间下界的Ricci曲率的广义概念。在此条件下证明了Alexandrov空间的一个拉普拉斯比较定理。作为一个应用，我们证明了拓扑分裂定理。这项工作是与T.Shioya教授一起完成的。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