关于若干模型热核估计，泛函不等式的随机分析及相关问题的研究

There are much interaction between the theory on probability theory, functional analysis, geometric analysis for the research about the heat kernel estimates , functional inequalities and related fields. In this program we will study the stochastic analysis on heat kernel estimates, functional inequalities and related problems for several models. In particular, the research subjects contain the Dirichlet heat kernel estimates on horn-shaped domains and cone-like domains, the super Poincare, weak Poincare inequalities and Wang type Harnack inequalities on a general (complete) Riemannian manifolds based on the integrable condition of Ricci curvature and sectional curvature, the characterization for properties of singularity points of a Ricci flow through limit behaviors for various diffusion processes with time-changing Riemannian metric.

热核估计与泛函不等式的研究涉及概率论、泛函分析、几何分析等多个领域的交叉。本项目拟研究关于若干模型的热核估计，泛函不等式的随机分析的相关问题及应用，horn-shaped区域及类锥形区域上关于一般跳过程的Dirichlet热核估计，一般(非紧)完备流形上由Ricci曲率或截面曲率的积分条件导出的整体超庞加莱，弱庞加莱不等式以及王型Harnack不等式，利用各种度量随时间变化的扩散过程的极限性质来刻画Ricci flow奇异点的性质等等。

本项目中我们对几个模型热核估计，泛函不等式，随机分析及相关问题进行了研究，包括(i)马鞍型区域上稳定过程的Dirichlet热核估计(ii)积分条件导出的泛函不等式(iii)随机方法在微分几何研究中的应用。特别的，我们引入一些新的方法，研究了马鞍型区域上稳定过程Dirichlet热核双边热核估计，随机环境中长距离随机游动的不变原理，黎曼流形上倒向随机微分方程等问题。相关结果发表在AOP, PTRF, Math Ann, JFA, AAP等期刊上。