关于若干模型泛函不等式及其应用的研究

There are many applications of the theory of functional inequalities in the areas of functional analysis, probability theory, partial differential equation and differential geometry. We will mainly study functional inequalities and their applications for two different models, including functional inequalities for path and loop space over a Riemannian manifold, and functional inequalities corresponding to a large class of non-local Dirichlet forms. There are lots of backgrounds on stochastic analysis for the two models mentioned above. In particular, there are deep connections between the first model and Malliavin calculus, and the second model is much related to the study of jump processes. For path and loop space over a Riemannian manifold, we intend to study functional inequalities for the probability measure and O-U Dirichlet form induced by Brownian motion and Brownian bridge associated with a time-changing Riemannian metric respectively. At the same time, we will consider various related problems, including interwining formula, integration by parts formula, gradient estimate for (time- inhomogeneous) heat kernel and so on. For the second model, we will investigate functional inequalities corresponding to a large class of non-local Dirichlet forms on a general metric measure space, and apply them to the study of the intrinsic ultracontractivity of associated semigroup, heat kernel estimate and estimate for the speed of associated jump diffusion converging to the equilibrium.

泛函不等式的理论在泛函分析，概率论，偏微分方程，微分几何等领域有着广泛的应用。本项目主要研究关于两类模型的泛函不等式及其应用，包括黎曼轨道空间和环空间上的泛函不等式以及关于一大类非局部狄氏型的泛函不等式。上述两种模型都有很深的随机分析背景，特别第一类模型与Malliavin分析，第二类模型与跳过程都有着紧密联系。对于黎曼轨道空间与环空间，我们打算研究(黎曼流形上)关于随时间变化黎曼度量的布朗运动与布朗桥导出的概率测度和O-U狄氏型对应的泛函不等式，并且同时研究与其密切相关的缠绕公式，分部积分公式，(非时齐)热核梯度估计等问题。对于第二种模型，我们拟研究在一般测度度量空间上关于一大类非局部狄氏型的泛函不等式，并且将其应用于对应半群本质超压缩性，热核估计以及对应跳扩散过程收敛到平衡态速率估计等问题。

本项目研究了若干模型的泛函不等式与随机分析相关问题，包括轨道空间上的泛函不等式，跳过程对应半群的本质超压缩性，随机电导电阻模型的泛函不等式与随机分析，可压缩Navier-Stokes方程的随机刻画等。特别的，我们得到了在底空间不同曲率条件下，时间无限轨道空间上的庞加莱不等式的不同结果，我们证明了对于一般跳过程导出的Feynman-Kac半群和horn-shaped区域上Dirichlet半群的本质超压缩性判别准则及基态估计，我们也证明了随机电导电阻模型上alpha-stable型随机游动的逐点不变原则以及长时间的热核双边估计，同时我们引入了随机变分法对可压Navier-Stokes方程进行了刻画。