Geometry of subelliptic diffusions
Geometry of subelliptic diffusions
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亚椭圆扩散的几何形状
DOI：
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发表时间：
2016
期刊：
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通讯作者：
Anton Thalmaier
中科院分区：
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作者：
Anton Thalmaier
这些讲座侧重于一些概率方面有关的次黎曼几何。主要目的是介绍亚椭圆和次椭圆扩散。这些笔记是从几何学的角度写的，试图最大限度地减少遵循论点所必需的“概率行李”的重量。我们讨论特别是以下主题：随机流二阶微分算子;光滑的转移概率下霍曼德括号条件;控制理论和Stroock-Varadhan的支持定理; Malliavin演算;霍曼德定理。这些笔记从几何随机分析中的众所周知的事实开始，并指导最近正在进行的研究课题，如亚椭圆热核估计;亚椭圆扩散半群的梯度估计和Harnack型不等式;与亚黎曼扩散相关的曲率概念。
These lectures focus on some probabilistic aspects related to sub-Riemannian geometry. The main intention is to give an introduction to hypoelliptic and subelliptic diffusions. The notes are written from a geometric point of view trying to minimize the weight of “probabilistic baggage” necessary to follow the arguments. We discuss in particular the following topics: stochastic flows to second order differential operators; smoothness of transition probabilities under Hormander’s brackets condition; control theory and Stroock-Varadhan’s support theorems; Malliavin calculus; Hormander’s theorem. The notes start from well-known facts in Geometric Stochastic Analysis and guide to recent on-going research topics, like hypoelliptic heat kernel estimates; gradient estimates and Harnack type inequalities for subelliptic diffusion semigroups; notions of curvature related to sub-Riemannian diffusions.