A convergence to Brownian motion on sub-Riemannian manifolds
A convergence to Brownian motion on sub-Riemannian manifolds
复制标题
亚黎曼流形上布朗运动的收敛
DOI：
10.1090/tran/6831
复制
发表时间：
2017
影响因子：
1.3
通讯作者：
Laetsch, Thomas
中科院分区：
文献类型：
--
作者：
Gordina, Maria;Laetsch, Thomas
本文考虑了在亚黎曼流形设置下通过随机游走近似布朗运动的经典问题。为了构建这样的随机游走，我们首先解决与这种流形的简并性相关的几个问题。特别是，我们定义了一系列与底层流形的几何结构自然连接的亚拉普拉斯算子。当 是黎曼（非简并）流形时，我们恢复 Laplace-Beltrami 算子。然后，我们构造相应的随机游走，并在亚拉普拉斯算子的标准假设下，我们证明该随机游走（在半群水平上）收敛到一个过程，即水平布朗运动，其无穷小生成器是亚拉普拉斯算子。详细考虑配备标准亚黎曼度量的海森堡群的示例，在这种情况下，我们引入的亚拉普拉斯算子被证明是平方和（Hörmander）算子。参考
This paper considers a classical question of approximation of Brownian motion by a random walk in the setting of a sub-Riemannian manifold. To construct such a random walk we first address several issues related to the degeneracy of such a manifold. In particular, we define a family of sub-Laplacian operators naturally connected to the geometry of the underlying manifold. In the case whenis a Riemannian (non-degenerate) manifold, we recover the Laplace-Beltrami operator. We then construct the corresponding random walk, and under standard assumptions on the sub-Laplacian andwe show that this random walk converges (at the level of semigroups) to a process, horizontal Brownian motion, whose infinitesimal generator is the sub-Laplacian. An example of the Heisenberg group equipped with a standard sub-Riemannian metric is considered in detail, in which case the sub-Laplacian we introduced is shown to be the sum of squares (Hörmander’s) operator. References