论文标题
在亚里曼尼亚歧管上的随机发展的cartan连接
Cartan connections for stochastic developments on sub-Riemannian manifolds
论文作者
论文摘要
类似于布朗运动在里曼尼亚歧管上的表征为欧几里得空间上的布朗运动的发展，我们通过开发一个规范的随机过程来构建伊曼尼亚式的次 - 利曼尼亚式扩散，这是布朗尼的提升，以达到相关模型空间的提升。我们为Equinilpotisisable Sub-Riemannian歧管引入随机发展的概念使用了cartan连接，这些连接代替了Riemannian几何形状中的Levi-Civita连接。我们首先得出了随机过程的发电机的一般表达，这是随机发育相对于布朗运动与模型空间的cartan连接的随机发展。我们进一步为存在的cartan连接提供了必要和充分的条件，该连接将规范的随机过程开发到与与POPP体积定义的亚拉普拉斯相关的亚riemannian扩散。我们说明了使用两个发电机的免费次摩曼结构的合适的cartan连接的构建，我们讨论了一个不满足条件的示例。
Analogous to the characterisation of Brownian motion on a Riemannian manifold as the development of Brownian motion on a Euclidean space, we construct sub-Riemannian diffusions on equinilpotentisable sub-Riemannian manifolds by developing a canonical stochastic process arising as the lift of Brownian motion to an associated model space. The notion of stochastic development we introduce for equinilpotentisable sub-Riemannian manifolds uses Cartan connections, which take the place of the Levi-Civita connection in Riemannian geometry. We first derive a general expression for the generator of the stochastic process which is the stochastic development with respect to a Cartan connection of the lift of Brownian motion to the model space. We further provide a necessary and sufficient condition for the existence of a Cartan connection which develops the canonical stochastic process to the sub-Riemannian diffusion associated with the sub-Laplacian defined with respect to the Popp volume. We illustrate the construction of a suitable Cartan connection for free sub-Riemannian structures with two generators and we discuss an example where the condition is not satisfied.
