arXiv 1503.007252015-03math.DGmath.APmath.OCmath.PR

次Riemannian geometry中的内禀随机游动和次拉普拉斯算子

Intrinsic random walks and sub-Laplacians in sub-Riemannian geometry

Ugo Boscain, Robert Neel, Luca Rizzi

备注：: 42 pages, 1 figure. v2: minor revisions; v3: minor typos corrected; v4: final version, to appear on Advances in Mathematics
发表信息：: Advances in Mathematics, 314 (2017), 124-184
DOI：: 10.1016/j.aim.2017.04.024

摘要：

在次黎曼流形上，我们定义了两类拉普拉斯算子. \emph{macroscopic Laplacian} $Δ_ω$，作为水平梯度的散度，一旦体积$ω$是固定的，以及\emph{microscopic Laplacian}，作为与测地线随机游动序列相关联的算子。我们考虑一类一般的随机游动，其中考虑了\emph{all}次黎曼测地线。该算子仅取决于子黎曼分布的补充$\mathbf{c}$的选择，并表示为$L^c$。我们解决这两个运营商的等价性问题。这个问题是有趣的，因为在等正则次黎曼流形，总是有一个内在的体积（例如波普的一个$P$），但不是一个典型的选择补充。结果在很大程度上取决于所调查的结构类型。在接触结构上，对于每个体积$ω$，存在唯一的互补$c$，使得$Δ_ω=L^c$。在卡诺群上，如果$H$是Haar体积，则总是存在一个补$c$，使得$Δ_H=L^c$。然而，这种补充一般不是唯一的。对于准接触结构，一般情况下，$Δ_P \neq L^c$为任意选择的$c$ .特别地，$L^c$不是关于. Popp的方法这是令人惊讶的，特别是在4维，在适当的意义上，$Δ_P$是唯一的内在宏观拉普拉斯算子。我们在这里引入的一个重要概念是N-内禀体积，即只依赖于幂零近似的参数集的体积。当幂零近似不依赖于点时，N-内禀体积在常数的尺度下是唯一的，相应的N-内禀次拉普拉斯算子也是唯一的。这是对于小于或等于4的维度发生的情况，并且特别是在上述4维准接触结构中。

英文摘要：

On a sub-Riemannian manifold we define two type of Laplacians. The \emph{macroscopic Laplacian} $Δ_ω$, as the divergence of the horizontal gradient, once a volume $ω$ is fixed, and the \emph{microscopic Laplacian}, as the operator associated with a sequence of geodesic random walks. We consider a general class of random walks, where \emph{all} sub-Riemannian geodesics are taken in account. This operator depends only on the choice of a complement $\mathbf{c}$ to the sub-Riemannian distribution, and is denoted $L^c$. We address the problem of equivalence of the two operators. This problem is interesting since, on equiregular sub-Riemannian manifolds, there is always an intrinsic volume (e.g. Popp's one $P$) but not a canonical choice of complement. The result depends heavily on the type of structure under investigation. On contact structures, for every volume $ω$, there exists a unique complement $c$ such that $Δ_ω=L^c$. On Carnot groups, if $H$ is the Haar volume, then there always exists a complement $c$ such that $Δ_H=L^c$. However this complement is not unique in general. For quasi-contact structures, in general, $Δ_P \neq L^c$ for any choice of $c$. In particular, $L^c$ is not symmetric w.r.t. Popp's measure. This is surprising especially in dimension 4 where, in a suitable sense, $Δ_P$ is the unique intrinsic macroscopic Laplacian. A crucial notion that we introduce here is the N-intrinsic volume, i.e. a volume that depends only on the set of parameters of the nilpotent approximation. When the nilpotent approximation does not depend on the point, a N-intrinsic volume is unique up to a scaling by a constant and the corresponding N-intrinsic sub-Laplacian is unique. This is what happens for dimension smaller or equal than 4, and in particular in the 4-dimensional quasi-contact structure mentioned above.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

摘要：

英文摘要：

相关论文