Uniqueness for the signature of a path of bounded variation and the reduced path group
Uniqueness for the signature of a path of bounded variation and the reduced path group
复制标题
DOI：
10.4007/annals.2010.171.109
复制
发表时间：
2010-01-01
影响因子：
4.9
通讯作者：
Lyons, Terry
中科院分区：
文献类型：
--
作者：
Hambly, Ben;Lyons, Terry
本文引入了树型路和路间树型等价的概念，并证明了后者是有限长路的等价关系。我们表明，等价类形成一个自由组有一定的相似性，在每个类有一个唯一的路径，是树减少。这些路径的集合是简化路径组。这是一个连续的模拟组减少的话。路的签名是一个幂级数，其系数是路的某个张量值的定积分，我们将具有平凡签名的路定义为树型路，并证明了两条路树型等价的充要条件是它们具有相同的签名.通过这种方式，我们延长陈的定理的序列的唯一性的迭代积分与一个分段正规的路径到有限长度的路径，并确定适当的扩展意义参数化的一般设置。这是暗示认为这一结果作为一个非交换模拟的结果，可积函数的圆圈确定，直到勒贝格空集，由他们的傅立叶系数。作为第二个主题，我们给出了定量版本的陈定理的情况下，格路径和路径连续导数，并作为推论得出结果的平凡性指数产品的张量代数。
We introduce the notions of tree-like path and tree-like equivalence between paths and prove that the latter is an equivalence relation for paths of finite length. We show that the equivalence classes form a group with some similarity to a free group, and that in each class there is a unique path that is tree reduced. The set of these paths is the Reduced Path Group. It is a continuous analogue of the group of reduced words. The signature of the path is a power series whose coefficients are certain tensor valued definite iterated integrals of the path. We identify the paths with trivial signature as the tree-like paths, and prove that two paths are in tree-like equivalence if and only if they have the same signature. In this way, we extend Chen's theorems on the uniqueness of the sequence of iterated integrals associated with a piecewise regular path to finite length paths and identify the appropriate extended meaning for parametrisation in the general setting. It is suggestive to think of this result as a noncommutative analogue of the result that integrable functions on the circle are determined, up to Lebesgue null sets, by their Fourier coefficients. As a second theme we give quantitative versions of Chen's theorem in the case of lattice paths and paths with continuous derivative, and as a corollary derive results on the triviality of exponential products in the tensor algebra.