论文标题
无参数路径空间上的拓扑
Topologies on unparameterised path space
论文作者
论文摘要
由K.T.引入的路径的签名Chen [5] $ 1954美元$，近年来已广泛研究。 Hambly和Lyons的2010年$ Paper [12]表明，该签名是在连续有限变化路径的空间上进行的，直到一般的重新聚集概念称为树状等价。该签名已被广泛用于应用中，并得到结果[15]的基础，该结果保证了由签名的线性函数设置的紧凑型在紧凑型函数上均匀的近似。 我们详细研究了三个候选拓扑的特性在一组未参数的路径（类似树状的等效类）上。这些是通过签名的属性获得的，是：（1）产品拓扑，通过为张量代数配备产品拓扑结构，并要求$ s $作为嵌入，（2）从基础路径空间上与$ d（3）$ d（3）的商学拓扑衍生的商拓扑，以及[3）] γ^* - σ^*\ vert \ vert_ {1} $使用合适的代表$γ^*$和等价类的$σ^*$。拓扑是通过严格的包容来排序的，（1）是最弱的，（3）最强的。每个都是可分离的，（1）既可以计数器，又是$σ$ - 盖，但不是贝尔的空间，因此既不是抛光剂也不是局部紧凑。商拓扑（2）是不可测量的，公制$ d $尚未完成。 （未参数）路径空间上的一个重要功能是受控微分方程的（固定时间）解决方案图。对于一系列此类方程式，我们证明了该地图的每个拓扑的可测量性。在更牢固的规律性假设下，我们在产品拓扑的显式紧凑子集上显示了连续性（1）。我们将这些结果与[15]的预期签名模型联系起来。
The signature of a path, introduced by K.T. Chen [5] in $1954$, has been extensively studied in recent years. The $2010$ paper [12] of Hambly and Lyons showed that the signature is injective on the space of continuous finite-variation paths up to a general notion of reparameterisation called tree-like equivalence. The signature has been widely used in applications, underpinned by the result [15] that guarantees uniform approximation of a continuous function on a compact set by a linear functional of the signature. We study in detail, and for the first time, the properties of three candidate topologies on the set of unparameterised paths (the tree-like equivalence classes). These are obtained through properties of the signature and are: (1) the product topology, obtained by equipping the tensor algebra with the product topology and requiring $S$ to be an embedding, (2) the quotient topology derived from the 1-variation topology on the underlying path space, and (3) the metric topology associated to $d( [ γ] ,[ σ] ) := \vert\vert γ^*-σ^*\vert\vert_{1}$ using suitable representatives $γ^*$ and $σ^*$ of the equivalence classes. The topologies are ordered by strict inclusion, (1) being the weakest and (3) the strongest. Each is separable and Hausdorff, (1) being both metrisable and $σ$-compact, but not a Baire space and so neither Polish nor locally compact. The quotient topology (2) is not metrisable and the metric $d$ is not complete. An important function on (unparameterised) path space is the (fixed-time) solution map of a controlled differential equation. For a broad class of such equations, we prove measurability of this map for each topology. Under stronger regularity assumptions, we show continuity on explicit compact subsets of the product topology (1). We relate these results to the expected signature model of [15].
