论文标题
SL_2（R） - 具有有限变化的平面路径的开发和签名渐近学
SL_2(R)-developments and Signature Asymptotics for Planar Paths with Bounded Variation
论文作者
论文摘要
由正式的全局迭代路径积分定义的签名变换是路径空间与张量代数之间的同态，它在几何学，控制理论，数理论以及随机分析中进行了研究。一个优雅的等轴测指标指出，有界变化路径$γ$的长度可以从其归一化签名的渐近学中恢复：$ \ text {length}（γ）= \ lim_ {n \ rightarrow \ rightarrow \ rightarrow \ infty} \ big big big \ vert n！\ int_ {0 <t_ {1} <\ cdots <t_ {n} <t}dγ_{t_ {1}} \ otimes \ cdots \ cdots \ otimesdγ_{t_ {t_ {t_ {n}} 该属性取决于一个关键的拓扑非分类概念，称为树状，即没有树状的碎片）。现有参数至关重要地依赖于$γ$在单位速度参数化下具有连续导数的$γ$。在本文中，我们通过仅在$γ'$的角度上假设局部界限（确保没有树状的碎片）来证明平面路径的上述轴测猜想。我们的技术基于将路径提升到特殊线性组$ {\ rm sl} _ {2}（\ mathbb {r}）$并分析相关角度动力学在微观级别上的行为。
The signature transform, defined by the formal tensor series of global iterated path integrals, is a homomorphism between the path space and the tensor algebra that has been studied in geometry, control theory, number theory as well as stochastic analysis. An elegant isometry conjecture states that the length of a bounded variation path $γ$ can be recovered from the asymptotics of its normalised signature: $\text{Length}(γ)=\lim_{n\rightarrow\infty}\big\Vert n!\int_{0<t_{1}<\cdots<t_{n}<T}dγ_{t_{1}}\otimes\cdots\otimes dγ_{t_{n}}\big\Vert^{\frac{1}{n}}$. This property depends on a key topological non-degeneracy notion known as tree-reducedness (namely, with no tree-like pieces). Existing arguments have relied crucially on $γ$ having a continuous derivative under the unit speed parametrisation. In this article, we prove the above isometry conjecture for planar paths by assuming only local bounds on the angle of $γ'$ (which ensures the absence of tree-like pieces). Our technique is based on lifting the path onto the special linear group ${\rm SL}_{2}(\mathbb{R})$ and analysing the behaviour of the associated angle dynamics at a microscopic level.
