Topologies on unparameterised path space
Thomas Cass, William F. Turner

TL;DR
This paper investigates three different topologies on the space of unparameterised paths, analyzing their properties, relationships, and implications for the measurability and continuity of the controlled differential equation solution map.
Contribution
It provides the first detailed study of these topologies on unparameterised path space, clarifying their properties and relevance for signature-based analysis.
Findings
The three topologies are strictly ordered by inclusion, with distinct properties.
All topologies are separable and Hausdorff; some are metrizable, others are not.
The solution map of controlled differential equations is measurable under these topologies.
Abstract
The signature of a path, introduced by K.T. Chen [5] in , has been extensively studied in recent years. The 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 to be an embedding, (2) the quotient…
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Taxonomy
TopicsMetabolism, Diabetes, and Cancer · Hyperglycemia and glycemic control in critically ill and hospitalized patients
