Topologies on unparameterised path space

Abstract

The signature of a path, introduced by K.T. Chen [10] in 1954, has been extensively studied in recent years. The fundamental 2010 paper [20] of Hambly and Lyons showed that the signature is an injective function on the space of continuous, finite-variation paths up to a general notion of reparameterisation called tree-like equivalence. This result has been extended to geometric rough paths by Boedihardjo et al. [5]. More recently, the approximation theory of the signature has been widely used in the literature in applications. The archetypal instance of these results, see e.g. [24], guarantees uniform approximation, on compact sets, of a continuous function by a linear functional on the (extended) tensor algebra acting on the signature.In this paper we study in detail, and for the first time, the properties of three natural candidate topologies on the set of unparameterised paths, i.e. the tree-like equivalence classes. These are obtained by privileging different properties of the signature and are: (1) the product topology, obtained by equipping the range of the signature with the (subspace topology of the) product topology in the extended tensor algebra and then 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]):=||γ⁎−σ⁎||1 using the (constant-speed) tree-reduced representatives γ⁎ and σ⁎ of the respective equivalence classes. We evaluate these spaces from the point of view of their suitability when it comes to studying (probability) measures on them. We prove that the respective collections of open sets are ordered by strict inclusion, (1) being the weakest and (3) the strongest. Our other conclusions can be summarised as follows. All three topological spaces are separable and Hausdorff, (1) being both metrisable and σ-compact, but not a Baire space and hence being neither Polish nor locally compact. The completion of (1), in any metric inducing the product topology, is the subspace G⁎ of group-like elements. The quotient topology (2) is not metrisable and the metric d is not complete. We also discuss some open problems related to these spaces.We consider finally the implications of the selection of the topology for uniform approximation results involving the signature. A stereotypical model for a continuous function on (unparameterised) path space is the solution of a controlled differential equation. We thus prove, for a broad class of these equations, well-definedness and measurability of the (fixed-time) solution map with respect to the Borel sigma-algebra of each topology. Under stronger regularity assumptions, we further show continuity of this same map on explicit compact subsets of the product topology (1). We relate these results to the expected signature model of [24].