In the context of controlled differential equations, the signature is the exponential function on paths. B. Hambly and T. Lyons proved that the signature of a bounded variation path is trivial if and only if the path is tree-like. We extend Hambly-Lyons' result and their notion of tree-like paths to the setting of weakly geometric rough paths in a Banach space. At the heart of our approach is a new definition for reduced path and a lemma identifying the reduced path group with the space of signatures.where x : [0, T ] → R d is a path with bounded variation and X 1 , . . . , X d are formal non-commutative indeterminates. After proving a homomorphism property of the map S ([8], see (2.1) below), he gave an argument [10] that the map S restricted to appropriate classes of paths is, up to translation and reparametrisation, injective. Hambly and Lyons [16], motivated by the application of the map S in rough path theory, posed the following problem:How to characterise the kernel of the map S?Hambly and Lyons [16] proved that for a bounded variation path x, S (x) = 1 if and only if x is tree-like. They conjectured that the result extends to weakly geometric rough paths, a fundamental class of control paths for which controlled differential equations can be defined. Their result directly implies that the space *
The goal of this paper is to simplify and strengthen the Le Jan-Qian approximation scheme of studying the uniqueness of signature problem to the non-Markov setting. We establish a general framework for a class of multidimensional stochastic processes over [0, 1] under which with probability one, the signature (the collection of iterated path integrals in the sense of rough paths) is well-defined and determines the sample paths of the process up to reparametrization. In particular, by using the Malliavin calculus we show that our method applies to a class of Gaussian processes including fractional Brownian motion with Hurst parameter H > 1/4, the Ornstein-Uhlenbeck process and the Brownian bridge.
We exhibit an explicit natural isomorphism between spaces of branched and geometric rough paths. This provides a multi-level generalisation of the isomorphism of Lejay-Victoir [LV06] as well as a canonical version of the Itô-Stratonovich correction formula of Hairer-Kelly [HK15]. Our construction is elementary and uses the property that the Grossman-Larson algebra is isomorphic to a tensor algebra.We apply this isomorphism to study signatures of branched rough paths. Namely, we show that the signature of a branched rough path is trivial if and only if the path is tree-like, and construct a non-commutative Fourier transform for probability measures on signatures of branched rough paths. We use the latter to provide sufficient conditions for a random signature to be determined by its expected value, thus giving an answer to the uniqueness moment problem for branched rough paths.2010 Mathematics Subject Classification. Primary 60H10; Secondary 16T05, 60B15.
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