Uniqueness for the signature of a path of bounded variation and the reduced path group
We introduce the notions of tree-like path and tree-like equivalence between paths and prove that the latter is an equivalence relation for paths of finite length. We show that the equivalence classes form a group with some similarity to a free group, and that in each class there is one special tree...
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| Format: | Journal article |
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2005
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| _version_ | 1826291506857639936 |
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| author | Hambly, B Lyons, T |
| author_facet | Hambly, B Lyons, T |
| author_sort | Hambly, B |
| collection | OXFORD |
| description | We introduce the notions of tree-like path and tree-like equivalence between paths and prove that the latter is an equivalence relation for paths of finite length. We show that the equivalence classes form a group with some similarity to a free group, and that in each class there is one special tree reduced path. The set of these paths is the Reduced Path Group. It is a continuous analogue to the group of reduced words. The signature of the path is a power series whose coefficients are definite iterated integrals of the path. We identify the paths with trivial signature as the tree-like paths, and prove that two paths are in tree-like equivalence if and only if they have the same signature. In this way, we extend Chen's theorems on the uniqueness of the sequence of iterated integrals associated with a piecewise regular path to finite length paths and identify the appropriate extended meaning for reparameterisation in the general setting. It is suggestive to think of this result as a non-commutative analogue of the result that integrable functions on the circle are determined, up to Lebesgue null sets, by their Fourier coefficients. As a second theme we give quantitative versions of Chen's theorem in the case of lattice paths and paths with continuous derivative, and as a corollary derive results on the triviality of exponential products in the tensor algebra. |
| first_indexed | 2024-03-07T03:00:25Z |
| format | Journal article |
| id | oxford-uuid:b0c6388e-28a7-4476-9fa8-c3884d84cd27 |
| institution | University of Oxford |
| last_indexed | 2024-03-07T03:00:25Z |
| publishDate | 2005 |
| record_format | dspace |
| spelling | oxford-uuid:b0c6388e-28a7-4476-9fa8-c3884d84cd272022-03-27T03:58:54ZUniqueness for the signature of a path of bounded variation and the reduced path groupJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:b0c6388e-28a7-4476-9fa8-c3884d84cd27Symplectic Elements at Oxford2005Hambly, BLyons, TWe introduce the notions of tree-like path and tree-like equivalence between paths and prove that the latter is an equivalence relation for paths of finite length. We show that the equivalence classes form a group with some similarity to a free group, and that in each class there is one special tree reduced path. The set of these paths is the Reduced Path Group. It is a continuous analogue to the group of reduced words. The signature of the path is a power series whose coefficients are definite iterated integrals of the path. We identify the paths with trivial signature as the tree-like paths, and prove that two paths are in tree-like equivalence if and only if they have the same signature. In this way, we extend Chen's theorems on the uniqueness of the sequence of iterated integrals associated with a piecewise regular path to finite length paths and identify the appropriate extended meaning for reparameterisation in the general setting. It is suggestive to think of this result as a non-commutative analogue of the result that integrable functions on the circle are determined, up to Lebesgue null sets, by their Fourier coefficients. As a second theme we give quantitative versions of Chen's theorem in the case of lattice paths and paths with continuous derivative, and as a corollary derive results on the triviality of exponential products in the tensor algebra. |
| spellingShingle | Hambly, B Lyons, T Uniqueness for the signature of a path of bounded variation and the reduced path group |
| title | Uniqueness for the signature of a path of bounded variation and the
reduced path group |
| title_full | Uniqueness for the signature of a path of bounded variation and the
reduced path group |
| title_fullStr | Uniqueness for the signature of a path of bounded variation and the
reduced path group |
| title_full_unstemmed | Uniqueness for the signature of a path of bounded variation and the
reduced path group |
| title_short | Uniqueness for the signature of a path of bounded variation and the
reduced path group |
| title_sort | uniqueness for the signature of a path of bounded variation and the reduced path group |
| work_keys_str_mv | AT hamblyb uniquenessforthesignatureofapathofboundedvariationandthereducedpathgroup AT lyonst uniquenessforthesignatureofapathofboundedvariationandthereducedpathgroup |