The Geometry of Rough Path Space
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
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2026
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| _version_ | 1866918299366850560 |
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| author | Geller, Martin Lyons, Terry |
| author_facet | Geller, Martin Lyons, Terry |
| contents | We describe $H^p(V)$, a subset of $p$-rough path space $Ω_p(V)$ which is a vector space under an addition operation $\boxplus$ and a scalar multiplication $\odot$. We show that the domain of $\boxplus$ can be extended to $Ω_p(V)\times H^p(V)$, allowing any $p$-rough path $X$ to be additively perturbed by an $H\in H^p(V)$. We prove associativity $(X\boxplus H)\boxplus \tilde H = X\boxplus (H\boxplus \tilde H)$ and trivial kernel $X\boxplus H = X \Leftrightarrow H = 1$, where $1$ is the additive zero in $(H^p(V),\boxplus,\odot)$. Finally, we show that enlarging $H^p(V)$ to almost rough paths $H^{am,p}(V)$ does not enlarge the set of displacements of a given $X$, i.e. $\{X\boxplus H: H\in H^p(V)\}=\{X\boxplus H: H\in H^{am,p}(V)\}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_15402 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The Geometry of Rough Path Space Geller, Martin Lyons, Terry Classical Analysis and ODEs Probability 60L20 (Primary), 60H10 (Secondary) We describe $H^p(V)$, a subset of $p$-rough path space $Ω_p(V)$ which is a vector space under an addition operation $\boxplus$ and a scalar multiplication $\odot$. We show that the domain of $\boxplus$ can be extended to $Ω_p(V)\times H^p(V)$, allowing any $p$-rough path $X$ to be additively perturbed by an $H\in H^p(V)$. We prove associativity $(X\boxplus H)\boxplus \tilde H = X\boxplus (H\boxplus \tilde H)$ and trivial kernel $X\boxplus H = X \Leftrightarrow H = 1$, where $1$ is the additive zero in $(H^p(V),\boxplus,\odot)$. Finally, we show that enlarging $H^p(V)$ to almost rough paths $H^{am,p}(V)$ does not enlarge the set of displacements of a given $X$, i.e. $\{X\boxplus H: H\in H^p(V)\}=\{X\boxplus H: H\in H^{am,p}(V)\}$. |
| title | The Geometry of Rough Path Space |
| topic | Classical Analysis and ODEs Probability 60L20 (Primary), 60H10 (Secondary) |
| url | https://arxiv.org/abs/2601.15402 |