The Geometry of Rough Path Space
Martin Geller, Terry Lyons
TL;DR
The paper develops a canonical, internal perturbation theory for rough paths by introducing an $H$-space $\mathscr{H}^p(V)$ of genuine rough-path perturbations and a primary operation $X \boxplus H$ defined via Lyons' sewing lemma. It proves key algebraic properties: trivial kernel and associativity of the action, and, by relating $\mathscr{H}^p(V)$ to an $I$-space $\mathfrak{I}^p(V)$ via the bijection $\mathbb{1}^{(\cdot)}$ with inverse $\mathrm{dev}$, provides a robust vector-space structure of displacements inside rough path space $\Omega^p(V)$. Extending to almost rough paths $\mathscr{H}^{\mathrm{am},p}(V)$ shows these enlargements do not create new displacements, i.e., displacements from a base rough path are exhausted by $\mathscr{H}^p(V)$. The work situates an internal tangent-like calculus within rough path space, connects with existing translations and extensions, and yields a precise, level-by-level construction that applies to both weakly geometric and non-weakly geometric rough paths without smoothness assumptions.
Abstract
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. $\{X\boxplus H: H\in H^p(V)\}=\{X\boxplus H: H\in H^{am,p}(V)\}$.