The Geometry of Rough Path Space

Abstract

We describe $H^p(V)$, a subset of $p$-rough path space $\hat{I}{\text{c◯}}_p(V)$ which is a vector space under an addition operation $⊞$ and a scalar multiplication $\odot$. We show that the domain of $⊞$ can be extended to $\hat{I}{\text{c◯}}_p(V)\times H^p(V)$, allowing any $p$-rough path $X$ to be additively perturbed by an $H\in H^p(V)$. We prove associativity $(X⊞H)⊞\tilde{H}=X⊞(H⊞\tilde{H})$ and trivial kernel $X⊞H=X\iff H=1$, where $1$ is the additive zero in $(H^p(V),⊞,\odot )$. 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. $\{X⊞H:H\in H^p(V)\}=\{X⊞H:H\in H^{am,p}(V)\}$.

82 pages; no figures