Locality of rough path lifts

Abstract

Every Hölder continuous path $X$ admits a geometric rough path lift $\mathbf{X}$ by the Lyons--Victoir extension theorem. A natural question that emerges when lifting more than one path segment at once is that of \emph{locality}, namely whether the lift ${\mathbf{X}}_{s,t}$ only depends on the increments $X_{s,u}$, $u\in [s,t]$. We investigate the locality of rough path lifts in deterministic and stochastic settings. On the deterministic side, we show that no local, homogeneous rough path lift can be defined on $\gamma$-Hölder paths for all $\gamma \le 1/2$. More strongly, we show that no Lévy area can be defined which is at the same time bounded, with no further regularity assumptions, and either local and homogeneous or time translation--invariant. We moreover show that the boundedness requirement is sharp: an unbounded, local, time translation--invariant, and bilinear Lévy area can be defined on all continuous paths. On the stochastic side, we classify all local, square-integrable rough path lifts of $d$-dimensional fractional Brownian motion with Hurst parameter $H\in (0,1/2]$. For $H\le 1/4$, we show that no such lifts exist when $d\ge 2$, while for $H>1/4$, we show that all such lifts are stochastic translations of the canonical rough path. We further refine the classification by requiring invariance in law under time translation, scaling, and coordinate permutation, and show that only the canonical lift satisfies these constraints except at \(H=1/3\), for which there is a one-parameter family of lifts. Finally, we present an argument showing that scale-invariance forces a local $\eta$-Hölder rough path lift of fractional Brownian motion to be square-integrable, allowing scale-invariance to replace square-integrability in our classification and non-existence results.

We included a proof that local, time translation-invariant RPLs of fBm are in L2, a partial answer to the conjecture that square-integrability does not need to be assumed