Flow Equations on Spaces of Rough Paths

Given an Itô vector fieldM, there is a unique solutionξt(h) to the differential equationdξt(h)dt=M(ξt(h)), ξ0(h)=hfor any continuous and piece-wisely smooth pathh. We show that for anyt∈R, the maph→ξt(h) is continuous in thep-variation topology for anyp⩾1, so that it uniquely extends to a solution flow on the space of all geometric rough paths. Applying this result to the Driverʹs geometric flow equation on the path space over a closed Riemannian manifolddζtdt=Xh(ζt), ξ0=id,whereXhis the vector field defined by parallel translatinghvia a connection, our result especially yields a deterministic construction of the Driverʹs flow.