Flow equations on spaces of rough paths

Abstract

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 Driver's 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 Driver's flow. © 1997 Academic Press.