In this article, we study different scaling rough path limit regimes in space and time for the Langevin dynamics on a quasi-planar fluctuating Helfrich surface. The convergence results of the processes were already proven in [Citation1]. We extend this work by proving the convergence of the Itô and Stratonovich rough path lift. For the rough path limit, there appears, typically, an area correction term to the Itô iterated integrals and, in certain regimes, to the Stratonovich iterated integrals. This yields additional information on the homogenization limit and enables us to conclude on homogenization results for diffusions driven by the Brownian motion on the membrane using the continuity of the Itô-Lyons map in rough paths topology.
