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In this paper, we study different scaling rough path limit regimes in spaceand time for the Langevin dynamics on a quasi-planar fluctuating Helfrichsurfaces. The convergence results of the processes were already proven in thework by Duncan, Elliott, Pavliotis and Stuart (2015). We extend this work byproving the convergence of the It\^o and Stratonovich rough path lift. For therough path limit, there appears, typically, an area correction term to theIt\^o iterated integrals, and in certain regimes to the Stratonovich iteratedintegrals. This yields additional information on the homogenization limit andenables to conclude on homogenization results for diffusions driven by theBrownian motion on the membrane using the continuity of the It\^o-Lyons map inrough paths topology.

Rough homogenization for Langevin dynamics on fluctuating Helfrich surfaces