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