Rough homogenization for Langevin dynamics on fluctuating Helfrich surfaces

TL;DR

This paper extends previous homogenization results for Langevin dynamics on fluctuating Helfrich surfaces by proving convergence of rough path lifts, revealing area correction terms and enhancing understanding of diffusion limits on membranes.

Contribution

It introduces the convergence of Itô and Stratonovich rough path lifts for Langevin dynamics on Helfrich surfaces, including correction terms, advancing homogenization theory in this context.

Findings

Proved convergence of rough path lifts for Langevin dynamics.

Identified area correction terms in the rough path limit.

Enabled homogenization results for membrane-driven diffusions.

Abstract

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.