# Convergence of Diffusion Generated Motion to Motion by Mean Curvature

## TL;DR

This paper presents an elementary proof demonstrating that the Merriman-Bence-Osher thresholding algorithm converges to motion by mean curvature, providing a convergence rate without relying on the maximum principle.

## Contribution

It offers a new, simpler proof of convergence for the MBO scheme to MMC, including a convergence rate and weaker assumptions on the heat kernel.

## Key findings

- Proof of convergence without maximum principle
- Establishment of a convergence rate
- Applicable under weak heat kernel assumptions

## Abstract

We provide a new proof of convergence to motion by mean curvature (MMC) for the Merriman-Bence-Osher (MBO) thresholding algorithm. The proof is elementary and does not rely on maximum principle for the scheme. The strategy is to construct a natural ansatz of the solution and then estimate the error. The proof thus also provides a convergence rate. Only some weak integrability assumptions of the heat kernel, but not its positivity, is used. Currently the result is proved in the case when smooth and classical solution of MMC exists.