# Stochastic Cahn-Hilliard equation in higher space dimensions: The motion of bubbles

## TL;DR

This paper investigates the stochastic dynamics of droplets governed by the Cahn-Hilliard equation in higher dimensions, establishing a rigorous SDE for droplet motion and analyzing the stability of a deterministic slow manifold under small noise.

## Contribution

It introduces a rigorous derivation of the stochastic differential equation describing droplet motion and proves the stability of the slow manifold in the stochastic Cahn-Hilliard model.

## Key findings

- Derived a stochastic differential equation for droplet motion.
- Proved stability of the slow manifold under small stochastic perturbations.
- Established the behavior of droplets in higher-dimensional stochastic Cahn-Hilliard equations.

## Abstract

We study the stochastic motion of a droplet in a stochastic Cahn-Hilliard equation in the sharp interface limit for sufficiently small noise. The key ingredient in the proof is a deterministic slow manifold, where we show its stability for long times under small stochastic perturbations. We also give a rigorous stochastic differential equation for the motion of the center of the droplet.