Brownian earthworm

TL;DR

This paper proves that two reflected Brownian motions driven by the same noise in a 3D torus do not converge when outside a small enough sphere, highlighting non-convergence behavior in stochastic processes.

Contribution

It establishes a new non-convergence result for coupled reflected Brownian motions in a specific geometric setting.

Findings

Distance between motions does not tend to zero almost surely

Non-convergence occurs when the sphere radius is sufficiently small

Results depend on the geometry of the torus and sphere

Abstract

We prove that the distance between two reflected Brownian motions, driven by the same white noise, outside a sphere in a 3-dimensional flat torus does not converge to 0, a.s., if the radius of the sphere is sufficiently small, relative to the size of the torus.