[Paper Review] Convergence of random walks to Brownian motion on cubical complexes

Tom M. W. Nye|arXiv (Cornell University)|Aug 12, 2015

Stochastic processes and statistical mechanics29 references8 citations

TL;DR

This paper establishes that random walks on cubical complexes converge to Brownian motion in the limit, providing a computationally tractable approximation for statistical inference on stratified spaces. By projecting Brownian paths onto a standard cube and using a reflection-based random walk algorithm that respects cube boundaries, the authors prove convergence of transition kernels, enabling statistical modeling on complex spaces like evolutionary tree spaces.

ABSTRACT

Cubical complexes are metric spaces constructed by gluing together unit cubes in an analogous way to the construction of simplicial complexes. We construct Brownian motion on such spaces, define random walks, and prove that the transition kernels of the random walks converge to that for Brownian motion. The proof involves pulling back onto the complex the distribution of Brownian sample paths on the standard cube, and combining this with a distribution on walks between cubes in the complex. The main application lies in analysing sets of evolutionary trees: several tree spaces are cubical complexes and we briefly describe our results and some applications in this context. Our results extend readily to a class of polyhedral complex in which every cell of maximal dimension is isometric to a given fixed polyhedron.

Motivation & Objective

• To develop a tractable stochastic process for statistical inference on cubical complexes, which are increasingly used in evolutionary biology and data analysis.
• To address the challenge of intractable normalizing constants in probability distributions on stratified spaces by using transition kernels of Markov processes.
• To establish a rigorous convergence result between discrete random walks and continuous Brownian motion on cubical complexes.
• To extend the framework to a broader class of polyhedral complexes where each maximal cell is isometric to a fixed polyhedron.
• To provide a foundation for constructing statistical models—such as means and variances—on non-Euclidean, stratified spaces like BHV tree space.

Proposed method

• Define Brownian motion on cubical complexes by extending Euclidean Brownian motion within each cube and allowing uniform transition across shared codimension-1 faces.
• Construct a random walk algorithm that simulates Brownian motion by moving in a direction vector δ, reflecting at cube boundaries with equal probability into adjacent cubes.
• Use a projection map P to pull back Brownian sample paths from the complex onto a single standard cube, enabling measure transfer and explicit kernel construction.
• Prove convergence of the transition kernels of the random walks to that of Brownian motion using the pull-back measure and careful analysis of infinite boundary crossings.
• Generalize the result to a class of polyhedral complexes where maximal cells are isometric to a fixed polyhedron, maintaining symmetry for analytical control.
• Handle technical challenges arising from infinite boundary crossings by leveraging the structure of the complex and probabilistic coupling techniques.

Experimental results

Research questions

• RQ1 Can random walks on cubical complexes be constructed such that their transition kernels converge to those of Brownian motion?
• RQ2 How can Brownian motion be rigorously defined on cubical complexes with non-trivial stratification and boundary interactions?
• RQ3 What is the impact of the initial point’s location (e.g., interior vs. codimension-2 face) on the convergence of random walks to Brownian motion?
• RQ4 To what extent can the convergence result be extended to more general polyhedral complexes beyond cubical complexes?
• RQ5 Can this framework support the construction of statistical models—such as Fréchet means and variances—on non-Euclidean, stratified spaces like BHV tree space?

Key findings

• The transition kernels of the proposed random walks converge to the transition kernel of Brownian motion on cubical complexes, establishing a fundamental link between discrete and continuous stochastic processes.
• The convergence holds when the initial point lies in the interior of a maximal-dimensional cube, and can be extended to points in lower-codimension faces, excluding codimension-2 faces where the projection map becomes non-canonical.
• The method relies on a pull-back construction via a projection map P that transfers measures from the complex to a standard cube, enabling explicit kernel computation.
• The random walk algorithm incorporates reflection at codimension-1 boundaries with equal probability into adjacent cubes, mimicking the behavior of Brownian motion at junctions.
• The result generalizes to a class of polyhedral complexes where each maximal cell is isometric to a fixed polyhedron, though extension to general simplicial complexes remains challenging due to lack of symmetry.
• The framework provides a foundation for statistical inference on stratified spaces, such as evolutionary tree spaces, by enabling the use of tractable transition kernels as parametric models.