Diffusion on Some Simple Stratified Spaces

Abstract

A variety of different imaging techniques produce data which naturally lie in stratified spaces. These spaces consist of smooth regions of maximal dimension glued together along lower dimensional boundaries. Diffusion processes are important as they can be used to represent noise in statistical models on spaces for which standard parametric probability distributions do not exist. We consider particles undergoing Brownian motion in some low dimensional stratified spaces, and obtain analytic solutions to the heat equation specifying the distribution of particles. These solutions play the role of prototypical distributions for studying behaviour near singularities. While probabilistic reasoning can be used to solve the heat equation in some straightforward cases, more generally we construct solutions from eigenfunctions of the Laplacian. Specifically, we solve the heat equation on: open books; two-dimensional cones; the Petersen graph with unit edge length; and the cone of this graph which corresponds to a space of evolutionary trees.