The Wolfram Demonstration Project has an excellent example by Yu-Sung Chang showing construction of a Voronoi diagram using distance transform on a set of points in 2D plane. Here same method is used on a set of points which perform random walks. All calculations and animation are done in Mathematica.

Yu-Sung Chang Demonstration: demonstrations.wolfram.com/VoronoiImage/

Voronoi Image of Random Walks – Slow from Vitaliy Kaurov on Vimeo.