Brownian Bridges on Polygons

There are no straight lines or sharp corners in nature, said the famous Catalan architect Antoni Gaudí. The famous Polish–French–American mathematician Benoit B. Mandelbrot went even further and asserted that the curves in nature are fractal. Inspired by Mandelbrot’s assertion, we consider regular shapes with edges replaced by stochastic fractals. In particular, we consider equilateral triangles, squares, and pentagons where the edges are replaced by realizations of different Brownian bridges: a “normal” Brownian bridge, a reflective Brownian bridge, and a sticky Brownian bridge.