We introduce and study Brownian motion on spaces of discrete regular curves in Euclidean space, such as polygons in the plane. Previously, it has been established when these spaces of discrete regular curves are geodesically complete. I will present joint work with Emmanuel Hartman which relies on a general result by Grigor'yan and shows that all spaces of discrete regular curves that are geodesically complete are also stochastically complete, that is, the associated Brownian motion exists for all times. This provides a rigorous footing for performing data statistics, such as data inference and data imputation, on these spaces. We include simulations for sample paths of Brownian motion on spaces of discrete regular curves.