Sample path properties of G-Brownian motion

The present article is devoted to careful study of sample pathwise properties of G-Brownian motion, i.e., those which quasi-surely hold for all paths. We develop a comparison machinery to analyze the irregular functional of G-Brownian paths and to characterize their zero sets. The key ingredient of our approach is an estimate of solutions to G-heat equations. As a result, we prove that the G-Brownian motion path is nowhere differentiable.