p-variation of a path ($\Vert x{\Vert }_{p\text{-var};[0,T]}={\left({\sup }_D{\sum }_{[u,v]\in D}d(x_u,x_v{)}^p\right)}^{1/p}$)

For $p\ge 1$, the displayed quantity measures the accumulated oscillation of a path in a metric space. For $p=1$ it is total variation. A Hölder continuous path of exponent $\alpha$ has finite $p$-variation on a compact interval if $p\alpha \ge 1$. Applying this construction to the Carnot-Carathéodory distance gives the defining regularity of a weak geometric p-rough path.