= p-variation of a path
{title2=$\|x\|_{p\text{-var};[0,T]}=\left(\sup_D\sum_{[u,v]\in D}d(x_u,x_v)^p\right)^{1/p}$}
= p-variation
{synonym}
For $p\ge1$, 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 $p\alpha\ge1$. Applying this construction to the <Carnot-Carathéodory distance> gives the defining regularity of a <weak geometric p-rough path>.
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