= Rough path metric
{title2=$d_p(\mathbf X,\mathbf Y)=\max_{1\le k\le\lfloor p\rfloor}\left(\sup_D\sum_{[u,v]\in D}|X^{(k)}_{u,v}-Y^{(k)}_{u,v}|^{p/k}\right)^{k/p}$}
This inhomogeneous variation metric compares the tensor increments level by level. A common alternative is the Hölder metric $\rho_\alpha=\max_k\sup_{s<t}|X^{(k)}_{s,t}-Y^{(k)}_{s,t}|/|t-s|^{k\alpha}$. On a finite interval convergence in this metric implies $p$-variation convergence whenever $p\alpha\ge1$. Both topologies control the iterated integrals needed by the <universal limit theorem>.
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