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  "body": "\n## Definition\n\nLet $T \u003e 0$, let $d \\ge 1$, and let $\\alpha \\in (1/3, 1/2]$.\nFor a path $X \\colon [0, T] \\to \\R^d$ write $X_{s,t} = X_t - X_s$.\nAn *$\\alpha$-Hölder rough path* on $[0, T]$ with values in $\\R^d$ is a pair $(X, \\mathbb{X})$ of a path $X \\colon [0, T] \\to \\R^d$ and a map $\\mathbb{X} \\colon [0, T]^2 \\to \\R^d \\otimes \\R^d$ such that:\n\n1. (Chen's relation) $\\mathbb{X}_{s,t} - \\mathbb{X}_{s,u} - \\mathbb{X}_{u,t} = X_{s,u} \\otimes X_{u,t}$ for all $s, u, t \\in [0, T]$;\n2. $\\sup_{s \\ne t} \\abs{X_{s,t}} / \\abs{t - s}^{\\alpha} \u003c \\infty$;\n3. $\\sup_{s \\ne t} \\abs{\\mathbb{X}_{s,t}} / \\abs{t - s}^{2\\alpha} \u003c \\infty$.\n\nHere $\\otimes$ is the [[tensor product|0BF2DE2SYK]], and $\\R^d \\otimes \\R^d$ is identified with the $d \\times d$ matrices.\n\n## Remarks\n\nThe map $\\mathbb{X}_{s,t}$ stands for the iterated integral $\\int_s^t X_{s,r} \\otimes dX_r$.\nClause 2 says that $X$ is [[Hölder continuous|012PBA0FMD]] with exponent $\\alpha$.\nExample: for a continuously differentiable path $X$, set $\\mathbb{X}_{s,t} = \\int_s^t X_{s,r} \\otimes \\dot{X}_r \\, dr$.\nSplitting $\\int_s^t$ at $u$ and writing $X_{s,r} = X_{s,u} + X_{u,r}$ for $r \\ge u$ gives Chen's relation, and clauses 2 and 3 hold with exponents $1$ and $2$.\nExample: for a [[Brownian motion|22735FGJV6]] $B$, the [[Itô integrals|JA95NH206E]] $\\mathbb{B}_{s,t} = \\int_s^t B_{s,r} \\otimes dB_r$ give a rough path almost surely, for each $\\alpha \\in (1/3, 1/2)$.\nNon-example: a smooth path $X$ with $\\mathbb{X} = 0$ fails Chen's relation when $X_{s,u} \\otimes X_{u,t} \\ne 0$.\nThe second level is not determined by $X$: if $g \\colon [0, T] \\to \\R^d \\otimes \\R^d$ is $2\\alpha$-Hölder, then $\\mathbb{X}_{s,t} + g_t - g_s$ also satisfies the three clauses.\n",
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