Pure-area rough path
= Pure-area rough path
{title2=$\mathbf y_t=\exp(t[e_1,e_2])$}
The first level of this group path is zero but its second level grows linearly. Its <Carnot-Carathéodory distance> increments are a constant times $|t-s|^{1/2}$. It therefore has finite $p$-variation precisely when $p\geq2$, and is a step-two weak geometric rough path at $p=2$, even though its first-level trace is constant. For higher truncations the same Lie exponential defines a compatible lift.