Weak geometric p-rough path ($\mathbf{X}:[0,T]\to G^{\lfloor p\rfloor }({\mathbb{R}}^d)$)

A weak geometric $p$-rough path is a continuous path in the free step-N nilpotent Lie group, with $N=\lfloor p\rfloor$, whose increments have finite $p$-variation for the Carnot-Carathéodory distance. Its multiplicative increments are ${\mathbf{X}}_{s,t}={\mathbf{X}}_s^{-1}\otimes {\mathbf{X}}_t$. They obey the Chen identity and the algebraic symmetries of path signatures.

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 $p\ge 2$, 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.