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A geometric $p$-rough path is a limit, in the rough path metric, of path signatures of sufficiently regular paths. It is a weak geometric p-rough path. At the same exponent these definitions need not coincide; weak geometric paths can be approximated after slightly weakening the variation exponent. The Brownian enhancement is geometric for every $p>2$.

Let $\mathbf{X}$ be a weak geometric p-rough path, $N=\lfloor p\rfloor$, with increments $X_{s,t}^I$ indexed by words $I=(i_1,\dots ,i_k)$ of length at most $N$. Choose a control function for rough paths $\omega$ with $|X_{s,t}^I|\le \omega (s,t{)}^{|I|/p}$. We use the standard sufficient vector-field hypothesis $V_i\in {\mathrm{Lip}}^{\gamma }$, $\gamma >p$, with globally bounded regularity norms; smooth bounded fields are a special case. Regard $V_i$ as the differential operator $V_i\cdot \nabla$, and put $F_I=V_{i_1}\cdots V_{i_k}\mathrm{id}$.

A pathlevel solution of a rough differential equation is a continuous state path $y:[0,T]\to {\mathbb{R}}^e$, with prescribed $y_0$, satisfying a local expansionfor some $\theta >1$ on sufficiently small control intervals. One can take $\theta =\min (\gamma ,N+1)/p>1$ under the stated regularity. This definition specifies the state path, without requiring its full rough-path enhancement as additional data. For $2<p<3$, the displayed expansion readsRepeated indices are summed.

We use the following discrete form of the Davie lemma. For a finite ordered grid and a continuous superadditive control, suppose nonnegative errors $a_{s,t}$ vanish on adjacent grid points and, for every grid triple $s<u<t$, satisfyThere are $\delta >0$ and $C$, depending on $L,A,p,\theta$ but not on the grid, such that $a_{s,t}\le C\omega (s,t{)}^{\theta }$ whenever $\omega (s,t)\le \delta$. This is the discrete higher-order error estimate: splitting close to half the control size makes the two $\theta$-power errors contract; a crossing single grid interval contributes zero adjacent error.

Define the local Euler mapFor small $\omega (s,t)$ it has Lipschitz constant at most $1+L\omega (s,t{)}^{1/p}$. Taylor expansion, the Chen identity and the shuffle identity for iterated integrals give the composition estimateHere is the cancellation underlying the estimate. Assign degree $k$ to a level-$k$ increment. Expanding each $F_I(E_{s,u}(z))$ and grouping terms through degree $N$ gives the differential-operator coefficients $F_{IJ}(z)$ multiplying $X_{s,u}^I{X}_{u,t}^J$. Products of coefficients from the first increment are combined by the shuffle identity, which is the iterated Leibniz rule. Chen's identity says that the sum of these products is $X_{s,t}^{IJ}$. Thus every term of degree at most $N$ agrees with the corresponding term in $E_{s,t}$. The remaining Taylor terms have control order at least $\theta$; bounded ${\mathrm{Lip}}^{\gamma }$ norms make the estimate uniform in $z$. For smooth fields the remainder has order $(N+1)/p$.

On a partition $\pi$, construct the Euler sequence by $y_{t_{j+1}}^{\pi }=E_{t_j,t_{j+1}}(y_{t_j}^{\pi })$. For grid endpoints letIt vanishes on adjacent points. Insert $y_u^{\pi }=E_{s,u}(y_s^{\pi })+I_{s,u}^{\pi }$ into the next local map. The Lipschitz and composition estimates giveThe Davie lemma therefore yields a bound $|I_{s,t}^{\pi }|\le C\omega (s,t{)}^{\theta }$ independent of $\pi$ on small control intervals. Together with the Euler expansion this gives $|y_t^{\pi }-y_s^{\pi }|\le C^{\prime }\omega (s,t{)}^{1/p}$. A finite subdivision into small-control intervals gives a uniform global bound as well.

Interpolate each Euler sequence linearly in time, and take partitions with mesh tending to zero. Continuity of the control makes these paths uniformly equicontinuous: the grid estimates apply to the interval enlarged to its neighboring grid endpoints, and the enlargement tends to zero uniformly with the mesh. The paths are uniformly bounded. The Arzelà-Ascoli theorem supplies a uniformly convergent subsequence with continuous limit $y$.

For arbitrary $s<t$ in a small control interval, choose neighboring grid endpoints converging to $s,t$. Continuity of $\mathbf{X}$, the coefficient functions and the control lets us pass to the limit in the grid error estimate. We obtain the defining expansion with $|R_{s,t}|\le C\omega (s,t{)}^{\theta }$, and $y(0)=y_0$. This constructs a pathlevel RDE solution. No continuity theorem for the RDE solution map was used to obtain existence.

In the truncated tensor algebra, let ${\mathfrak{g}}^N(V)$ be the linear span of all iterated Lie brackets of elements of $V$ of length at most $N$, with bracket $[a,b]=a\otimes b-b\otimes a$. Brackets of length greater than $N$ vanish. The free step-N nilpotent Lie group isThe exponential and logarithm are the finite seriesIts multiplication is the tensor multiplication above, its identity is $e=(1,0,\dots ,0)$, and its inverse isThe truncated Baker--Campbell--Hausdorff formula shows that this multiplication stays in the exponential of the Lie algebra. This is the connected simply connected Lie group of the free step-$N$ nilpotent Lie algebra, rather than a discrete free nilpotent group.

Give the first layer $V$ its usual Euclidean norm. An absolutely continuous curve $\gamma$ in $G^N(V)$ is horizontal iffor an absolutely continuous control $x$ in $V$. In the truncated tensor algebra this equation is ${\dot{\gamma }}_t={\gamma }_t\otimes {\dot{x}}_t$, truncated at degree $N$. Its horizontal length is $\int |{\dot{x}}_t|dt$. The Carnot-Carathéodory distance isThe first-layer Lie brackets span the full Lie algebra, so horizontal curves reach every endpoint. Thus this infimum is finite; it is a left-invariant distance inducing the usual manifold topology. Dilations ${\delta }_{\lambda }(a{)}^{(k)}={\lambda }^k{a}^{(k)}$ give $d_{CC}({\delta }_{\lambda }g,{\delta }_{\lambda }h)=\lambda d_{CC}(g,h)$ for $\lambda >0$. Equivalently $d_{CC}(e,g)$ is the least length of a bounded variation control whose step-$N$ path signature is $g$.

For $p\ge 1$ set $N=\lfloor p\rfloor$. A weak geometric p-rough path is a continuous path $\mathbf{X}:[0,T]\to G^N(V)$, usually based at $e$, withIts increments ${\mathbf{X}}_{s,t}={\mathbf{X}}_s^{-1}\otimes {\mathbf{X}}_t$ obey ${\mathbf{X}}_{s,u}={\mathbf{X}}_{s,t}\otimes {\mathbf{X}}_{t,u}$. Equivalently their degree-$k$ tensor components have finite $p/k$-variation and satisfy the path signature algebraic constraints. This is a group-valued definition; being a limit of smooth path signatures in the same $p$-variation topology is the stronger definition of a geometric p-rough path.

For $p\ge 1$, 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 $p\alpha \ge 1$. Applying this construction to the Carnot-Carathéodory distance gives the defining regularity of a weak geometric p-rough path.