## Lecture 19. Geometric rough paths

In this Lecture, the geometric concepts introduced in the previous lectures are now used to revisit the notion of $p$-rough path that was introduced before. We will see that using Carnot groups gives a perfect description of the space of $p$-rough paths through the notion of geometric rough path.

Definition: Let $p \ge 1$. An element $x \in C_0^{p-var} ([0,T], \mathbb{G}_{[p]} (\mathbb{R}^d))$ is called a geometric $p$-rough path if there exists a sequence $x_n \in C_0^{1-var} ([0,T], \mathbb{G}_{[p]} (\mathbb{R}^d))$ that converges to $x$ in the $p$-variation distance. The space of geometric $p$-rough paths will be denoted by $\mathbf{\Omega G}^p([0,T],\mathbb{R}^d)$.

To have it in mind, we recall the definition of a $p$-rough path.

Definition: Let $p \ge 1$ and $x \in C_0^{p-var}([0,T],\mathbb{R}^d)$. We say that $x$ is a $p$-rough path if there exists a sequence $x_n \in C_0^{1-var}([0,T],\mathbb{R}^d)$ such that $x_n\to x$ in $p$-variation and such that for every $\varepsilon > 0$, there exists $N \ge 0$ such that for $m,n \ge N$,
$\sum_{j=1}^{[p]} \left\| \int dx_n^{\otimes j}- \int dx_m^{\otimes j} \right\|^{1/j}_{\frac{p}{j}-var, [0,T]} \le \varepsilon.$
The space of $p$-rough paths is denoted $\mathbf{\Omega}^p([0,T],\mathbb{R}^d)$.

Our first goal is of course to relate the notion of geometric rough path to the notion of rough path.

Proposition: Let $y \in C_0^{p-var} ([0,T], \mathbb{G}_{[p]} (\mathbb{R}^d))$ be a geometric $p$-rough path, then the projection of $y$ onto $\mathbb{R}^d$ is a $p$-rough path.

Proof: Let $y \in C_0^{p-var} ([0,T], \mathbb{G}_{[p]} (\mathbb{R}^d))$ be a geometric $p$-rough path and let us consider a sequence $y_n \in C_0^{1-var} ([0,T], \mathbb{G}_{[p]} (\mathbb{R}^d))$ that converges to $y$ in the $p$-variation distance. Denote by $x$ the projection of $y$ onto $\mathbb{R}^d$ and by $x_n$ the projection of $y_n$. From a previous theorem $y_n=S_{[p]}(x_n)$. It is clear that $x_n$ converges to $x$ in $p$-variation. So, we want to prove that for every $\varepsilon > 0$, there exists $N \ge 0$ such that for $m,n \ge N$,
$\sum_{j=1}^{[p]} \left\| \int dx_n^{\otimes j}- \int dx_m^{\otimes j} \right\|^{1/j}_{\frac{p}{j}-var, [0,T]} \le \varepsilon.$
Let us now keep in mind that
$d_{p-var; [s,t]}(y_n,y_m)=\left(\sup_{ \Pi \in \mathcal{D}[s,t]} \sum_{k=0}^{n-1} d( y_n(t_k)^{-1}y_n(t_{k+1}) , y_m(t_k)^{-1}y_m(t_{k+1}))^{p}\right)^{1/p}.$
and consider the control
$\omega(s,t)=\left( \frac{ d_{p-var; [s,t]}(y_n,y_m)}{d_{p-var; [0,T]}(y_n,y_m) } \right)^p+\left( \frac{ d_{p-var; [s,t]}(0,y_m)}{d_{p-var; [0,T]}(0,y_m) } \right)^p.$
We have
$\left\| \int dx_n^{\otimes k}- \int dx_m^{\otimes k} \right\|_{\frac{p}{k}-var, [0,T]}$
$= \left(\sup_{ \Pi \in \mathcal{D}[0,T]} \sum_{j=0}^{n-1} \left\| \int_{\Delta^k [t_j,t_{j+1}]} dx_n^{\otimes k}- \int _{\Delta^k [t_j,t_{j+1}]} dx_m^{\otimes k} \right\|^{p/k}\right)^{k/p}$
$\le \left( \sup_{0 \le s \le t \le T} \frac{ \left\| \int_{\Delta^k [s,t]} dx_n^{\otimes k}- \int _{\Delta^k [s,t]} dx_m^{\otimes k} \right\|}{\omega(s,t)^{k/p} } \right) \omega(0,T)^{k/p}$
From the ball-box estimate, there is a constant $C$ such that for $x,y \in \mathbb{G}_{[p]}(\mathbb{R}^d)$:
$\| x-y \| \le C \max \{ d(x,y) \max \{ 1, d(0,x)^{N-1} \}, d(x,y)^N \}.$
We deduce
$\frac{ \left\| \int_{\Delta^k [s,t]} dx_n^{\otimes k}- \int _{\Delta^k [s,t]} dx_m^{\otimes k} \right\|}{\omega(s,t)^{k/p} }$
$\le C \max \left\{ d_{p-var; [0,T]}(y_n,y_m) \max \{ 1, d_{p-var; [0,T]}(0,y_m)^{N-1} \}, d_{p-var; [0,T]}(y_n,y_m)^N \right\}$
and thus
$\left\| \int dx_n^{\otimes k}- \int dx_m^{\otimes k} \right\|_{\frac{p}{k}-var, [0,T]} \le C' d_{p-var; [0,T]}(y_n,y_m)$
This is the estimate we were looking for $\square$

Conversely, any $p$-rough path admits at least one lift as a geometric $p$-rough path.

Proposition: Let $x \in C_0^{p-var} ([0,T], \mathbb{R}^d)$ be a $p$-rough path. There exists a geometric $p$-rough path $y \in \mathbf{\Omega G}^p([0,T],\mathbb{R}^d)$ such that the projection of $y$ onto $\mathbb{R}^d$ is $x$.

Proof: Consider a sequence $x_n \in C_0^{1-var}([0,T],\mathbb{R}^d)$ such that $x_n\to x$ in $p$-variation and such that for every $\varepsilon > 0$, there exists $N \ge 0$ such that for $m,n \ge N$,
$\sum_{j=1}^{[p]} \left\| \int dx_n^{\otimes j}- \int dx_m^{\otimes j} \right\|^{1/j}_{\frac{p}{j}-var, [0,T]} \le \varepsilon.$
We claim that $y_n=S_{[p]} (x_n)$ is a sequence that converges in $p$-variation to some $y \in \mathbf{\Omega G}^p([0,T],\mathbb{R}^d)$ such that the projection of $y$ onto $\mathbb{R}^d$ is $x$. Let us consider the control
$\omega(s,t)= \left(\frac{ \sum_{j=1}^{[p]} \left\| \int dx_n^{\otimes j}- \int dx_m^{\otimes j} \right\|^{1/j}_{\frac{p}{j}-var, [s,t]}} { \sum_{j=1}^{[p]} \left\| \int dx_n^{\otimes j}- \int dx_m^{\otimes j} \right\|^{1/j}_{\frac{p}{j}-var, [0,T]}} \right)^p+\left( \frac{ d_{p-var; [s,t]}(0,y_m)}{d_{p-var; [0,T]}(0,y_m) } \right)^p.$
We have
$d_{p-var; [0,T]}(y_n,y_m) \le \left( \sup_{0 \le s \le t \le T} \frac{ d \left( y_n(s)^{-1}y_n(t), y_m(s)^{-1}y_m(t) \right) }{\omega(s,t)^{1/p} } \right) \omega(0,T)^{1/p}$
and argue as above to get, thanks to the ball-box theorem, an estimate like
$d_{p-var; [0,T]}(y_n,y_m) \le C \left( \sum_{j=1}^{[p]} \left\| \int dx_n^{\otimes j}- \int dx_m^{\otimes j} \right\|^{1/j}_{\frac{p}{j}-var, [0,T]}\right)^{1/N}$
$\square$

In general, we stress that there may be several geometric rough paths with the same projection onto $\mathbb{R}^d$. The following proposition is useful to prove that a given path is a geometric rough path.

Proposition: If $q < p$, then $C_0^{q-var} ([0,T], \mathbb{G}_{[p]} (\mathbb{R}^d)) \subset \mathbf{\Omega G}^p([0,T],\mathbb{R}^d)$.

Proof: As in Euclidean case, it is not difficult to prove that $x \in \mathbf{\Omega G}^p([0,T],\mathbb{R}^d)$ if and only if
$\lim_{\delta \to 0} \sup_{ \Pi \in \Delta[s,t], | \Pi | \le \delta } \sum_{k=0}^{n-1} d( x(t_k), x(t_{k+1}) )^p=0,$
which is easy to check when $x \in C_0^{q-var} ([0,T], \mathbb{G}_{[p]} (\mathbb{R}^d))$ $\square$

If $y \in \mathbf{\Omega G}^p([0,T],\mathbb{R}^d)$, then as we just saw, the projection $x$ of $y$ onto $\mathbb{R}^d$ is a $p$-rough path and we can write
$y(t)=1+\sum_{k=1}^{[p]} \int_{\Delta^k[0,t]} dx^{\otimes k}.$
This is a convenient way to write geometric rough paths that we will often use in the sequel. For $N \ge [p]$ we can then define the lift of $y$ in $\mathbf{\Omega G}^N([0,T],\mathbb{R}^d)$ as:
$S_N(y)(t) =1+\sum_{k=1}^{N} \int_{\Delta^k[0,t]} dx^{\otimes k}.$
The following result is then easy to prove by using the previous results.

Proposition: Let $p \ge 1$ and $N \ge [p]$. There exist constants $C_1,C_2 > 0$ such that for every $y \in \mathbf{\Omega G}^p([0,T],\mathbb{R}^d)$,
$\| y \|_{p-var,[0,T]} \le \| S_{N} (y) \|_{p-var; [0,T]} \le C_2 \| y \|_{p-var,[0,T]}.$

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