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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