Lecture 18. Paths with bounded p-variation in Carnot groups

In this Lecture, we go one step further to understand $p$-rough paths from paths in Carnot groups. The connection is made through the study of paths with bounded p-variation in Carnot groups.

Definition: A continuous path $x : [s,t] \to \mathbb{G}_N(\mathbb{R}^d)$ is said to have a bounded variation on $[s,t]$, if the 1-variation of $x$ on $[s,t]$, which is defined as
$\| x \|_{1-var; [s,t]} :=\sup_{ \Pi \in \mathcal{D}[s,t]} \sum_{k=0}^{n-1} d( x(t_{k+1}) , x(t_k)),$
is finite, where $d$ is the Carnot-Caratheodory distance on $\mathbb{G}_N(\mathbb{R}^d)$. The space of continuous bounded variation paths $x : [s,t] \to \mathbb{R}^d$, will be denoted by $C^{1-var} ([s,t], \mathbb{G}_N(\mathbb{R}^d))$.

The 1-variation distance between $x,y \in C^{1-var} ([s,t], \mathbb{G}_N(\mathbb{R}^d))$ is then defined as
$d_{1-var; [s,t]}=\sup_{ \Pi \in \mathcal{D}[s,t]} \sum_{k=0}^{n-1} d( x(t_k)^{-1}x(t_{k+1}), y(t_k)^{-1}y(t_{k+1})).$

As for the linear case the following proposition is easy to prove:

Proposition: Let $x \in C^{1-var} ([0,T], \mathbb{G}_N(\mathbb{R}^d))$. The function $(s,t)\to \| x \|_{1-var, [s,t]}$ is additive, i.e for $0 \le s \le t \le u \le T$,
$\| x \|_{1-var, [s,t]}+ \| x \|_{1-var, [t,u]}= \| x \|_{1-var, [s,u]},$
and controls $x$ in the sense that for $0 \le s \le t \le T$,
$d(x(s),x(t))\le \| x \|_{1-var, [s,t]}.$
The function $s \to \| x \|_{1-var, [0,s]}$ is moreover continuous and non decreasing.

We will denote $C_0^{1-var} ([0,T], \mathbb{G}_N(\mathbb{R}^d)$ the space of continuous bounded variation paths that start at 0. It turns out that $C_0^{1-var} ([0,T], \mathbb{G}_N(\mathbb{R}^d))$ is always isometric to $C_0^{1-var} ([0,T], \mathbb{R}^d)$. Remember that for $x \in C^{1-var} ([0,T], \mathbb{R}^d)$, the lift of $x$ in $\mathbb{G}_N(\mathbb{R}^d)$ is denoted by $S_N(x)$.

Definition: For every, $x \in C_0^{1-var} ([0,T], \mathbb{R}^d)$, we have
$\| S_N(x) \|_{1-var; [0,T]}=\| x \|_{1-var; [0,T]}.$
Moreover, for every $y \in C_0^{1-var} ([0,T], \mathbb{G}_N(\mathbb{R}^d)$, there exists one and only one $x \in C_0^{1-var} ([0,T], \mathbb{R}^d)$ such that $y=S_N(x).$

Proof: Let $x \in C_0^{1-var} ([0,T], \mathbb{R}^d)$. From the very definition of the Carnot-Caratheodory distance, for $0 \le s \le t \le T$, we have
$d(S_N(x)(s),S_N(x)(t)) \ge \| x \|_{1-var, [s,t]}.$
As a consequence we obtain,
$\| S_N(x) \|_{1-var; [0,T]} \ge \| x \|_{1-var; [0,T]}.$
On the other hand, $S_N(x)$ is the solution of the differential equation
$S_N(x)(t)=\sum_{i=1}^d \int_0^t X_i( S_N(x)(s)) dx^i(s), \quad 0 \le t \le T.$
This implies,
$d\left(S_N(x)(s), S_N(x)(t)\right) \le \int_s^t \| dx(u)\|= \| x \|_{1-var, [s,t]}.$
Finally, let $y \in C_0^{1-var} ([0,T], \mathbb{G}_N(\mathbb{R}^d)$. Let $x$ be the projection of $y$ onto $\mathbb{R}^d$. From the theorem of equivalence of homogeneous norms, we deduce that $x$ has a bounded variation in $\mathbb{R}^d$. We claim that $y=S_N(x)$. Consider the path $z=y S_N(x)^{-1}$. This is a bounded variation path whose projection on $\mathbb{R}^d$ is 0. We want to prove that it implies that $z=0$. Denote by $z_2$ the projection of $z$ onto $\mathbb{G}_2(\mathbb{R}^d)$. Again from the equivalence of homogeneous norms, we see that $z_2$ has a bounded variation in $\mathbb{G}_2(\mathbb{R}^d)$. Since the projection of $z_2$ on $\mathbb{R}^d$ is 0, we deduce that $z_2$ is in the center of $\mathbb{G}_2(\mathbb{R}^d)$, which implies that $z_2(s)^{-1}z_2(t)=z_2(t)-z_2(s)$. From the equivalence of homogeneous norms, we have then
$d(z_2(s),z_2(t)) \simeq \| z_2(t)-z_2(s) \|^{1/2}.$
Since $z_2$ has a bounded variation in $\mathbb{G}_2(\mathbb{R}^d)$, it has thus a $1/2$-variation for the Euclidean norm. This implies $z_2=0$. Using the same argument inductively shows that for $n \le N$, the projection of $z$ onto $\mathbb{G}_n(\mathbb{R}^d)$ will be 0. We conclude $z=0$ $\square$

As a conclusion, bounded variation paths in Carnot groups are the lifts of the bounded variation paths in $\mathbb{R}^d$. As we will see, the situation is very different for paths with bounded $p$-variation when $p \ge 2$.

Definition: Let $p \ge 1$. A continuous path $x : [s,t] \to \mathbb{G}_N(\mathbb{R}^d)$ is said to have a bounded $p$-variation on $[s,t]$, if the p-variation of $x$ on $[s,t]$, which is defined as
$\| x \|_{p-var; [s,t]} :=\left( \sup_{ \Pi \in \mathcal{D}[s,t]} \sum_{k=0}^{n-1} d( x(t_{k+1}) , x(t_k))^p\right)^{1/p},$
is finite. The space of continuous paths $x : [s,t] \to \mathbb{R}^d$ with a $p$-bounded variation will be denoted by $C^{p-var} ([s,t], \mathbb{G}_N(\mathbb{R}^d))$.

The $p$-variation distance between $x,y \in C^{p-var} ([s,t], \mathbb{G}_N(\mathbb{R}^d))$ is then defined as
$d_{p-var; [s,t]}=\left(\sup_{ \Pi \in \mathcal{D}[s,t]} \sum_{k=0}^{n-1} d( x(t_k)^{-1}x(t_{k+1}) , y(t_k)^{-1}y(t_{k+1}))^p\right)^{1/p}.$

As for $\mathbb{R}^d$ valued paths, we restrict our attention to $p \ge 1$ because any path with a $p$-bounded variation, $p < 1$ needs to be constant. We have then the following theorem that extends the previous result. The proof is somehow similar to the previous result, so we let the reader fill the details.

Theorem: Let $1 \le p < 2$. For every $y \in C_0^{p-var} ([0,T], \mathbb{G}_N(\mathbb{R}^d)$, there exists one and only one $x \in C_0^{p-var} ([0,T], \mathbb{R}^d)$ such that
$y=S_N(x).$
Moreover, we have
$\| x \|_{p-var; [0,T]} \le \| S_N(x) \|_{p-var; [0,T]}\le C\| x \|_{p-var; [0,T]}.$

For $p \ge 2$, the situation is different as we are going to explain in the next Lectures. This can already be understood by using the estimates on iterated integrals that were obtained in a previous Lecture. Indeed, we have the following very important proposition that already shows the connection between $p$-rough paths and paths with a bounded $p$-variation in Carnot groups:

Proposition: Let $p \ge 1$ and $N \ge [p]$. There exist constants $C_1,C_2 > 0$ such that for every $x \in C_0^{1-var} ([0,T], \mathbb{G}_N(\mathbb{R}^d)$,
$C_1 \left( \sum_{j=1}^{[p]} \left\| \int dx^{\otimes j}\right\|^{1/j}_{\frac{p}{j}-var, [s,t]} \right) \le \| S_{N} (x) \|_{p-var; [s,t]} \le C_2 \left( \sum_{j=1}^{[p]} \left\| \int dx^{\otimes j}\right\|^{1/j}_{\frac{p}{j}-var, [s,t]} \right).$

Proof: This is a consequence of the theorem about the equivalence of homogeneous norms on Carnot groups. Write the stratification of $\mathfrak{g}_N(\mathbb{R}^d)$ as:
$\mathfrak{g}_N(\mathbb{R}^d)=\mathcal{V}_1 \oplus \cdots \oplus \mathcal{V}_N$
and denote by $\pi_i$ the projection onto $\mathcal{V}_i$. Let us denote by $\| \cdot \|$ the norm on $\mathfrak{g}_N(\mathbb{R}^d)$ that comes from the norm on formal series. Then,
$\rho(g)=\sum_{i=1}^N \| \pi_i (g) \|^{1/i}$
is an homogeneous norm on $\mathbb{G}_N(\mathbb{R}^d)$. Thus, there exist constants $C_1,C_2 > 0$ such that for every $g \in \mathbb{G}_N(\mathbb{R}^d)$,
$C_1 \rho(g) \le d(0,g) \le C_2 \rho (g).$
In particular, we get
$C_1 \rho\left(S_N(x)(s)^{-1}S_N(x)(t)\right) \le d\left(S_N(x)(s), S_N(x)(t)\right) \le C_2 \rho \left(S_N(x)(s)^{-1}S_N(x)(t)\right).$
Let us now observe that
$\rho\left( (S_N(x)(s))^{-1} S_N(x)(t) \right)=\sum_{k=1}^N \left\| \int_{\Delta^k [s,t]} dx^{\otimes k} \right\|^{1/k}$
and that, from a previous lecture $k \ge [p]$,
$\left\| \int_{\Delta^k [s,t]} dx^{\otimes k} \right\| \le \frac{C^k}{\left( \frac{k}{p}\right)!} \left( \sum_{j=1}^{[p]} \left\| \int dx^{\otimes j}\right\|^{1/j}_{\frac{p}{j}-var, [s,t]} \right)^k, \quad 0 \le s \le t \le T.$
The conclusion easily follows $\square$

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