Lecture 17. The Carnot Carathéodory distance

In this Lecture we introduce a canonical distance on a Carnot group. This distance is naturally associated to the sub-Riemannian structure which is carried by a Carnot group. It plays a fundamental role in the rough paths topology. Let \mathbb{G}_N(\mathbb{R}^d) be the free Carnot group over \mathbb{R}^d. Remember that if x \in C^{1-var}([0,T],\mathbb{R}^d), then we denote by S_N(x) the lift of x in \mathbb{G}_N(\mathbb{R}^d). The first important concept is the notion of horizontal curve.

Definition: A curve y: [0,1] \to \mathbb{G}_N(\mathbb{R}^d) is said to be horizontal if there exists x \in C^{1-var}([0,T],\mathbb{R}^d) such that y=S_N(x).

It is remarkable that any two points of \mathbb{G}_N(\mathbb{R}^d) can be connected by a horizontal curve.

Proposition: Given two points g_1 and g_2 \in \mathbb{G}_N(\mathbb{R}^d), there is at least one x \in C^{1-var}([0,T],\mathbb{R}^d) such that g_1S_N(x)(1)=g_2.

Proof: Let us denote by G the subgroup of diffeomorphisms \mathbb{G}_N(\mathbb{R}^d) \rightarrow \mathbb{G}_N(\mathbb{R}^d) generated by the one-parameter subgroups corresponding to X_1, \cdots , X_d. The Lie algebra of G can be identified with the Lie algebra generated by X_1, \cdots , X_d, i.e. \mathfrak{g}_N(\mathbb{R}^d). We deduce that G can be identified with \mathbb{G}_N(\mathbb{R}^d) itself, so that it acts transitively on \mathbb{G}. It means that for every x \in \mathbb{G}_N(\mathbb{R}^d), the map G \rightarrow \mathbb{G}_N(\mathbb{R}^d), g \rightarrow g(x) is surjective. Thus, every two points in \mathbb{G} can be joined by a piecewise smooth horizontal curve where each piece is a segment of an integral curve of one of the vector fields \mathbf{e}_i \square

In the above proof, the horizontal curve constructed to join the two points is not smooth. Nevertheless, it can be shown that it is always possible to connect two points with a smooth horizontal curve.

Let us also remark that this theorem is a actually a very special case of the so-called Chow-Rashevski theorem which is one of the cornerstones of sub-Riemannian geometry. We now are ready for the definition of the Carnot-Carathéodory distance.

Definition For g_1,g_2 \in \mathbb{G}_N(\mathbb{R}^d), we define
d(g_1,g_2)=\inf_{\mathcal{S}(g_1,g_2)} \| x \|_{1-var,[0,1]},
where
\mathcal{S}(g_1,g_2)=\{ x \in C^{1-var}([0,1],\mathbb{R}^d), g_1S_N(x)(1)=g_2 \}.
d(g_1,g_2) is called the Carnot-Carathéodory distance between g_1 and g_2.

The first thing to prove is that d is indeed a distance.
Lemma: The Carnot-Carathéodory distance is indeed a distance.

Proof: The symmetry and the triangle inequality are easy to check and we let the reader find the arguments. The last thing to prove is that d(g_1,g_2)=0 implies g_1=g_2. From the definition of d it clear that d_R \le d where d_R is the Riemmanian measure on \mathbb{G}_N(\mathbb{R}^d). It follows that d(g_1,g_2)=0 implies g_1=g_2 \square

We then observe the following properties of d:

Proposition:

  • For g_1,g_2 \in \mathbb{G}_N(\mathbb{R}^d), d(g_1,g_2)=d(g_2,g_1)=d(0,g_1^{-1} g_2).
  • Let (\Delta_t)_{t \ge 0} be the one parameter family of dilations on \mathbb{G}_N(\mathbb{R}^d). For g_1,g_2 \in \mathbb{G}_N(\mathbb{R}^d), and t \ge 0, d(\Delta_t g_1,\Delta_t g_2)=t d(g_1,g_2).

Proof: The first part of the proposition stems from the fact that for every x \in C^{1-var}([0,T],\mathbb{R}^d), S_N(x)^{-1}= S_N(-x), so that g_1S_N(x)(1)=g_2 is equivalent to g_2S_N(-x)(1)=g_1 which also equivalent to S_N(x)(1)=g_1^{-1} g_2. For the second part, we observe that for t \ge 0, \Delta_t S_N(x)=S_N(tx) \square

The Carnot-Carathéodory distance is pretty difficult to explicitly compute in general. It is often much more convenient to estimate using a so-called homogeneous norm.

Definition: A homogeneous norm on \mathbb{G}_N(\mathbb{R}^d) is a continuous function \parallel \cdot \parallel : \mathbb{G}_N(\mathbb{R}^d) \rightarrow  [0,+\infty) , such that:

  • \parallel \Delta_t x \parallel=t \parallel x \parallel, t \ge 0 , x \in \mathbb{G}_N(\mathbb{R}^d);
  • \parallel  x^{-1} \parallel= \parallel x \parallel, x \in \mathbb{G}_N(\mathbb{R}^d);
  • \parallel x \parallel=0 if and only if x=0.

It turns out that the Carnot-Carathéodory distance is equivalent to any homogeneous norm in the following sense:

Theorem: Let \parallel \cdot \parallel be a homogeneous norm on \mathbb{G}_N(\mathbb{R}^d). There exist two positive constants C_1 and C_2 such that for every x,y \in \mathbb{G}_N(\mathbb{R}^d),
A \| x^{-1}y \| \le d(x,y) \le B \| x^{-1}y \|.

By using the left invariance of d, it is of course enough to prove that for every x \in \mathbb{G}_N(\mathbb{R}^d),
A \| x \| \le d(0,x) \le B \| x \|.
We first prove that the function x\to d(0,x) is bounded on compact sets (of the Riemannian topology of the Lie group \mathbb{G}_N(\mathbb{R}^d)). As we have seen before, every x \in \mathbb{G}_N(\mathbb{R}^d) can be written as a product:
x=\prod_{i=1}^N e^{t_i X_{k_i}}.
From the very definition of the distance, we have then
d(0,x)\le d\left(0,\prod_{i=1}^N e^{t_i X_{k_i}}\right)\le \sum_{i=1}^N |t_i|.
It is not difficult to see that \sum_{i=1}^N |t_i| can uniformly be bounded on compact sets, therefore d(0,x) is bounded on compact sets. Consider now the compact set
\mathbf{K}= \{ x \in \mathbb{G}_N(\mathbb{R}^d), \| x \|=1 \}.
Since d(0,x) is bounded on K, we deduce that there exists a constant B such that for every x \in \mathbf{K},
d(0,x) \le B.
Since d_R \le d, where d_R is the Riemannian distance, we deduce that there exists a constant A such that for every x \in \mathbf{K},
d(0,x) \ge A.
Now, for every x \in \mathbb{G}_N(\mathbb{R}^d), x \neq 0, we deduce that
A \le d\left( 0,  \Delta_{1/\| x \|} x  \right) \le B
This yields the expected result \square

Let us give an example of a homogeneous norm which is particularly adapted to rough paths theory. 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, it is easily checked that
\rho(g)=\sum_{i=1}^N \| \pi_i (g) \|^{1/i}
is an homogeneous norm on \mathbb{G}_N(\mathbb{R}^d). This homogeneous norm is particulary adapted to the study of paths because if x \in C^{1-var}([0,T], \mathbb{R}^d), then one has:
\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}

We finally quote the following result, not difficult to prove which is often referred to as the ball-box estimate.

Proposition: There exists a constant C such that for every x,y \in \mathbb{G}_N(\mathbb{R}^d),
d(x,y) \le C \max \{ \| x-y \|, \| x -y \|^{1/N} \max \{ 1, d(0,x)^{1-1/N} \} \}.
and
\| x-y \| \le C \max \{ d(x,y)  \max \{ 1, d(0,x)^{N-1} \}, d(x,y)^N \}.
In particular, for every compact set K \subset \mathbb{G}_N(\mathbb{R}^d), there is a constant C_K such that for every x,y \in K,
\frac{1}{C_K} \| x-y\|  \le d(x,y) \le C_K \| x -y \|^{1/N}.

Proof: See the book by Friz-Victoir, page 152 \square

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