In this Lecture, we go one step further to understand -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 is said to have a bounded variation on , if the 1-variation of on , which is defined as
is finite, where is the Carnot-Caratheodory distance on . The space of continuous bounded variation paths , will be denoted by .
The 1-variation distance between is then defined as
As for the linear case the following proposition is easy to prove:
Proposition: Let . The function is additive, i.e for ,
and controls in the sense that for ,
The function is moreover continuous and non decreasing.
We will denote the space of continuous bounded variation paths that start at 0. It turns out that is always isometric to . Remember that for , the lift of in is denoted by .
Definition: For every, , we have
Moreover, for every , there exists one and only one such that
Proof: Let . From the very definition of the Carnot-Caratheodory distance, for , we have
As a consequence we obtain,
On the other hand, is the solution of the differential equation
Finally, let . Let be the projection of onto . From the theorem of equivalence of homogeneous norms, we deduce that has a bounded variation in . We claim that . Consider the path . This is a bounded variation path whose projection on is 0. We want to prove that it implies that . Denote by the projection of onto . Again from the equivalence of homogeneous norms, we see that has a bounded variation in . Since the projection of on is 0, we deduce that is in the center of , which implies that . From the equivalence of homogeneous norms, we have then
Since has a bounded variation in , it has thus a -variation for the Euclidean norm. This implies . Using the same argument inductively shows that for , the projection of onto will be 0. We conclude
As a conclusion, bounded variation paths in Carnot groups are the lifts of the bounded variation paths in . As we will see, the situation is very different for paths with bounded -variation when .
Definition: Let . A continuous path is said to have a bounded -variation on , if the p-variation of on , which is defined as
is finite. The space of continuous paths with a -bounded variation will be denoted by .
The -variation distance between is then defined as
As for valued paths, we restrict our attention to because any path with a -bounded variation, 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 . For every , there exists one and only one such that
Moreover, we have
For , 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 -rough paths and paths with a bounded -variation in Carnot groups:
Proposition: Let and . There exist constants such that for every ,
Proof: This is a consequence of the theorem about the equivalence of homogeneous norms on Carnot groups. Write the stratification of as:
and denote by the projection onto . Let us denote by the norm on that comes from the norm on formal series. Then,
is an homogeneous norm on . Thus, there exist constants such that for every ,
In particular, we get
Let us now observe that
and that, from a previous lecture ,
The conclusion easily follows