In this Lecture, the geometric concepts introduced in the previous lectures are now used to revisit the notion of -rough path that was introduced before. We will see that using Carnot groups gives a perfect description of the space of -rough paths through the notion of geometric rough path.
Definition: Let . An element is called a geometric -rough path if there exists a sequence that converges to in the -variation distance. The space of geometric -rough paths will be denoted by .
To have it in mind, we recall the definition of a -rough path.
Definition: Let and . We say that is a -rough path if there exists a sequence such that in -variation and such that for every , there exists such that for ,
The space of -rough paths is denoted .
Our first goal is of course to relate the notion of geometric rough path to the notion of rough path.
Proposition: Let be a geometric -rough path, then the projection of onto is a -rough path.
Proof: Let be a geometric -rough path and let us consider a sequence that converges to in the -variation distance. Denote by the projection of onto and by the projection of . From a previous theorem . It is clear that converges to in -variation. So, we want to prove that for every , there exists such that for ,
Let us now keep in mind that
and consider the control
From the ball-box estimate, there is a constant such that for :
This is the estimate we were looking for
Conversely, any -rough path admits at least one lift as a geometric -rough path.
Proposition: Let be a -rough path. There exists a geometric -rough path such that the projection of onto is .
Proof: Consider a sequence such that in -variation and such that for every , there exists such that for ,
We claim that is a sequence that converges in -variation to some such that the projection of onto is . Let us consider the control
and argue as above to get, thanks to the ball-box theorem, an estimate like
In general, we stress that there may be several geometric rough paths with the same projection onto . The following proposition is useful to prove that a given path is a geometric rough path.
Proposition: If , then .
Proof: As in Euclidean case, it is not difficult to prove that if and only if
which is easy to check when
If , then as we just saw, the projection of onto is a -rough path and we can write
This is a convenient way to write geometric rough paths that we will often use in the sequel. For we can then define the lift of in as:
The following result is then easy to prove by using the previous results.
Proposition: Let and . There exist constants such that for every ,