In this Lecture, we prove one of the fundamental estimates of rough paths theory. This estimate is due to Davie. It provides a basic estimate for the solution of the differential equation
in terms of the -variation of the lift of in the free Carnot group of step .
We first introduce the somehow minimal regularity requirement on the vector fields ‘s to study rough differential equations.
Definition. A vector field on is called -Lipschitz if it is times continuously differentiable and there exists a constant such that the supremum norm of its th derivatives and the Holder norm of its th derivative are bounded by . The smallest that satisfies the above condition is the -Lipschitz norm of and will be denoted .
The fundamental estimate by Davie is the following;
Definition: Let . Assume that are -Lipschitz vector fields in . Let . Let be the solution of the equation
There exists a constant depending only on and such that for every ,
where is the lift of in .
We start with two preliminary lemmas, the first one being interesting in itself.
Lemma: Let . Assume that are -Lipschitz vector fields in . Let . Let be the solution of the equation
There exists a constant depending only on such that,
where is the identity map.
Proof: For notational simplicity, we denote . An iterative use of the change of variable formula leads to
Since are -Lipschitz, we deduce that
we deduce that
The result follows then easily by plugging this estimate into the integral
The second lemma is an analogue of a result already used in previous lectures (Young-Loeve estimate, estimates on iterated integrals).
Lemma: Let . Let us assume that:
- There exists a control such that
- There exists a control and such that for ,
Then, for all ,
For , consider then the control
If is such that , we can find a such that , . Indeed, the continuity of forces the existence of a such that . We obtain therefore
which implies by maximization,
We have and an iteration easily gives
and the result follows by letting