We now turn to the proof of Lyons’ continuity theorem.

**Theorem:** *Let . Assume that are -Lipschitz vector fields in . Let such that
with .
Let be the solutions of the equations
There exists a constant depending only on and such that for ,
where is the control
*

**Proof:** We may assume , and for conciseness of notations, we set . Let

We have,

and, in the same way,

Therefore, there exist and a constant such that

and

and

We define then as the concatenation of and . As in the proof of Davie’s lemma, we denote by the solution of the equation

and consider the functionals

and

From the proof of Davie’s estimate, it is seen that

and thus

On the other hand, by estimating

as in the proof of Davie’s lemma, that is by inserting which is the solution of the equation driven by the concatenation of and , and then by using the two lemmas of the previous lecture, we obtain the estimate

It remains to bound . For this let us observe that

can then be estimated by using classical estimates on differential equations driven by bounded variation paths. This gives,

By denoting , we can summarize the two above estimates as follows:

and

From a lemma already used in the proof of Davie’s estimate, the first estimate implies

Using now the second estimate we obtain that for any interval included in ,

Using the fact that and picking a subdivision such that

we see that it implies

Coming back to the estimate

concludes the proof