We now turn to the proof of the continuity theorem. We start with several lemmas, which are not difficult but a little technical. The first one is geometrically very intuitive.

**Lemma:*** Let such that with and with . Then, there exists and a constant such that , and
and
*

**Proof:**

See the book by Friz-Victoir, page 161

The next ingredient is the following estimate.

**Lemma:** *Let . Assume that are -Lipschitz vector fields in . Let such that
*

*Let be the solutions of the equations
and
If
and
then, for some constant depending only on ,
*

**Proof:** Let us first observe that it is enough to prove the result when . Indeed, suppose that we can prove the result in that case. Define then the path to be the concatenation of and reparametrized so that . It is seen that the solution of the equation

satisfies

We thus assume that . In that case, from the assumption, we have

Taylor’s expansion gives then, with ,

and similarly

The result is then easily obtained by using classical estimates for Riemann-Stieltjes integrals (details can be found page 230 in the book by Friz-Victoir)

Finally, the last lemma is an easy consequence of Gronwall’s lemma

**Lemma:*** Let . Assume that are -Lipschitz vector fields in . Let . Let be the solutions of the equations
and
If
and
then, for some constant depending only on ,
*

**Proof:** See the book by Friz-Victoir page 232