In the previous lecture, we proved the Chen’s expansion formula which establishes the fact that the signature of a path is the exponential of a Lie series. This expansion is of course formal but analytically makes sense in a number of situations that we now describe. The first case of study are linear equations.

Let us consider matrices and let be the solution of the differential equation

where . The solution admits a representation as an absolutely convergent Volterra series

The formal analogy between this expansion and the signature leads to the following result:

**Proposition:** *There exists such that for ,
where
is the iterated Lie bracket and
*

**Proof:** We only give the sketch of the proof. Details can be found in this paper by Strichartz. First, we observe that a combinatorial argument shows that

On the other hand, we have the estimate

As a consequence, we obtain

For the matrix norm we have the estimate so we conclude that for some constant ,

We deduce that if is such that , then the series

is absolutely convergent on the interval . At this point, we can observe that the Chen's expansion formula is a purely algebraic statement, thus expanding the exponential

and rearranging the terms leads to

which is equal to

Another framework, close to this linear case, in which the Chen's expansion makes sense are Lie groups. Let be a Lie group acting on . Let us denote by the Lie algebra of . Elements of can be seen as vector fields on . Indeed, for , we can define

where is the exponential mapping on the Lie group . With this identification, it is easily checked that the Lie bracket in the Lie algebra coincides with the Lie bracket of vector fields and that the exponential map in the group corresponds to the flow generated by the vector field . As above we get then the following result:

**Proposition:** *Let and . Let us consider the differential equation
There exists such that for ,
*

A special case will be of interest for us: The case where the Lie group is nilpotent. Let us recall that a Lie group is said to be nilpotent of order if every bracket of length greater or equal to is 0. In that case, the sum in the exponential is finite and the representation is then of course valid on the whole time interval .