In this lecture we introduce the central notion of the signature of a path which is a convenient way to encode all the algebraic information on the path which is relevant to study differential equations driven by . The motivation for the definition of the signature comes from formal manipulations on Taylor series.
Let us consider a differential equation
where the ‘s are smooth vector fields on .
If is a function, by the change of variable formula,
Now, a new application of the change of variable formula to leads to
We can continue this procedure to get after steps
for some remainder term , where we used the notations:
- If is a word with length ,
If we let , assuming (which is by the way true for small enough if the ‘s are analytic), we are led to the formal expansion formula:
This shows, at least at the formal level, that all the information given by on is contained in the iterated integrals .
Let be the non commutative algebra over of the formal series with indeterminates, that is the set of series
Definition: Let . The signature of (or Chen’s series) is the formal series:
As we are going to see in the next few lectures, the signature is a fascinating algebraic object. At the source of the numerous properties of the signature lie the following so-called Chen’s relations
Lemma: Let . For any word and any ,
where we used the convention that if is a word with length 0, then .
Proof: It follows readily by induction on by noticing that
To avoid heavy notations, it will be convenient to denote
This notation actually reflects a natural algebra isomorphism between and . With this notation, observe that the signature writes then
and that the Chen’s relations become
The Chen’s relations imply the following flow property for the signature:
Proposition: Let . For any ,