Left-Invariance for Smooth Vector Fields and Applications

Let X = {X_0,..., X_m} be a family of smooth vector fields on an open set A ⊆ R^N. Motivated by applications to the PDE theory of Hörmander operators, for a suitable class of open sets A, we find necessary and sufficient conditions on X for the existence of a Lie group (A, ∗) such that the operator L = X_1^2 + ... + X_m^2 + X_0 is left-invariant with respect to the operation ∗. Our approach is constructive, as the group law is constructed by means of the solution of a suitable ODE naturally associated to vector fields in X. We provide an application to a partial differential operator appearing in the Finance.