Left-Invariance for Smooth Vector Fields and Applications

Let X={X0,…,Xm} be a family of smooth vector fields on an open set Ω⊆RN. Motivated by applications to the PDE theory of Hörmander operators, for a suitable class of open sets Ω, we find necessary and sufficient conditions on X for the existence of a Lie group (Ω,∗) such that the operator L=∑i=1mXi2+X0 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.