Left-Invariance for Smooth Vector Fields and Applications

Let X = ₀,… ,Xₘ\ X = { X 0 , … , X m } be a family of smooth vector fields on an open set Ω ⊆ RN Ω ⊆ R N . Motivated by applications to the PDE theory of Hörmander operators, for a suitable class of open sets Ω Ω , we find necessary and sufficient conditions on X for the existence of a Lie group (Ω ,*) ( Ω , ∗ ) such that the operator L=∑ i = 1ᵐXᵢ²+X₀ L = ∑ i = 1 m X i 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.