Left-Invariance for Smooth Vector Fields and Applications

Let \(X = \{X_0,\ldots ,X_m\}\) $X=\{X_0,\dots ,X_m\}$ be a family of smooth vector fields on an open set \(\Omega \subseteq \mathbb {R}^N\) $\mathrm{\Omega }\subseteq {\mathbb{R}}^N$ . Motivated by applications to the PDE theory of Hörmander operators, for a suitable class of open sets \(\Omega \) $\mathrm{\Omega }$ , we find necessary and sufficient conditions on X for the existence of a Lie group \((\Omega ,*)\) $(\mathrm{\Omega },\ast )$ such that the operator \(L=\sum _{i = 1}^mX_i^2+X_0\) $L={\sum }_{i=1}^m{X}_i^2+X_0$ is left-invariant with respect to the operation \(*\) $\ast$ . 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.