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.
Left-Invariance for Smooth Vector Fields and Applications / Biagi, Stefano; Bonfiglioli, Andrea; Polidoro, Sergio. - In: THE JOURNAL OF GEOMETRIC ANALYSIS. - ISSN 1050-6926. - 34:9(2024), pp. 1-21. [10.1007/s12220-024-01733-3]
Left-Invariance for Smooth Vector Fields and Applications
Polidoro, Sergio
2024
Abstract
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.File | Dimensione | Formato | |
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