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We introduce a notion of subunit vector field for fully nonlinear degenerateelliptic equations. We prove that an interior maximum of a viscositysubsolution of such an equation propagates along the trajectories of subunitvector fields. This implies strong maximum and minimum principles when theoperator has a family of subunit vector fields satisfying the H\"ormandercondition. In particular these results hold for a large class of nonlinearsubelliptic PDEs in Carnot groups. We prove also a strong comparison principlefor degenerate elliptic equations that can be written inHamilton-Jacobi-Bellman form, such as those involving the Pucci's extremaloperators over H\"ormander vector fields.

New strong maximum and comparison principles for fully nonlinear degenerate elliptic PDEs