Analyticity of the semigroup in a degenerate case

We consider the following class of second order differential operators d d where a(x) is a positive semi-definite symmetric matrix which has quadratic growth and b(x) is a vector field of class C ² which has linear growth. We assume that the mapping a : ℝ^d → ℒ(ℝ^d) is also of class C ² with bounded second derivatives, so that it can be written as a(x) = ½σxσ^* x, x∈ℝ^d, for some matrix valued function σ : ℝ^d → ℒ(ℝ^d) which is Lipschitz-continuous (for a proof of this fact see [66] and also [114]. Here we assume further regularity for o; namely we assume that a can be factorized by some a which is twice differentiable with bounded derivatives.