THE SECTOR OF ANALYTICITY OF THE ORNSTEIN–UHLENBECK SEMIGROUP ON Lᵖ SPACES WITH RESPECT TO INVARIANT MEASURE

The sector of analyticity of the Ornstein–Uhlenbeck semigroup is computed on the space L μ p := Lp (RN; μ) with respect to its invariant measure μ. If A=Δ + Bx· ∇ denotes the generator of the Ornstein–Uhlenbeck semigroup, then the angle Θ2 of the sector of analyticity in L μ 2 is π/2 minus the spectral angle of BQ∞, Q∞ being the matrix determining the Gaussian measure μ. The angle of analyticity in L μ p is then given by the formula cot θ p = ( p − 2 ) 2 + p 2 ( cot θ 2 ) 2 2 p − 1