Generation of analytic semigroups by generalized Ornstein–Uhlenbeck operators with potentials

Abstract

It is shown that the generalized Ornstein–Uhlenbeck operator “with potential” A Φ , G , V u : = Δ u − ∇ Φ ⋅ ∇ u + G ⋅ ∇ u − V u with suitable domain generates an analytic semigroup on the weighted space L p ( R N , μ ) , 1 < p < ∞ , where μ ( d x ) = e − Φ d x . The result extends the generation theorem established by Metafune–Prüss–Rhandi–Schnaubelt when V ≡ 0 to the case where V ≢ 0 , while the proof is based on their useful theorem that a class of second order elliptic operators generates an analytic semigroup on the unweighted space L p ( R N ) .