Feller generators with singular drifts in the critical range

We consider diffusion operator − Δ + b ⋅ ∇ in R d , d ≥ 3 , with drift b in a large class of locally unbounded vector fields that can have critical-order singularities. Covering the entire range of admissible magnitudes of singularities of b , we construct a strongly continuous Feller semigroup on the space of continuous functions vanishing at infinity, thus completing a number of results on well-posedness of SDEs with singular drifts. Our approach uses De Giorgi's method ran in L p for p sufficiently large, hence the gain in the assumptions on singular drift. For the critical borderline value of the magnitude of singularities of b , we construct a strongly continuous semigroup in a “critical” Orlicz space on R d whose topology is stronger than the topology of L p for any 2 ≤ p < ∞ but is slightly weaker than that of L ∞ .