Feller generators with singular drifts in the critical range

We consider diffusion operator -Δ + b · ∇ in Rᵈ, 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 (but excluding the borderline value), 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. The previous results on Feller semigroups employed strong elliptic gradient bounds and hence required the magnitude of the singularities to be less than a small dimension-dependent constant. Our approach is different and uses De Giorgi's method ran in Lᵖ 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 Rᵈ whose local topology is stronger than the local topology of Lᵖ for any 2 ≤ p<∞ but is slightly weaker than that of L^∞.