Feller generators with singular drifts in the critical range

Abstract

We consider diffusion operator $-\Delta + b \cdot \nabla$ in $\mathbb R^d$, $d \geq 3$, 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 $L^p$ 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 $\mathbb R^d$ whose local topology is stronger than the local topology of $L^p$ for any $2 \leq p<\infty$ but is slightly weaker than that of $L^\infty$.