Feller generators with singular drifts in the critical range

Abstract

We consider diffusion operator $-\Delta +b\cdot \nabla$ in $R^d$, $d\ge 3$, 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 $2\le p<\infty$ but is slightly weaker than that of $L^{\infty }$.