Feller generators with singular drifts in the critical range

TL;DR

This paper establishes the existence of Feller semigroups for diffusion operators with highly singular drifts in the critical range, extending previous results by employing De Giorgi's method and constructing semigroups in specialized function spaces.

Contribution

It introduces a novel approach using De Giorgi's method to handle critical singularities in drifts, broadening the class of well-posed diffusion processes.

Findings

Constructed Feller semigroups for singular drifts with large admissible singularities.

Extended the range of singularities beyond previous small constant bounds.

Developed semigroup theory in a critical Orlicz space for borderline singularities.

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 $L^\infty$.