Heat kernel estimates for stable-driven SDEs with distributional drift

Abstract

We consider the formal SDE) is a time-inhomogeneous Besov drift and Z t is a symmetric d-dimensional α-stable process, α ∈ (1, 2), whose spectral measure is absolutely continuous w.r.t. the Lebesgue measure on the sphere. Above, L r and B β p,q respectively denote Lebesgue and Besov spaces. We show that, when β >, the martingale solution associated with the formal generator of (E) admits a density which enjoys two-sided heat kernel bounds as well as gradient estimates w.r.t. the backward variable. Our proof relies on a suitable mollification of the singular drift aimed at using Duhamel expansion. We then use a normalization method combined with Besov space properties (thermic characterization, duality and product rules) to derive estimates.