Heat kernel estimates for stable-driven SDEs with distributional drift

Abstract

We consider the formal SDE dX t = b(t, X t)dt + dZ t , X 0 = x $\in$ R d , (E) where b $\in$ L r ([0, T ], B $\beta$ p,q (R d , R d)) is a time-inhomogeneous Besov drift and Z t is a symmetric d-dimensional $\alpha$-stable process, $\alpha$ $\in$ (1, 2), whose spectral measure is absolutely continuous w.r.t. the Lebesgue measure on the sphere. Above, L r and B $\beta$ p,q respectively denote Lebesgue and Besov spaces. We show that, when $\beta$ > (1--$\alpha$+ $\alpha$/r + d/p)/2 , 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.