Heat Kernel Estimates for Stable-driven SDEs with Distributional Drift

We consider the formal SDE

\(\textrm{d} X_t = b(t,X_t)\textrm{d} t + \textrm{d} Z_t, \qquad X_0 = x \in \mathbb {R}^d, (\text {E})\)

where \(b\in L^r ([0,T],\mathbb {B}_{p,q}^\beta (\mathbb {R}^d,\mathbb {R}^d))\) 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 \(\mathbb {B}_{p,q}^\beta \) respectively denote Lebesgue and Besov spaces. We show that, when \(\beta > \frac{1-\alpha + \frac{\alpha }{r} + \frac{d}{p}}{2}\), 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 a Duhamel-type expansion. We then use a normalization method combined with Besov space properties (thermic characterization, duality and product rules) to derive estimates.