Heat Kernel Estimates for Stable-driven SDEs with Distributional Drift

We consider the formal SDEdX(t)=b(t,X-t)dt+dZ(tt),X0=x is an element of R-d,(E)where b is an element of L (R)([0,T],B-p,q(beta)(R-d,R-d)) is a time-inhomogeneous Besov drift and Z(t )is a symmetric d-dimensional alpha-stable process, alpha is an element of (1,2), whose spectral measure is absolutely continuous w.r.t. the Lebesgue measure on the sphere. Above, Lr 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 a Duhamel-type expansion. We then use a normalization method combined with Besov space properties (thermic characterization, duality and product rules) to derive estimates.

Automated summary

This paper investigates a formal stochastic differential equation with a time-inhomogeneous Besov drift and a symmetric alpha-stable process. We prove that the martingale solution associated with this equation has a special density that satisfies heat kernel bounds and gradient estimates.