On Heat kernel Estimtes for Brownian SDEs with Distributional Drift

TL;DR

This paper derives heat kernel bounds and regularity estimates for multidimensional Brownian SDEs with singular, distributional drift, covering both non-degenerate and degenerate noise scenarios.

Contribution

It introduces a novel parametrix method using the transition density of the unperturbed SDE, establishing weak well-posedness and regularity for singular SDEs.

Findings

Established heat kernel bounds for singular drift SDEs.

Proved regularity estimates and Schauder bounds for associated PDEs.

Demonstrated irreducibility and strong Feller property of solutions.

Abstract

We establish heat-kernel bounds and regularity estimates for the transition densities of the diffusion associated with the martingale problem corresponding to the generator of a formal multidimensional Brownian SDE with singular drift. As a by-product, we also derive Schauder estimates for the associated Kolmogorov (kinetic) Cauchy problem. We consider both the cases of non-degenerate and degenerate noise (e.g. kinetic-type models), in the so-called Young regime. Namely, we consider a time inhomogeneous drift in L $\infty$ [0,T ] C $\beta$ o for some fixed time horizon T , where ), with o standing for an underlying distance, namely the usual Euclidean one in the non degenerate setting, and the scale-homogeneous one in the kinetic case. Importantly, the estimates are obtained by employing as parametrix the transition density of the SDE (with variable coefficients) without singular perturbation, as opposed to the standard Levi parametrix obtained by freezing the noise. Finally, since the noise is multiplicative, the weak well-posedness of the singular SDE is a novel result in itself, and the density estimates directly imply irreducibility and strong Feller property of its solutions.