Density and gradient estimates for non degenerate Brownian SDEs with unbounded measurable drift

Abstract: We consider non degenerate Brownian SDEs with H{\"o}lder continuous in space diffusion coefficient and unbounded drift with linear growth. We derive two sided bounds for the associated density and pointwise controls of its derivatives up to order two under some additional spatial H{\"o}lder continuity assumptions on the drift. Importantly, the estimates reflect the transport of the initial condition by the unbounded drift through an auxiliary, possibly regularized, flow.