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

We consider non degenerate Brownian SDEs with Holder 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 Wilder 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. (C) 2020 Elsevier Inc. All rights reserved.

Automated summary

This study deals with non degenerate Brownian stochastic differential equations with Holder continuous spatial diffusion coefficients and unbounded drift with linear growth. Two-sided bounds for density and pointwise controls of derivatives up to order two are derived under additional spatial Wilder continuity assumptions on the drift. The estimates show the transport of initial conditions by unbounded drift through an auxiliary, possibly regularized, flow.