Absolute continuity under flows generated by SDE with measurable drift coefficients

We consider the Itô SDE with a non-degenerate diffusion coefficient and a measurable drift coefficient. Under the condition that the gradient of the diffusion coefficient and the divergences of the diffusion and drift coefficients are exponentially integrable with respect to the Gaussian measure, we show that the stochastic flow leaves the reference measure absolutely continuous.

► We prove an a priori estimate of the Radon–Nikodym density of the stochastic flow.
► We obtain a limit theorem for SDE when the diffusion coefficient is non-degenerate.
► The Lebesgue measure is shown to be absolutely continuous under the stochastic flow.