Absolute continuity under flows generated by SDE with measurable drift coefficient

TL;DR

This paper proves that stochastic flows generated by certain SDEs with measurable drift coefficients preserve absolute continuity of the reference measure, under specific integrability conditions on the coefficients.

Contribution

It establishes absolute continuity of stochastic flows for SDEs with measurable drifts under exponential integrability conditions, extending previous results to less regular coefficients.

Findings

Stochastic flow preserves absolute continuity under specified conditions.

Exponential integrability of coefficients ensures measure preservation.

Results apply to SDEs with non-degenerate diffusion and measurable drift.

Abstract

We consider the It\^{o} SDE with non-degenerate diffusion coefficient and 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.