In this paper we obtain necessary and sufficient conditions for the reversibility of the diffusion property, assuming the existence of a density at every time $t$. The proofs are based on techniques of the stochastic calculus of variations.
"Integration by Parts and Time Reversal for Diffusion Processes." Ann. Probab. 17 (1) 208 - 238, January, 1989. https://doi.org/10.1214/aop/1176991505