The Langevin diffusion is **reversible** with respect to the density $\rho(x)=e^{-U(x)}$.  This property is more transparent when the SDE is written as $$
dX_t = \nabla \log \rho(X_t) dt + \sqrt{2} dW_t
$$ and is the basis of MCMC methods based on Langevin diffusions; see e.g.

[Exponential convergence of Langevin diffusions and their discrete approximations][1]


  [1]: https://projecteuclid.org/journalArticle/Download?urlid=bj%2F1178291835