The reason a Langevin diffusion leaves $\nu(x)=e^{-U(x)}$ invariant is because it is reversible with respectsymmetric or reversible with respect to $\nu$. In comparison to general diffusion processes, the ergodic properties of symmetric diffusions are not only easier to analyze, but it is often possible to obtain an explicit formula for the stationary density $\rho(x)=e^{-U(x)}$of a symmetric diffusion. This symmetry property is a bit more transparent when the SDELangevin equation is written as $$ dX_t = \nabla \log \rho(X_t) dt + \sqrt{2} dW_t $$ and is$$ dX_t = \nabla \log \nu(X_t) dt + \sqrt{2} dW_t \;. \tag{$\star$} $$ The symmetry property can then be easily verified by checking that the basisinfinitesimal generator of MCMC methods based onthe Langevin diffusions; see ediffusion is symmetric in an $L^2$-inner product weighted by $\nu$.g This property of the Langevin diffusion is analogous to the detailed balance condition of a Markov chain.
Exponential convergence of Langevin diffusions and their discrete approximations
It is also the basis of standard MCMC methods based on Langevin diffusions. In fact, many random-walk based MCMC methods (like random walk Metropolis) with target probability density proportional to $\rho(x)$$\nu(x)$, can be viewed as weak approximations toweakly approximate the corresponding Langevin diffusion; seediffusion ($\star$). This is remarkable because, e.g. Theorem 5.1 of
Metropolis integration schemes for self-adjoint diffusions
In particular, the random walk Metropolis doesn’t even explicitlyalgorithm does not involve the gradient of $U$, but by virtue of being reversible.
To read more about this connection, can weakly approximate the Langevin diffusionsee the introduction and Theorem 5.2 of this paper.