Suppose the following SDE for $X_t$ is well-posed:
$$dX_t =  dB_t - \nabla\Phi(X_t)\,dt.$$

For what $\Phi\in C^1(R^d)$  will $X$ have stationary distribution $u_{\infty}$? For what $\Phi$ will log-Sobolev inequality holds for $u_{\infty}$ ? 

Conditions like $exp{(-\Phi)}\in L^1$ and $Hess(\Phi)>0$ respectively will work, but I would love to know the strongest reference, results or method. In particular, if $exp{(-\Phi)}\notin L^1$, can $X$ still have stationary distribution ? Thanks!