Suppose the following SDE for $X_t$ is well-posed: $$dX_t = \sqrt{2}\, dB_t - \nabla\Phi(X_t)\,dt.$$

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 references, results or methods. In particular, if $exp{(-\Phi)}\notin L^1$, can $X$ still have stationary distribution ? Thanks!

Suppose the following SDE for $X_t$ is well-posed: $$dX_t = \sqrt{2}\, dB_t - \nabla\Phi(X_t)\,dt.$$

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 references, results or methods. Thanks!

Remark: Seems like by Girsanov transform, $exp{(-\Phi)}\in L^1$ is necessary and sufficient condition for my first question.

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!

Suppose the following SDE for $X_t$ is well-posed: $$dX_t = \sqrt{2}\, dB_t - \nabla\Phi(X_t)\,dt.$$

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 references, results or methods. In particular, if $exp{(-\Phi)}\notin L^1$, can $X$ still have stationary distribution ? Thanks!