We consider the gradient dynamics $ d X_{t} = d B_{t} - \nabla(U(X_{t}))dt $ in $\mathbb{R}^{d}$.
G.Royer in the book "An initiation to logarithmic sobolev inequalities" (p29,30) says that if
(1) U is $C^{2}$,
(2) the sde doesn't explode in finite time almost surely, and
(3) $\exp(-U)$ is integrable on $\mathbb{R}^{d}$,

then the corresponding Boltzmann-Gibbs measure defined by $d\mu(x)=\frac{1}{\zeta} \exp(-U(x)) dx$ is reversible for the process (and consequently stationary distribution for the process).

Now, consider U which doesn't satisfy the 3rd assumption (integrability). Then, according to the theory above, we can't deduce stationarity of the Boltzmann-Gibbs measure written above.

My question is what can we say now about the stationary distribution of the process? Can the sde have stationary distribution of some other form ?( ie.other than the Boltzmann-Gibbs measure) OR The sde doesn't have any stationary distribution at all ? (implying that if the sde has stationary distribution then it has to be of the Boltzmann-Gibbs form )

I would really appreciate some help here because I am confused.

Consider the over damped Langevin dynamics: $d X_{t} = d B_{t} - \nabla U(X_{t}) dt $ on $\mathbb{R}^{d}$ where $B_t$ is a standard Brownian motion. On pages 29 and 30 of the following book

Royer, Gilles, An introduction to logarithmic Sobolev inequalities, Cours Spécialisés (Paris). 5. Paris: Société Mathématique de France. 114 p. (1999). ZBL0927.60006.

Royer, Gilles, An initiation to logarithmic Sobolev inequalities. Transl. from the French by Donald Babbitt, SMF/AMS Texts and Monographs 14; Cours Spécialisés (Paris) 5. Providence, RI: American Mathematical Society (AMS); Paris: Société Mathématique de France (ISBN 978-0-8218-4401-4/pbk). viii, 119 p. (2007). ZBL1138.60007.

the author says that if

1. $U$ is $C^{2}$,
2. the corresponding SDE doesn't explode in finite time almost surely, and
3. $\exp(-U)$ is integrable on $\mathbb{R}^{d}$.

then the corresponding Boltzmann-Gibbs measure defined by $d\mu(x)\propto \exp(-U(x)) dx$ is reversible for the process (and consequently the stationary distribution for the process).

Now, consider $U$ which doesn't satisfy the third assumption (integrability). Then, according to the theory above, we can't deduce stationarity of the Boltzmann-Gibbs measure written above.

In this context, my question is:

What can we say now about the stationary distribution of the process? Can this SDE have a stationary distribution of some other form (i.e. other than the Boltzmann-Gibbs)? OR does the SDE have no stationary distribution at all (implying that if the SDE has a stationary distribution then it has to be of the Boltzmann-Gibbs form)?

I would really appreciate some help here because I am confused.