Stationary distribution of overdamped Langevin dynamics

Question

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.

Answer 1

As explained here Convergence of a continuous time stochastic gradient descent algorithm even for weak regularity for $U$ (Lipschitz and L2 integrability), they indeed converge to Brownian motion as $t\to +\infty$.

here "An SDE perspective on stochastic convex optimization" they study these type of SDEs (Stochastic gradient descent) and basically ask bounded and an L2 condition for gamma_t in theorem 3.1.

Here in (H0) they also have extra constraints to $f$ that might be sharp.

As shown here a function can be convex and C1 but its gradient not be Lipschitz

The map $f:\Bbb R\to \Bbb R$, $f(x)=\frac23\lvert x\rvert^{3/2}=\int_0^x \lvert t\rvert^{1/2}\operatorname{sgn}t\,dt$ is convex and $C^1$, but $f'(x)=\lvert x\rvert^{1/2}\operatorname{sgn}x$ is not Lipschitz continuous in any neighbourhood of $0$. More generally, integrate your favourite monotone increasing continuous function which is not locally Lipschitz and you'll obtain a counterexample in $\Bbb R$.

And as shown here without this Lipschitz condition, we don't even have convergence for the deterministic gradient descent.