Consider the $n$-dimensional stochastic differential equation $$ \mathrm{d}X_t = f(X_t)\mathrm{d}t +\mathrm{d}W_t, $$ where $f\colon \mathbb{R}^n\to \mathbb{R}^n$ is some smooth function and $W_t$ is a Wiener process.
If the vector field $f(X)$ is a gradient flow, that is, if it can be written as $f(X)=-\nabla U(X)$ for some potential function $U\colon \mathbb{R}^n\to \mathbb{R}$, it is well-known that $x$ has a steady-state density (i.e. the stationary solution to the Fokker-Planck equation) which is given by (up to normalizing constants) $$ p_{ss}(X) = \exp(-U(X)), $$
Consider now a vector field $f$ of the form: $$ f(X)=-\nabla U(Y)|_{Y=g(X)}\ \ \ \ \ (\ast) $$ where $g\colon \mathbb{R}^n \to \mathbb{R}^n$ is some smooth function. In this case $f(X)$ is still a gradient flow, but with respect to the variable $Y=g(X)$.
My question : What it can be said about the steady-state density $p_{ss}(X)$ when $(\ast)$ holds?
My approach would be that of considering an extended state space in which the drift can be written as a gradient flow, and then marginalizing the resulting steady-state density. Does it make sense? Is there any more clever/easier approach?
Any comment is welcome, as well as pointers to literature. Thanks.