The usual Langevin equation for a particle in a 1D harmonic potential

$dq(t) = p(t)~dt$

$dp(t) = -q(t)~dt + a ~dW(t) - b~p(t)~dt$

admits as an invariant measure the Gibbs measure ${1\over Z}\exp(-{2b\over a^2}{q^2+p^2\over 2})$. (We assume here $a, b > 0$, and that $W$ is a Brownian motion.)

Now, assume that the damping coefficient $b$ depends on $q$. More precisely, let $b: {\mathbb R} \to [c, d]$ with $ 0 < c < d < \infty$ be a (smooth) function, and consider the SDE

$dq(t) = p(t)~dt$

$dp(t) = -q(t)~dt + a ~dW(t) - b(q(t))~p(t)~dt$

It seems natural that, since the damping term is bounded below, there exists an invariant probability measure. Is it true? If yes, what is the proof, or what tools should I use to prove it?

For example, if $b(q) = 10 + \sin(q)$, I expect the system to be at least as "well-behaved" as the usual Langevin equation with $b = 9$. But even for this simple example, I fail to find a proof.

Thanks a lot!