We consider the following diffusion processes on $d$-dim torus,
$$
dX^{(c)}(t)=c b(X^{(c)}(t))dt + dB(t),
$$
$B(t)$ is the Brownian motion, $b(\cdot)$ is a divergence free ($div(b)=0$)
smooth vector field with period 1, $c$ is a large parameter.
This diffusion process has Lebsegue measure (on torus) as invariance measure
due to the condition $div(b)=0$.
This is a particular example of more general class of diffusion processes,
$$
dX^{(c)}(t)=(-\nabla U(X^{(c)}(t))+ c b(X^{(c)}(t))dt + dB(t),
$$
with $U, b$ periodic and satisfying
$$
div(b\exp(-2U))=0,
$$
such that they have $\mu$ as the invariance measure,
$$
d\mu=\frac 1Z \exp(-2U(x)) dx.
$$
Such diffusion processes appear in MCMC(Markoc Chain Monte Carlo)
that one chooses particular $b$ to simulate the underlying distribution $\mu$.
A main concern is how well the distribution of $X^{(c)}(t)$ approximate $\mu$
and how to choose a better $b$.
We are able to say some quantitative behaviors of such processes by taking $c$ large.