I am trying to prove that: $\lim_{t \to \infty} W_2(\mu_t, \nu_t) = 0 $ where we have that $\mu_t = Law(X_t)$ and $\nu_t = Law(Z_t)$ with $$dX_t = -h(X_t)dt + \sqrt(\frac{2}{\beta})dB_t$$ $$dZ_t = -h(Z_t)dt+\sqrt(\frac{2}{\beta})dB_t$$
I know that the Wasserstein distance in this case is given by: $$W_2(\mu, \nu) := \iint|\theta-\theta'|^2\zeta(d\theta d\theta'))^\frac{1}{2}$$ However, I am not too sure where to go from here.