Assume we have a reflected Brownian motion in a smooth bounded domain $D \subseteq \mathbb R^d$. It can have nonzero (but constant) drift, non-identity (but constant) covariance matrix, and oblique (=not normal) reflection.
We want to prove exponential ergodicity, that is, $$ ||P^t(\cdot, x) - \pi(\cdot)||_{TV} \le a(x)e^{-kt}, $$ for some constant $k > 0$, which is called {\it the exponent of ergodicity}. TV is the total variation distance, $\pi$ is the invariant distribution (provided it exists and is unique).
There is a following theorem (see, e.g. the article "Lyapunov implies Poincare" by Arnaud Guillin):
if we construct a function V (Lyapunov function) such that $$ LV \le -kV + c1_B, $$ where $k, c > 0$, $B$ is some {\it petite} set (in practice, it means compact set), and $L$ is the generator, then exponential ergodicity holds (with the same exponent $k$).
This works perfectly fine in domains like $\mathbb R^d_+$ (posiitve orthant). However, if $D$ is bounded, then we can take $B = D$, and the condition is trivially fulfilled with ANY k (just take any smooth $V$ in the domain of the generator and adjust $c$, taking it large enough!) So it means that exponent of ergodicity can be taken as large as possible??
Thank you in advance.
