The order-1/d exponential ergodicity conjecture for generalized reflected Brownian motion

Let $\operatorname{GRBM}(R,\mu,\Gamma,U)$ be the generalized reflected Brownian motion defined by the strong solution of the stochastic differential equation in the paper, with transition kernel $P^t$ and stationary distribution $\pi$. Assume the hypotheses of Theorem. Order-$1/d$ exponential ergodicity conjecture. There exist a function

$$W:\mathbb{R}^d\rightarrow[1,\infty)$$

and a constant $C>0$ such that

$$\left\|P^t(x,\cdot)-\pi(\cdot)\right\|_{TV}\leq W(x)\exp\left(-\frac{Ct}{d}\right).$$

The preceding corollary gives uniform exponential ergodicity and integrability of the Lyapunov function under the stationary distribution; this conjecture predicts that the exponential rate has order $1/d$. The paper notes that obtaining the exact dependence of the rate on $d$ is difficult, and the conjectured order is not established there.