General large deviations principle for stationary reflected Brownian motion in the octant

Let $Z$ be a semimartingale reflected Brownian motion (SRBM) with data $(\theta,\Gamma,R)$, where $R$ is a completely-$\mathcal{S}$ matrix and there is a probability measure $\mathbb{P}_{\pi}$ under which $Z$ is stationary. For a measurable set $A\subset\mathbb{R}^d_+$, let $A^c$ and $A^o$ denote its closure and interior, respectively.

General large deviations principle. For every measurable $A\subset\mathbb{R}^d_+$,

$$\limsup_{u\to\infty}\frac{1}{u}\log\mathbb{P}_{\pi}(Z(0)/u\in A)\leq-\inf_{v\in A^c}I(v)$$

\nand

$$\liminf_{u\to\infty}\frac{1}{u}\log\mathbb{P}_{\pi}(Z(0)/u\in A)\geq-\inf_{v\in A^o}I(v).$$

The principle describes the exponential decay of the stationary distribution's probabilities at large spatial scales; the source notes that it had only been established in some special cases, so its general validity remains open.