Reflected Brownian motion with negative drift ($R(t)=\Gamma (\sigma B(t)-dt),d>0$)

Apply the Skorokhod reflection map on the half-line to Brownian motion with negative constant drift and positive variance parameter ${\sigma }^2$. The result is kept nonnegative by a regulator acting only at zero. It has a stationary distribution, unlike zero-drift reflection on the whole half-line.

The stationary law is the law of ${\sup }_{t\ge 0}(\sigma B(t)-dt)$. Let $\kappa =2d/{\sigma }^2$. The Exponential martingale for Brownian motion $e^{\kappa (\sigma B_t-dt)}$, stopped on first hitting $b$, is bounded by $e^{\kappa b}$. On no hitting, the driving path tends to minus infinity. Dominated convergence and optional stopping therefore give $1=e^{\kappa b}\mathbb{P}(\text{hit }b)$. The reflected process started at zero converges to this supremum law: its value at time $t$ has the law of the driving path's maximum up to $t$, by reversing the increments. The resulting density is $(2d/{\sigma }^2)e^{-2dx/{\sigma }^2}$ for $x\ge 0$.