The $t$-Martin boundary of a random walk on a half-space with reflected boundary conditions is identified. It is shown in particular that the $t$-Martin boundary of such a random walk is not stable in the following sense: for different values of $t$, the $t$-Martin compactifications are not equivalent.
"$t$-Martin boundary of reflected random walks on a half-space." Electron. Commun. Probab. 15 149 - 161, 2010. https://doi.org/10.1214/ECP.v15-1541