We study subcritical two-dimensional oriented percolation seen from its rightmost point on the set of infinite configurations which are bounded above. This a Feller process whose state space is not compact and has no invariant measures. We prove that it converges in distribution to a measure which charges only finite configurations.
"Convergence in distribution for subcritical 2D oriented percolation seen from its rightmost point." Ann. Probab. 42 (3) 1285 - 1296, May 2014. https://doi.org/10.1214/13-AOP841