Random walks in time-inhomogeneous random environment conditioned to stay positive

Abstract: We consider a random walk ${S_n}_{n\in \mathbb{N}}$ in time-inhomogeneous random environment $\xi$. For almost each realization of $\xi$, we formulate a quenched harmonic function, based on which we can define the random walk in random environment conditioned to stay positive by the Doob's $h$-transform. Furthermore, we prove a quenched invariance principle for the conditioned random walk for almost each realization of $\xi$.