We consider the persistence probabilities of an autoregressive chain of order one with continuous innovations. In the case of positive drifts, we show that these persistence probabilities are compound-geometric and satisfy a Baxter-Spitzer factorization generalizing that of the random walk. In the case of negative drifts, we exhibit a discrete Van Dantzig problem, which implies that the Baxter-Spitzer factorization never happens, except in a degenerate case. For positive drifts and log-concave innovations, we show that the first passage time in $(-\infty,0)$ has a log-convex distribution, whereas in the case of negative drifts and log-convex innovations on ${\mathbb R}^+$, it has a log-concave distribution. The case of the bi-exponential innovations is studied in detail, which leads for positive drifts to an additive factorization of the exponential law.