  We study the semi-discrete directed polymer model introduced by O'Connell-Yor
in its stationary regime, based on our previous work on the stationary
$q$-totally asymmetric simple exclusion process ($q$-TASEP) using a two-sided
$q$-Whittaker process. We give a formula for the free energy distribution of
the polymer model in terms of Fredholm determinant and show that the universal
KPZ stationary distribution appears in the long time limit. We also consider
the limit to the stationary KPZ equation and discuss the connections with
previously found formulas.
