  We prove that given any fixed asymptotic velocity, the finite length
O'Connell-Yor polymer has an infinite length limit satisfying the law of large
numbers with this velocity. By a Markovian property of the quenched polymer
this reduces to showing the existence of Busemann functions: almost sure limits
of ratios of random point-to-point partition functions. The key ingredients are
the Burke property of the O'Connell-Yor polymer and a comparison lemma for the
ratios of partition functions. We also show the existence of infinite length
limits in the Brownian last passage percolation model.
