  We develop a dynamical approach to infinite volume directed polymer measures
in random environments. We define polymer dynamics in 1+1 dimension as a
stochastic gradient flow on polymers pinned at the origin, for energy involving
quadratic nearest neighbor interaction and local interaction with random
environment. We prove existence and uniqueness of the solution, continuity of
the flow, the order-preserving property with respect to the coordinatewise
partial order, and the invariance of the asymptotic slope. We establish
ordering by noise which means that if two initial conditions have distinct
slopes, then the associated solutions eventually get ordered coordinatewise.
This, along with the shear-invariance property and existing results on static
infinite volume polymer measures, allows to prove that for a fixed asymptotic
slope and almost every realization of the environment, the polymer dynamics has
a unique invariant distribution given by a unique infinite volume polymer
measure, and, moreover, One Force -- One Solution principle holds. We also
prove that every polymer measure is concentrated on paths with well-defined
asymptotic slopes and give an estimate on deviations from straight lines.
