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The first goal of this paper is to prove multiple asymptotic results for atime-discrete and space-continuous polymer model of a random walk in a randompotential. These results include: existence of deterministic free energydensity in the infinite volume limit for every fixed asymptotic slope;concentration inequalities for free energy implying a bound on its fluctuationexponent; straightness estimates implying a bound on the transversalfluctuation exponent. The culmination of this program is almost sure existenceand uniqueness of polymer measures on one-sided infinite paths with givenendpoint and slope, and interpretation of these infinite-volume Gibbs measuresas thermodynamic limits. Moreover, we prove that marginals of polymer measureswith the same slope and different endpoints are asymptotic to each other.  The second goal of the paper is to develop ergodic theory of the Burgersequation with positive viscosity and random kick forcing on the real linewithout any compactness assumptions. Namely, we prove a One Force -- OneSolution principle, using the infinite volume polymer measures to construct afamily of stationary global solutions for this system, and proving that each ofthose solutions is a one-point pullback attractor on the initial conditionswith the same spatial average. This provides a natural extension of the sameprogram realized for the inviscid Burgers equation with the help of actionminimizers that can be viewed as zero temperature limits of polymer measures.

Thermodynamic limit for directed polymers and stationary solutions of the Burgers equation