Global solutions for continuous-time Burgers equations and infinite-volume polymer measures

Abstract

We study the stochastic Burgers equation on the whole space, driven by space-homogeneous Gaussian noises that are white in time and colored in space, for both positive and zero viscosity. We show that in dimension one, for any fixed average velocity, there is a unique ergodic global solution to the Burgers equation for all nonnegative viscosities, and that the One Force---One Solution principle holds. We also describe the basin of attraction and show that these global solutions are one-point pullback attractors. Moreover, the viscous global solution converges to the inviscid global solution as the viscosity tends to zero. Our technique relies on the study of polymer measures; accordingly, we establish the existence and uniqueness of infinite-volume polymer measures in a given asymptotic direction, as well as their convergence to semi-infinite minimizers in the zero-temperature limit. We also establish the existence part of all these results in arbitrary dimension larger than one.