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.
Related
- Global existence of smooth solutions to the Navier-Stokes equations for initial data with infinite energy
- Multidimensional Infinite-Horizon Markovian Backward Stochastic Differential Equations with Continuous Coefficients and Nonzero-Sum Differential Games
- Stationarity equations for fractional affine area measures: ellipsoid rigidity and polytopal solutions
- NESTED DISCONTINUOUS ASYMPTOTIC PROFILES FOR THE VISCOUS BURGERS EQUATION WITH INFINITE MASS
- A Necessary and Sufficient Condition for the Existence of Global Smooth Solutions to the One-Dimensional Inviscid Burgers-Vlasov System
BibTeX
@misc{li2026global,
title = {{Global solutions for continuous-time Burgers equations and infinite-volume polymer measures}},
author = {Liying Li},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.08355},
eprint = {2611.08355},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.08355},
}
RIS
TY - UNPB
TI - Global solutions for continuous-time Burgers equations and infinite-volume polymer measures
AU - Liying Li
PY - 2026
DA - 2026/11/05
PB - arXiv
DO - 10.48550/arxiv.2611.08355
UR - https://arxiv.org/abs/2611.08355
AB - 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.
ER -