# Zero Temperature Limit for Directed Polymers and Inviscid Limit for Stationary Solutions of Stochastic Burgers Equation

@article{Bakhtin2018ZeroTL, title={Zero Temperature Limit for Directed Polymers and Inviscid Limit for Stationary Solutions of Stochastic Burgers Equation}, author={Yuri Bakhtin and Liying Li}, journal={Journal of Statistical Physics}, year={2018}, volume={172}, pages={1358-1397} }

We consider a space-continuous and time-discrete polymer model for positive temperature and the associated zero temperature model of last passage percolation type. In our previous work, we constructed and studied infinite-volume polymer measures and one-sided infinite minimizers for the associated variational principle, and used these objects for the study of global stationary solutions of the Burgers equation with positive or zero viscosity and random kick forcing, on the entire real line. In…

## 11 Citations

Stationary Solutions to the Stochastic Burgers Equation on the Line

- Mathematics
- 2019

We consider invariant measures for the stochastic Burgers equation on $\mathbb{R}$, forced by the derivative of a spacetime-homogeneous Gaussian noise that is white in time and smooth in space. An…

Dynamic polymers: invariant measures and ordering by noise

- MathematicsProbability Theory and Related Fields
- 2021

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…

Busemann functions and semi-infinite O’Connell–Yor polymers

- MathematicsBernoulli
- 2020

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…

Zero temperature limit for the Brownian directed polymer among Poissonian disasters

- Mathematics
- 2019

We study a continuum model of directed polymer in random environment. The law of the polymer is defined as the Brownian motion conditioned to survive among space-time Poissonian disasters. This model…

On global solutions of the random Hamilton–Jacobi equations and the KPZ problem

- Mathematics
- 2017

In this paper, we discuss possible qualitative approaches to the problem of KPZ universality. Throughout the paper, our point of view is based on the geometrical and dynamical properties of…

On a factorization formula for the partition function of directed polymers

- Mathematics
- 2021

We prove a factorization formula for the point-to-point partition function associated with a model of directed polymers on the space-time lattice Z, subject to an i.i.d. random potential and in the…

Non-existence of bi-infinite polymers

- MathematicsElectronic Journal of Probability
- 2022

We show that nontrivial bi-infinite polymer Gibbs measures do not exist in typical environments in the inverse-gamma (or log-gamma) directed polymer model on the planar square lattice. The precise…

Non-existence of bi-infinite polymer Gibbs measures.

- Mathematics
- 2020

We show that nontrivial bi-infinite polymer Gibbs measures do not exist in typical environments in the inverse-gamma (or log-gamma) directed polymer model on the planar square lattice. The precise…

Random polymers in disastrous environments

- Mathematics
- 2019

We study a random walk on Z^d in a random environment of space-time disasters and consider the decay rate of its quenched survival probability. We then study the associated branching random walk and…

Invariant measures for stochastic conservation laws on the line

- Mathematics
- 2022

We consider a stochastic conservation law on the line with solution-dependent diffusivity, a super-linear, sub-quadratic Hamiltonian, and smooth, spatially-homogeneous kick-type random forcing. We…

## References

SHOWING 1-10 OF 46 REFERENCES

Thermodynamic Limit for Directed Polymers and Stationary Solutions of the Burgers Equation

- MathematicsCommunications on Pure and Applied Mathematics
- 2018

The first goal of this paper is to prove multiple asymptotic results for a time‐discrete and space‐continuous polymer model of a random walk in a random potential. These results include: existence of…

Space-time stationary solutions for the Burgers equation

- Mathematics
- 2012

We construct space-time stationary solutions of the 1D Burgers equation with random forcing in the absence of periodicity or any other compactness assumptions. More precisely, for the forcing given…

Localization and Perron--Frobenius theory for directed polymers

- Mathematics
- 2009

We consider directed polymers in a random potential given by a deterministic profile with a strong maximum at the origin taken with random sign at each integer time.
We study two main objects based…

Variational Formulas and Cocycle solutions for Directed Polymer and Percolation Models

- Mathematics
- 2013

We discuss variational formulas for the law of large numbers limits of certain models of motion in a random medium: namely, the limiting time constant for last-passage percolation and the limiting…

Inviscid Burgers equation with random kick forcing in noncompact setting

- Mathematics
- 2014

We develop ergodic theory of the inviscid Burgers equation with random kick forcing in noncompact setting. The results are parallel to those in our recent work on the Burgers equation with Poissonian…

The Burgers equation with Poisson random forcing

- Mathematics
- 2013

We consider the Burgers equation on the real line with forcing given by Poissonian noise with no periodicity assumption. Under a weak concentration condition on the driving random force, we prove…

Burgers Turbulence and Random Lagrangian Systems

- Mathematics
- 2003

Abstract: We consider a spatially periodic inviscid random forced Burgers equation in arbitrary dimension and the random time-dependent Lagrangian system related to it. We construct a unique…

Two results concerning asymptotic behavior of solutions of the Burgers equation with force

- Mathematics
- 1991

We consider the Burgers equation with an external force. For the case of the force periodic in space and time we prove the existence of a solution periodic in space and time which is the limit of a…

Stochastic Dissipative PDE's and Gibbs Measures

- Mathematics
- 2000

Abstract: We study a class of dissipative nonlinear PDE's forced by a random force ηomega(t,x), with the space variable x varying in a bounded domain. The class contains the 2D Navier–Stokes…

Ratios of partition functions for the log-gamma polymer

- Mathematics
- 2015

We introduce a random walk in random environment associated to an underlying directed polymer model in 1 + 1 dimensions. This walk is the positive temperature counterpart of the competition in-…