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Crossover to the stochastic Burgers equation for the WASEP with a slow bond
Probability · Submitted 22 Jun 2015
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim For $\beta<1$, we show that the limit density fluctuation field is an Ornstein-Uhlenbeck process defined on the Schwartz space if $\gamma>\frac12$, while for $\gamma = \frac12$ it is an energy solution of the stochastic Burgers equation.
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The abstract, sentence by sentence
We consider the weakly asymmetric simple exclusion process in the presence of a slow bond and starting from the invariant state, namely the Bernoulli product measure of parameter $\rho\in(0,1)$. The rate of passage of particles to the right (resp. left) is $\frac1{\vphantom{n^\beta}2}+\frac{a}{2n^{\vphantom{\beta}\gamma}}$ (resp. $\frac1{\vphantom{n^\beta}2}-\frac{a}{2n^{\vphantom{\beta}\gamma}}$) except at the bond of vertices $\{-1,0\}$ where the rate to the right (resp. left) is given by $\frac{\alpha}{2n^\beta}+\frac{a}{2n^{\vphantom{\beta}\gamma}}$ (resp. $\frac{\alpha}{2n^\beta}-\frac{a}{2n^{\vphantom{\beta}\gamma}}$). Above, $\alpha>0$, $\gamma\geq \beta\geq 0$, $a\geq 0$. For $\beta<1$, we show that the limit density fluctuation field is an Ornstein-Uhlenbeck process defined on the Schwartz space if $\gamma>\frac12$, while for $\gamma = \frac12$ it is an energy solution of the stochastic Burgers equation. For $\gamma\geq\beta=1$, it is an Ornstein-Uhlenbeck process associated to the heat equation with Robin's boundary conditions. For $\gamma\geq\beta> 1$, the limit density fluctuation field is an Ornstein-Uhlenbeck process associated to the heat equation with Neumann's boundary conditions.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We consider the weakly asymmetric simple exclusion process in the presence of a slow bond and starting from the invariant state, namely the Bernoulli product measure of parameter $\rho\in(0,1)$. | background 0.57 | 0.55 | 0.02 | asserted 0.81 |
| s1 | The rate of passage of particles to the right (resp. left) is $\frac1{\vphantom{n^\beta}2}+\frac{a}{2n^{\vphantom{\beta}\gamma}}$ (resp. $\frac1{\vphantom{n^\beta}2}-\frac{a}{2n^{\vphantom{\beta}\gamma}}$) except at the bond of vertices $\{-1,0\}$ where the rate to the right (resp. left) is given by $\frac{\alpha}{2n^\beta}+\frac{a}{2n^{\vphantom{\beta}\gamma}}$ (resp. $\frac{\alpha}{2n^\beta}-\frac{a}{2n^{\vphantom{\beta}\gamma}}$). | method 0.87 | 0.24 | 0.03 | asserted 0.99 |
| s2 | Above, $\alpha>0$, $\gamma\geq \beta\geq 0$, $a\geq 0$. | method 0.54 | 0.14 | 0.02 | asserted 0.67 |
| s3 | For $\beta<1$, we show that the limit density fluctuation field is an Ornstein-Uhlenbeck process defined on the Schwartz space if $\gamma>\frac12$, while for $\gamma = \frac12$ it is an energy solution of the stochastic Burgers equation. | result 1.00 | 0.94 | 0.55 | asserted 1.00 |
| s4 | For $\gamma\geq\beta=1$, it is an Ornstein-Uhlenbeck process associated to the heat equation with Robin's boundary conditions. | result 1.00 | 0.90 | 0.11 | asserted 1.00 |
| s5 | For $\gamma\geq\beta> 1$, the limit density fluctuation field is an Ornstein-Uhlenbeck process associated to the heat equation with Neumann's boundary conditions. | result 1.00 | 0.91 | 0.12 | asserted 1.00 |
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Original abstract
We consider the weakly asymmetric simple exclusion process in the presence of a slow bond and starting from the invariant state, namely the Bernoulli product measure of parameter $\rho\in(0,1)$. The rate of passage of particles to the right (resp. left) is $\frac1{\vphantom{n^\beta}2}+\frac{a}{2n^{\vphantom{\beta}\gamma}}$ (resp. $\frac1{\vphantom{n^\beta}2}-\frac{a}{2n^{\vphantom{\beta}\gamma}}$) except at the bond of vertices $\{-1,0\}$ where the rate to the right (resp. left) is given by $\frac{\alpha}{2n^\beta}+\frac{a}{2n^{\vphantom{\beta}\gamma}}$ (resp. $\frac{\alpha}{2n^\beta}-\frac{a}{2n^{\vphantom{\beta}\gamma}}$). Above, $\alpha>0$, $\gamma\geq \beta\geq 0$, $a\geq 0$. For $\beta<1$, we show that the limit density fluctuation field is an Ornstein-Uhlenbeck process defined on the Schwartz space if $\gamma>\frac12$, while for $\gamma = \frac12$ it is an energy solution of the stochastic Burgers equation. For $\gamma\geq\beta=1$, it is an Ornstein-Uhlenbeck process associated to the heat equation with Robin's boundary conditions. For $\gamma\geq\beta> 1$, the limit density fluctuation field is an Ornstein-Uhlenbeck process associated to the heat equation with Neumann's boundary conditions.
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