Crossover to the stochastic Burgers equation for the WASEP with a slow bond

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 $\frac{1}{2}+\frac{a}{2n^{\gamma }}$ (resp. $\frac{1}{2}-\frac{a}{2n^{\gamma }}$) except at the bond of vertices $\{-1,0\}$ where the rate to the right (resp. left) is given by $\frac{\alpha }{2n^{\beta }}+\frac{a}{2n^{\gamma }}$ (resp. $\frac{\alpha }{2n^{\beta }}-\frac{a}{2n^{\gamma }}$). Above, $\alpha >0$, $\gamma \ge \beta \ge 0$, $a\ge 0$. For $\beta <1$, we show that the limit density fluctuation field is an Ornstein-Uhlenbeck process defined on the Schwartz space if $\gamma >\frac{1}{2}$, while for $\gamma =\frac{1}{2}$ it is an energy solution of the stochastic Burgers equation. For $\gamma \ge \beta =1$, it is an Ornstein-Uhlenbeck process associated to the heat equation with Robin's boundary conditions. For $\gamma \ge \beta >1$, the limit density fluctuation field is an Ornstein-Uhlenbeck process associated to the heat equation with Neumann's boundary conditions.

References in corpus (10)

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• Anomalous fluctuations for a perturbed Hamiltonian system with exponential interactions
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• Hydrodynamic profiles for the totally asymmetric exclusion process with a slow bond
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Cited by in corpus (10)

• The Kardar-Parisi-Zhang equation as scaling limit of weakly asymmetric interacting Brownian motions
• Energy solutions of KPZ are unique
• ASEP(q,j) converges to the KPZ equation
• Open ASEP in the Weakly Asymmetric Regime
• Some recent progress in singular stochastic PDEs
• KPZ fluctuations in finite volume
• Second order Boltzmann-Gibbs principle for polynomial functions and applications
• Non-Stationary KPZ equation from ASEP with slow bonds
• Mixing time for the asymmetric simple exclusion process in a random environment
• Phase transition in the KMP model with Slow/Fast boundaries