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Communications in Mathematical Physics

, Volume 346, Issue 3, pp 801–838 | Cite as

Crossover to the Stochastic Burgers Equation for the WASEP with a Slow Bond

  • Tertuliano Franco
  • Patrícia GonçalvesEmail author
  • Marielle Simon
Article

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, \({\alpha > 0}\), \({\gamma \geq \beta \geq 0}\), \({a\geq 0}\). For \({\beta < 1}\), we show that the limit density fluctuation field is an Ornstein–Uhlenbeck process defined on the Schwartz space if \({\gamma > \frac{1}{2}}\), while for \({\gamma = \frac{1}{2}}\) 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.

Keywords

Dirichlet Form Uhlenbeck Process Interact Particle System Asymmetric Simple Exclusion Process Stochastic Burger Equation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  • Tertuliano Franco
    • 1
  • Patrícia Gonçalves
    • 2
    • 3
    Email author
  • Marielle Simon
    • 4
  1. 1.UFBAInstituto de MatemáticaSalvadorBrazil
  2. 2.Departamento de MatemáticaPUC-RIORio de JaneiroBrazil
  3. 3.CMATCentro de Matemática da Universidade do MinhoBragaPortugal
  4. 4.Équipe MEPHYSTOInria Lille, Nord EuropeVilleneuve d’AscqFrance

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