Derivation of the stochastic Burgers equation with Dirichlet boundary conditions from the WASEP
Annales Henri Lebesgue, Volume 3 (2020), pp. 87-167.

We consider the weakly asymmetric simple exclusion process on the discrete space {1,,n-1}(n), in contact with stochastic reservoirs, both with density ρ(0,1) at the extremity points, and starting from the invariant state, namely the Bernoulli product measure of parameter ρ. Under time diffusive scaling tn 2 and for ρ=1 2, when the asymmetry parameter is taken of order 1/n, we prove that the density fluctuations at stationarity are macroscopically governed by the energy solution of the stochastic Burgers equation with Dirichlet boundary conditions, which is shown to be unique and to exhibit different boundary behavior than the Cole–Hopf solution.

Nous considérons le processus d’exclusion simple faible asymétrique sur l’espace discret {1,,n-1} (n), en contact avec des réservoirs stochastiques à chaque extrémité, tous deux à densité fixée ρ(0,1). L’état initial du processus est sa mesure d’équilibre, c’est-à-dire la mesure de Bernoulli produit, de paramètre ρ. Dans l’échelle diffusive tn 2 , lorsque ρ=1 2, et lorsque le paramètre de faible asymétrie est de l’ordre de 1/n, nous prouvons que les fluctuations à l’équilibre de la densité sont régies macroscopiquement par la solution d’énergie de l’équation de Burgers stochastique, avec conditions au bord de type Dirichlet : en particulier, nous montrons que cette solution est unique et que son comportement au bord est différent de la solution de Cole–Hopf associée.

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DOI: 10.5802/ahl.28
Classification: 60K35, 60F17, 60H15
Keywords: Stochastic Burgers Equation, KPZ universality class, WASEP, Dirichlet boundary conditions
Gonçalves, Patrícia 1; Perkowski, Nicolas 2; Simon, Marielle 3

1 Center for Mathematical Analysis, Geometry and Dynamical Systems, Instituto Superior Técnico Universidade de Lisboa Av. Rovisco Pais 1049-001 Lisboa (Portugal)
2 Freie Universität Berlin FB Mathematik und Informatik Arnimallee 7 14195 Berlin (Germany)
3 Inria, Univ. Lille, CNRS UMR 8524 - Laboratoire Paul Painlevé 59000 Lille (France)
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Gonçalves, Patrícia; Perkowski, Nicolas; Simon, Marielle. Derivation of the stochastic Burgers equation with Dirichlet boundary conditions from the WASEP. Annales Henri Lebesgue, Volume 3 (2020), pp. 87-167. doi : 10.5802/ahl.28. http://archive.numdam.org/articles/10.5802/ahl.28/

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