We consider the weakly asymmetric simple exclusion process on the discrete space , in contact with stochastic reservoirs, both with density at the extremity points, and starting from the invariant state, namely the Bernoulli product measure of parameter . Under time diffusive scaling and for , when the asymmetry parameter is taken of order , 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 (), en contact avec des réservoirs stochastiques à chaque extrémité, tous deux à densité fixée . L’état initial du processus est sa mesure d’équilibre, c’est-à-dire la mesure de Bernoulli produit, de paramètre . Dans l’échelle diffusive , lorsque , et lorsque le paramètre de faible asymétrie est de l’ordre de , 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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Keywords: Stochastic Burgers Equation, KPZ universality class, WASEP, Dirichlet boundary conditions
@article{AHL_2020__3__87_0, author = {Gon\c{c}alves, Patr{\'\i}cia and Perkowski, Nicolas and Simon, Marielle}, title = {Derivation of the stochastic {Burgers} equation with {Dirichlet} boundary conditions from the {WASEP}}, journal = {Annales Henri Lebesgue}, pages = {87--167}, publisher = {\'ENS Rennes}, volume = {3}, year = {2020}, doi = {10.5802/ahl.28}, language = {en}, url = {http://archive.numdam.org/articles/10.5802/ahl.28/} }
TY - JOUR AU - Gonçalves, Patrícia AU - Perkowski, Nicolas AU - Simon, Marielle TI - Derivation of the stochastic Burgers equation with Dirichlet boundary conditions from the WASEP JO - Annales Henri Lebesgue PY - 2020 SP - 87 EP - 167 VL - 3 PB - ÉNS Rennes UR - http://archive.numdam.org/articles/10.5802/ahl.28/ DO - 10.5802/ahl.28 LA - en ID - AHL_2020__3__87_0 ER -
%0 Journal Article %A Gonçalves, Patrícia %A Perkowski, Nicolas %A Simon, Marielle %T Derivation of the stochastic Burgers equation with Dirichlet boundary conditions from the WASEP %J Annales Henri Lebesgue %D 2020 %P 87-167 %V 3 %I ÉNS Rennes %U http://archive.numdam.org/articles/10.5802/ahl.28/ %R 10.5802/ahl.28 %G en %F AHL_2020__3__87_0
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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