Derivation of the stochastic Burgers equation with Dirichlet boundary conditions from the WASEP

Gonçalves, Patrícia; Perkowski, Nicolas; Simon, Marielle

doi:10.5802/ahl.28

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.

Metadata

DOI10.5802/ahl.28

Keywords Stochastic Burgers Equation, KPZ universality class, WASEP, Dirichlet boundary conditions

Abstract

We consider the weakly asymmetric simple exclusion process on the discrete space ${1, \dots, n - 1} (n \in ℕ)$ , in contact with stochastic reservoirs, both with density $ρ \in (0, 1)$ at the extremity points, and starting from the invariant state, namely the Bernoulli product measure of parameter $ρ$ . Under time diffusive scaling $t n^{2}$ and for $ρ = \frac{1}{2}$ , when the asymmetry parameter is taken of order $1 / \sqrt{n}$ , 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.

References

[BG97] Bertini, Lorenzo; Giacomin, Giambattista Stochastic Burgers and KPZ equations from particle systems, Commun. Math. Phys., Volume 183 (1997) no. 3, pp. 571-607 | DOI | MR | Zbl

[BGS16] Blondel, Oriane; Gonçalves, Patrícia; Simon, Marielle Convergence to the stochastic Burgers equation from a degenerate microscopic dynamics, Electron. J. Probab., Volume 21 (2016), 69, 25 pages | DOI | MR | Zbl

[CLO01] Chang, Chih-Chung; Landim, Claudio; Olla, Stefano Equilibrium fluctuations of asymmetric simple exclusion processes in dimension $d \geq 3$ , Probab. Theory Relat. Fields, Volume 119 (2001) no. 3, pp. 381-409 | DOI | Zbl

[Cor12] Corwin, Ivan The Kardar–Parisi–Zhang equation and universality class, Random Matrices Theory Appl., Volume 1 (2012) no. 1, 1130001, 76 pages | DOI | MR | Zbl

[CS18] Corwin, Ivan; Shen, Hao Open ASEP in the weakly asymmetric regime, Commun. Pure Appl. Math., Volume 71 (2018) no. 10, pp. 2065-2128 | DOI | MR | Zbl

[CST18] Corwin, Ivan; Shen, Hao; Tsai, Li-Cheng ASEP(q,j) converges to the KPZ equation, Ann. Inst. Henri Poincaré, Probab. Stat., Volume 54 (2018) no. 2, pp. 995-1012 | DOI | MR | Zbl

[CT17] Corwin, Ivan; Tsai, Li-Cheng KPZ equation limit of higher-spin exclusion processes, Ann. Probab., Volume 45 (2017) no. 3, pp. 1771-1798 | DOI | MR | Zbl

[DGP17] Diehl, Joscha; Gubinelli, Massimiliano; Perkowski, Nicolas The Kardar–Parisi–Zhang equation as scaling limit of weakly asymmetric interacting Brownian motions, Commun. Math. Phys., Volume 354 (2017) no. 2, pp. 549-589 | DOI | MR | Zbl

[DT16] Dembo, Amir; Tsai, Li-Cheng Weakly asymmetric non-simple exclusion process and the Kardar–Parisi–Zhang equation, Commun. Math. Phys., Volume 341 (2016) no. 1, pp. 219-261 | DOI | MR | Zbl

[FGN13] Franco, Tertuliano; Gonçalves, Patrícia; Neumann, Adriana Phase transition in equilibrium fluctuations of symmetric slowed exclusion, Stochastic Processes Appl., Volume 123 (2013) no. 12, pp. 4156-4185 | DOI | MR | Zbl

[FGN17] Franco, Tertuliano; Gonçalves, Patrícia; Neumann, Adriana Equilibrium fluctuations for the slow boundary exclusion process, From Particle Systems to Partial Differential Equations (Springer Proceedings in Mathematics & Statistics) Volume 209 (2017), pp. 177-197 | MR | Zbl

[FGS16] Franco, Tertuliano; Gonçalves, Patrícia; Simon, Marielle Crossover to the stochastic Burgers equation for the WASEP with a slow bond, Commun. Math. Phys., Volume 346 (2016) no. 3, pp. 801-838 | DOI | MR | Zbl

[Fre85] Freidlin, Mark Functional integration and partial differential equations, Annals of Mathematics Studies, Volume 109, Princeton University Press, 1985, x+545 pages | DOI | MR | Zbl

[GH18] Gerencsér, Máté; Hairer, Martin Singular SPDEs in domains with boundaries, Probab. Theory Relat. Fields, Volume 173 (2018) no. 3-4, pp. 697-758 | DOI | MR | Zbl

[GIP15] Gubinelli, Massimiliano; Imkeller, Peter; Perkowski, Nicolas Paracontrolled distributions and singular PDEs, Forum Math. Pi, Volume 3 (2015), e6, 75 pages | MR | Zbl

[GJ13] Gubinelli, Massimiliano; Jara, Milton Regularization by noise and stochastic Burgers equations, Stoch. Partial Differ. Equ., Anal. Comput., Volume 1 (2013) no. 2, pp. 325-350 | MR | Zbl

[GJ14] Gonçalves, Patrícia; Jara, Milton Nonlinear fluctuations of weakly asymmetric interacting particle systems, Arch. Ration. Mech. Anal., Volume 212 (2014) no. 2, pp. 597-644 | DOI | MR | Zbl

[GJ17] Gonçalves, Patrícia; Jara, Milton Stochastic Burgers equation from long range exclusion interactions, Stochastic Processes Appl., Volume 127 (2017) no. 12, pp. 4029-4052 | DOI | MR | Zbl

[GJS15] Gonçalves, Patrícia; Jara, Milton; Sethuraman, Sunder A stochastic Burgers equation from a class of microscopic interactions, Ann. Probab., Volume 43 (2015) no. 1, pp. 286-338 | DOI | MR | Zbl

[GJS17] Gonçalves, Patrícia; Jara, Milton; Simon, Marielle Second order Boltzmann–Gibbs principle for polynomial functions and applications, J. Stat. Phys., Volume 166 (2017) no. 1, pp. 90-113 | DOI | MR | Zbl

[GLM17] Gonçalves, Patrícia; Landim, Claudio; Milanés, Aniura Nonequilibrium fluctuations of one-dimensional boundary driven weakly asymmetric exclusion processes, Ann. Appl. Probab., Volume 27 (2017) no. 1, pp. 140-177 | DOI | MR | Zbl

[Gon08] Gonçalves, Patrícia Central limit theorem for a tagged particle in asymmetric simple exclusion, Stochastic Processes Appl., Volume 118 (2008) no. 3, pp. 474-502 | DOI | MR | Zbl

[GP16] Gubinelli, Massimiliano; Perkowski, Nicolas The Hairer–Quastel universality result at stationarity, RIMS Kôkyûroku Bessatsu, Volume B59 (2016), pp. 101-115 | Zbl

[GP18a] Gubinelli, Massimiliano; Perkowski, Nicolas Energy solutions of KPZ are unique, J. Am. Math. Soc., Volume 31 (2018) no. 2, pp. 427-471 | DOI | MR | Zbl

[GP18b] Gubinelli, Massimiliano; Perkowski, Nicolas Probabilistic approach to the stochastic Burgers equation, Stochastic partial differential equations and related fields (Springer Proceedings in Mathematics & Statistics) Volume 229 (2018), pp. 515-527 | DOI | MR | Zbl

[Gär87] Gärtner, Jürgen Convergence towards Burger’s equation and propagation of chaos for weakly asymmetric exclusion processes, Stochastic Processes Appl., Volume 27 (1987) no. 2, pp. 233-260 | DOI | MR | Zbl

[Hai13] Hairer, Martin Solving the KPZ equation, Ann. Math., Volume 178 (2013) no. 2, pp. 559-664 | DOI | MR | Zbl

[Hai14] Hairer, Martin A theory of regularity structures, Invent. Math., Volume 198 (2014) no. 2, pp. 269-504 | DOI | MR | Zbl

[Jan97] Janson, Svante Gaussian Hilbert spaces, Cambridge Tracts in Mathematics, Volume 129, Cambridge University Press, 1997, x+340 pages | DOI | MR | Zbl

[KL99] Kipnis, Claude; Landim, Claudio Scaling limits of interacting particle systems, Grundlehren der Mathematischen Wissenschaften, Volume 320, Springer, 1999 | MR | Zbl

[KLO12] Komorowski, Tomasz; Landim, Claudio; Olla, Stefano Fluctuations in Markov processes, Grundlehren der Mathematischen Wissenschaften, Volume 345, Springer, 2012, xviii+494 pages | MR | Zbl

[KPZ86] Kardar, Mehran; Parisi, Giorgio; Zhang, Yi-Cheng Dynamic scaling of growing interfaces, Phys. Rev. Lett., Volume 56 (1986) no. 9, pp. 889-892 | DOI | Zbl

[LCL07] Lyons, Terry J.; Caruana, Michael; Lévy, Thierry Differential equations driven by rough paths, Lecture Notes in Mathematics, Volume 1908, Springer, 2007, xviii+109 pages | MR | Zbl

[LMO08] Landim, Claudio; Milanés, Aniura; Olla, Stefano Stationary and non-equilibrium fluctuations in boundary driven exclusion processes, Markov Process. Relat. Fields, Volume 14 (2008) no. 2, pp. 165-184 | Zbl

[Nua06] Nualart, David The Malliavin calculus and related topics, Probability and Its Applications, Springer, 2006, xiv+382 pages | MR | Zbl

[Pap90] Papanicolaou, Vassilis G. The probabilistic solution of the third boundary value problem for second order elliptic equations, Probab. Theory Relat. Fields, Volume 87 (1990) no. 1, pp. 27-77 | DOI | MR | Zbl

[Par19] Parekh, Shalin The KPZ Limit of ASEP with Boundary, Commun. Math. Phys., Volume 365 (2019) no. 2, pp. 569-649 | DOI | MR | Zbl

[QS15] Quastel, Jeremy; Spohn, Herbert The one-dimensional KPZ equation and its universality class, J. Stat. Phys., Volume 160 (2015) no. 4, pp. 965-984 | DOI | MR | Zbl

[Qua12] Quastel, Jeremy Introduction to KPZ, Current developments in mathematics, 2011, International Press., 2012, pp. 125-194 | MR | Zbl

[Spo17] Spohn, Herbert The Kardar–Parisi–Zhang equation: a statistical physics perspective, Stochastic processes and random matrices, Oxford University Press, 2017, pp. 177-227 | MR | Zbl

[Wal86] Walsh, John B. An introduction to stochastic partial differential equations, École d’été de probabilités de Saint-Flour, XIV—1984 (Lecture Notes in Mathematics) Volume 1180, Springer, 1986, pp. 265-439 | DOI | MR | Zbl

[You36] Young, Laurence C. An inequality of the Hölder type, connected with Stieltjes integration, Acta Math., Volume 67 (1936) no. 1, pp. 251-282 | DOI | Zbl

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