Electronic Journal of Probability

Convergence to the stochastic Burgers equation from a degenerate microscopic dynamics

Oriane Blondel, Patrícia Gonçalves, and Marielle Simon

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In this paper we prove the convergence to the stochastic Burgers equation from one-dimensional interacting particle systems, whose dynamics allow the degeneracy of the jump rates. To this aim, we provide a new proof of the second order Boltzmann-Gibbs principle introduced in [7]. The main technical difficulty is that our models exhibit configurations that do not evolve under the dynamics - the blocked configurations - and are locally non-ergodic. Our proof does not impose any knowledge on the spectral gap for the microscopic models. Instead, it relies on the fact that, under the equilibrium measure, the probability to find a blocked configuration in a finite box is exponentially small in the size of the box. Then, a dynamical mechanism allows to exchange particles even when the jump rate for the direct exchange is zero.

Article information

Electron. J. Probab., Volume 21 (2016), paper no. 69, 25 pp.

Received: 30 March 2016
Accepted: 12 November 2016
First available in Project Euclid: 1 December 2016

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Zentralblatt MATH identifier

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 60G60: Random fields 60F17: Functional limit theorems; invariance principles 35R60: Partial differential equations with randomness, stochastic partial differential equations [See also 60H15]

porous medium equation microscopic model degenerate rates density fluctuations stochastic Burgers equation weak KPZ universality

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Blondel, Oriane; Gonçalves, Patrícia; Simon, Marielle. Convergence to the stochastic Burgers equation from a degenerate microscopic dynamics. Electron. J. Probab. 21 (2016), paper no. 69, 25 pp. doi:10.1214/16-EJP15. https://projecteuclid.org/euclid.ejp/1480561218

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