×

zbMATH — the first resource for mathematics

Stochastic Burgers and KPZ equations from particle systems. (English) Zbl 0874.60059
Summary: We consider two strictly related models: a solid on solid interface growth model and the weakly asymmetric exclusion process, both on the one-dimensional lattice. It has been proven that, in the diffusive scaling limit, the density field of the weakly asymmetric exclusion process evolves according to the Burgers equation [see A. de Masi, E. Presutti and E. Scacciatelli, Ann. Inst. Henri Poincaré, Probab. Stat. 25, No. 1, 1-38 (1989; Zbl 0664.60110), J. Gärtner, Stochastic Processes Appl. 27, No. 2, 233-260 (1988; Zbl 0643.60094), and C. Kipnis, S. Olla and S. R. S. Varadhan, Commun. Pure Appl. Math. 42, No. 2, 115-137 (1989; Zbl 0644.76001)] and the fluctuation field converges to a generalized Ornstein-Uhlenbeck process [see de Masi et al., loc. cit., and P. Dittrich and J. Gärtner, Math. Nachr. 151, 75-93 (1991; Zbl 0732.60112)]. We analyze instead the density fluctuations beyond the hydrodynamical scale and prove that their limiting distribution solves the (nonlinear) Burgers equation with a random noise on the density current. For the solid on solid model, we prove that the fluctuation field of the interface profile, if suitably rescaled, converges to the Kardar-Parisi-Zhang equation. This provides a microscopic justification of the so-called kinetic roughening, i.e. the non-Gaussian fluctuations in some non-equilibrium processes. Our main tool is the Cole-Hopf transformation and its microscopic version. We also develop a mathematical theory for the macroscopic equations.

MSC:
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
35Q53 KdV equations (Korteweg-de Vries equations)
PDF BibTeX XML Cite
Full Text: DOI