Interface fluctuations in the two-dimensional weakly asymmetric simple exclusion process. (English) Zbl 0757.60105
Summary: We consider the two-dimensional weakly asymmetric simple exclusion process, where the asymmetry is along the \(X\)-axis. The generator for such a process can be written as \(\varepsilon^{-2} L_ 0+\varepsilon^{-1} L_ \alpha\), \(\varepsilon>0\), where \(L_ 0\) and \(L_ \alpha\) are the generators for the nearest neighbor symmetric simple exclusion and totally asymmetric simple exclusion, respectively. We prove propagation of chaos and convergence to Burgers equation with viscosity in the limit as \(\varepsilon\) goes to zero. The density fluctuation field converges to a generalized Ornstein-Uhlenbeck process. The covariance kernel for a class of travelling wave solutions is consistent with a phase boundary which fluctuates according to a linear stochastic partial differential equation.
MSC:
| 60K35 | Interacting random processes; statistical mechanics type models; percolation theory |
| 60J70 | Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.) |
Keywords:
Ornstein-Uhlenbeck process; weakly asymmetric simple exclusion process; propagation of chaos; Burgers equation; travelling wave solutions; stochastic partial differential equationReferences:
| [1] | Andjel, E. D.; Bramson, M. D.; Liggett, T. M., Shocks in the asymmetric exclusion process, Probab. Theory Rel. Fields, 78, 231-247 (1988) · Zbl 0632.60107 |
| [2] | Bennasi, A.; Fouque, J. P., Hydrodynamic limit for the asymmetric exclusion process, Ann. Probab., 15, 546-560 (1987) · Zbl 0623.60120 |
| [3] | Calderoni, P.; Pulvirenti, M., Propagation of chaos for Burgers equation, Ann. Inst. H. Poincaré Sect. A, Phys. Theor., 29, 85-97 (1983) · Zbl 0526.60057 |
| [4] | De Masi, A.; Presutti, E.; Scacciatelli, E., The weakly asymmetric simple exclusion process, Ann. Inst. H. Poincaré Probab. Statist., 25, 1-38 (1989) · Zbl 0664.60110 |
| [5] | De Masi, A.; Presutti, E., An introductory course on the collective behavior of particle systems, CARR Repts. in Math. Phys. (1989) |
| [6] | De Masi, A.; Janiro, N.; Pellegrinotti, A.; Presutti, E., A survey of hydrodynamical behavior in many particle systems, (Lebowitz, J. L.; Montroll, E. W., Nonequilibrium Phenomena, Vol. 11 (1984), North-Holland: North-Holland Amsterdam) · Zbl 0567.76006 |
| [7] | Ferrari, P. A.; Kipnis, C.; Saada, E., Microscopic structure of travelling waves in the asymmetric simple exclusion process, Ann. Probab., 19, 226-244 (1991) · Zbl 0725.60113 |
| [8] | Ferrari, P. A., Shock fluctuation in asymmetric simple exclusion, Probab. Theory Rel. Fields, 91, 81-101 (1992) · Zbl 0744.60117 |
| [9] | Ferrari, P. A.; Presutti, E.; Vares, M. E., Nonequilibrium fluctuations for a zero range process, Ann. Inst. H. Poincaré, 24, 237-268 (1988) · Zbl 0653.60099 |
| [10] | Gartner, J.; Presutti, E., Shock fluctuations in particle systems, Ann. Inst. H. Poincaré Sect. B, 53, 1-14 (1990) · Zbl 0705.76054 |
| [11] | Holley, R.; Stroock, D. W., Generalized Ornstein-Uhlenbeck processes and infinite branching Brownian motions, Kyoto Univ. Res. Inst. Math. Public., A14, 741 (1978) · Zbl 0412.60065 |
| [12] | Landim, C., Hydrodynamical equation for attractive particle systems on \(Z^d\), Ann. Probab., 19, 1537-1558 (1991) · Zbl 0798.60084 |
| [13] | Liggett, T. M., Interacting Particle Systems (1985), Springer: Springer New York · Zbl 0559.60078 |
| [14] | McKean, H., Propagation of chaos for a class of parabolic equations, (Lecture series in Diff. equ. (1967), Catholic Univ), 1-57 |
| [15] | Metivier, M., Sufficient conditions for tightness and weak convergence of a sequence of processes, Internal Rept. of Univ. of Minnesota (1980) |
| [16] | Mitoma, I., Tightness for probabilities in \(C\)([0, 1], S’) and \(D\)([0, 1], S’), Ann. Probab., 11, 989 (1983) · Zbl 0527.60004 |
| [17] | Ravishankar, K., Fluctuations from the hydrodynamical limit for the symmetric simple exclusion in \(Z^2\), Stochastic Process. Appl., 42, 31-37 (1992) · Zbl 0754.60127 |
| [18] | Ravishankar, K., Interface fluctuations in the two dimensional weakly asymmetric simple exclusion, CARR Reports in Math. Phys. (1989) |
| [19] | Rebolledo, R., Sur l’existence de solutions certains problèmes de sémimartingales, C.R. Acad. Sci. Paris, 290, 843 (1980) · Zbl 0435.60042 |
| [20] | Sznitmann, A., A propagation of chaos result for Burgers equation, Probab. Theory Rel. Fields, 71, 581-613 (1988) · Zbl 0597.60055 |
| [21] | Smoller, J., Shock Waves and Reaction-Diffusion Equations (1983), Springer: Springer New York · Zbl 0508.35002 |
| [22] | Spohn, H., Dynamics of Systems with many Particles (1991), Springer: Springer New York |
| [23] | Walsh, J. B., An introduction to stochastic partial differential equations, (Lecture Notes on Math. (1984), Springer: Springer New York), 266-437, No. 1180 · Zbl 0608.60060 |
| [24] | Wick, W. D., A dynamical phase transition in an infinite particle system, J. Statist. Phys., 38, 1005-1025 (1985) · Zbl 0625.76080 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
