The Annals of Applied Probability

Hydrodynamic limits and propagation of chaos for interacting random walks in domains

Zhen-Qing Chen and Wai-Tong (Louis) Fan

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A new non-conservative stochastic reaction–diffusion system in which two families of random walks in two adjacent domains interact near the interface is introduced and studied in this paper. Such a system can be used to model the transport of positive and negative charges in a solar cell or the population dynamics of two segregated species under competition. We show that in the macroscopic limit, the particle densities converge to the solution of a coupled nonlinear heat equations. For this, we first prove that propagation of chaos holds by establishing the uniqueness of a new BBGKY hierarchy. A local central limit theorem for reflected diffusions in bounded Lipschitz domains is also established as a crucial tool.

Article information

Ann. Appl. Probab., Volume 27, Number 3 (2017), 1299-1371.

Received: September 2014
Revised: February 2016
First available in Project Euclid: 19 July 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60F17: Functional limit theorems; invariance principles 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 92D15: Problems related to evolution

Hydrodynamic limit propagation of chaos interacting particle system random walk annihilation reflecting diffusion boundary local time heat kernel coupled nonlinear partial differential equation BBGKY hierarchy Duhamel tree expansion isoperimetric inequality


Chen, Zhen-Qing; Fan, Wai-Tong (Louis). Hydrodynamic limits and propagation of chaos for interacting random walks in domains. Ann. Appl. Probab. 27 (2017), no. 3, 1299--1371. doi:10.1214/16-AAP1208.

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  • [1] Bass, R. F. and Hsu, P. (1991). Some potential theory for reflecting Brownian motion in Hölder and Lipschitz domains. Ann. Probab. 19 486–508.
  • [2] Bass, R. F. and Kumagai, T. (2008). Symmetric Markov chains on $\mathbb{Z}^{d}$ with unbounded range. Trans. Amer. Math. Soc. 360 2041–2075.
  • [3] Boldrighini, C., De Masi, A. and Pellegrinotti, A. (1992). Nonequilibrium fluctuations in particle systems modelling reaction–diffusion equations. Stochastic Process. Appl. 42 1–30.
  • [4] Boldrighini, C., De Masi, A., Pellegrinotti, A. and Presutti, E. (1987). Collective phenomena in interacting particle systems. Stochastic Process. Appl. 25 137–152.
  • [5] Burdzy, K. and Chen, Z.-Q. (2008). Discrete approximations to reflected Brownian motion. Ann. Probab. 36 698–727.
  • [6] Burdzy, K. and Chen, Z.-Q. (2013). Reflecting random walk in fractal domains. Ann. Probab. 41 2791–2819.
  • [7] Burdzy, K., Hołyst, R. and March, P. (2000). A Fleming–Viot particle representation of the Dirichlet Laplacian. Comm. Math. Phys. 214 679–703.
  • [8] Burdzy, K. and Quastel, J. (2006). An annihilating-branching particle model for the heat equation with average temperature zero. Ann. Probab. 34 2382–2405.
  • [9] Carlen, E. A., Kusuoka, S. and Stroock, D. W. (1987). Upper bounds for symmetric Markov transition functions. Ann. Inst. Henri Poincaré Probab. Stat. 23 245–287.
  • [10] Chen, M.-F. (2004). From Markov Chains to Non-equilibrium Particle Systems, 2nd ed. World Scientific, River Edge, NJ.
  • [11] Chen, Z. Q. (1993). On reflecting diffusion processes and Skorokhod decompositions. Probab. Theory Related Fields 94 281–315.
  • [12] Chen, Z.-Q. and Fan, W.-T. (2017). Systems of interacting diffusions with partial annihilations through membranes. Ann. Probab. 45 100–146.
  • [13] Chen, Z.-Q. and Fukushima, M. (2012). Symmetric Markov Processes, Time Change, and Boundary Theory. Princeton Univ. Press, Princeton, NJ.
  • [14] Chen, Z. Q., Williams, R. J. and Zhao, Z. (1995). On the existence of positive solutions for semilinear elliptic equations with Neumann boundary conditions. Probab. Theory Related Fields 101 251–276.
  • [15] Cox, J. T., Durrett, R. and Perkins, E. A. (2013). Voter model perturbations and reaction diffusion equations. Astérisque 349 vi+113.
  • [16] David, G. (1988). Morceaux de graphes lipschitziens et intégrales singulières sur une surface. Rev. Mat. Iberoam. 4 73–114.
  • [17] David, G. and Semmes, S. (1991). Singular integrals and rectifiable sets in $\textbf{R}^{n}$: Beyond Lipschitz graphs. Astérisque 193 152.
  • [18] Delmotte, T. (1999). Parabolic Harnack inequality and estimates of Markov chains on graphs. Rev. Mat. Iberoam. 15 181–232.
  • [19] Dittrich, P. (1988). A stochastic model of a chemical reaction with diffusion. Probab. Theory Related Fields 79 115–128.
  • [20] Dittrich, P. (1988). A stochastic particle system: Fluctuations around a nonlinear reaction–diffusion equation. Stochastic Process. Appl. 30 149–164.
  • [21] Durrett, R. and Levin, S. (1994). The importance of being discrete (and spatial). Theor. Popul. Biol. 46 363–394.
  • [22] Erdös, L., Schlein, B. and Yau, H.-T. (2007). Derivation of the cubic non-linear Schrödinger equation from quantum dynamics of many-body systems. Invent. Math. 167 515–614.
  • [23] Ethier, S. N. and Kurtz, T. G. (1986). Markov Processes: Characterization and Convergence. Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics. Wiley, New York.
  • [24] Fan, W.-T. (2014). Systems of reflected diffusions with interactions through membranes. Ph.D. thesis.
  • [25] Funaki, T. (1999). Free boundary problem from stochastic lattice gas model. Ann. Inst. Henri Poincaré Probab. Stat. 35 573–603.
  • [26] Golse, F. (2005). Hydrodynamic limits. In European Congress of Mathematics 699–717. Eur. Math. Soc., Zürich.
  • [27] Guo, M. Z., Papanicolaou, G. C. and Varadhan, S. R. S. (1988). Nonlinear diffusion limit for a system with nearest neighbor interactions. Comm. Math. Phys. 118 31–59.
  • [28] Gyrya, P. and Saloff-Coste, L. (2011). Neumann and Dirichlet heat kernels in inner uniform domains. Astérisque 336 viii+144.
  • [29] Kipnis, C. and Landim, C. (1999). Scaling Limits of Interacting Particle Systems. Springer, Berlin.
  • [30] Kipnis, C., Olla, S. and Varadhan, S. R. S. (1989). Hydrodynamics and large deviation for simple exclusion processes. Comm. Pure Appl. Math. 42 115–137.
  • [31] Kotelenez, P. (1986). Law of large numbers and central limit theorem for linear chemical reactions with diffusion. Ann. Probab. 14 173–193.
  • [32] Kotelenez, P. (1988). High density limit theorems for nonlinear chemical reactions with diffusion. Probab. Theory Related Fields 78 11–37.
  • [33] Kurtz, T. G. (1971). Limit theorems for sequences of jump Markov processes approximating ordinary differential processes. J. Appl. Probab. 8 344–356.
  • [34] Kurtz, T. G. (1981). Approximation of Population Processes. CBMS-NSF Regional Conference Series in Applied Mathematics 36. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA.
  • [35] Lang, R. and Nguyen, X.-X. (1980). Smoluchowski’s theory of coagulation in colloids holds rigorously in the Boltzmann–Grad-limit. Z. Wahrsch. Verw. Gebiete 54 227–280.
  • [36] May, R. M. and Nowak, M. A. (1992). Evolutionary games and spatial chaos. Nature 359 826–829.
  • [37] Maz’ja, V. G. (1985). Sobolev Spaces. Springer, Berlin.
  • [38] Saloff-Coste, L. (1997). Lectures on finite Markov chains. In Lectures on Probability Theory and Statistics (Saint-Flour, 1996). Lecture Notes in Math. 1665 301–413. Springer, Berlin.
  • [39] Stanley, R. P. (1999). Enumerative Combinatorics. Vol. 2. Cambridge Univ. Press, Cambridge.
  • [40] Stroock, D. W. (1988). Diffusion semigroups corresponding to uniformly elliptic divergence form operators. In Séminaire de Probabilités, XXII. Lecture Notes in Math. 1321 316–347. Springer, Berlin.
  • [41] Stroock, D. W. and Zheng, W. (1997). Markov chain approximations to symmetric diffusions. Ann. Inst. Henri Poincaré Probab. Stat. 33 619–649.
  • [42] Yau, H.-T. (1991). Relative entropy and hydrodynamics of Ginzburg–Landau models. Lett. Math. Phys. 22 63–80.