## The Annals of Applied Probability

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

#### Abstract

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

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

Dates
Revised: February 2016
First available in Project Euclid: 19 July 2017

https://projecteuclid.org/euclid.aoap/1500451224

Digital Object Identifier
doi:10.1214/16-AAP1208

Mathematical Reviews number (MathSciNet)
MR3678472

Zentralblatt MATH identifier
1372.60133

#### Citation

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. https://projecteuclid.org/euclid.aoap/1500451224

#### References

• [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.