The Annals of Applied Probability

Brownian coagulation and a version of Smoluchowski’s equation on the circle

Inés Armendáriz

Full-text: Open access


We introduce a one-dimensional stochastic system where particles perform independent diffusions and interact through pairwise coagulation events, which occur at a nontrivial rate upon collision. Under appropriate conditions on the diffusion coefficients, the coagulation rates and the initial distribution of particles, we derive a spatially inhomogeneous version of the mass flow equation as the particle number tends to infinity. The mass flow equation is in one-to-one correspondence with Smoluchowski’s coagulation equation. We prove uniqueness for this equation in a broad class of solutions, to which the weak limit of the stochastic system is shown to belong.

Article information

Ann. Appl. Probab., Volume 20, Number 2 (2010), 660-695.

First available in Project Euclid: 9 March 2010

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 82C21: Dynamic continuum models (systems of particles, etc.)

Coagulating particle systems hydrodynamic limit Smoluchowski’s equations


Armendáriz, Inés. Brownian coagulation and a version of Smoluchowski’s equation on the circle. Ann. Appl. Probab. 20 (2010), no. 2, 660--695. doi:10.1214/09-AAP633.

Export citation


  • [1] Aldous, D. J. (1999). Deterministic and stochastic models for coalescence (aggregation and coagulation): A review of the mean-field theory for probabilists. Bernoulli 5 3–48.
  • [2] Billingsley, P. (1999). Convergence of Probability Measures, 2nd ed. Wiley, New York.
  • [3] Deaconu, M. and Fournier, N. (2002). Probabilistic approach of some discrete and continuous coagulation equations with diffusion. Stochastic Process. Appl. 101 83–111.
  • [4] Eibeck, A. and Wagner, W. (2001). Stochastic particle approximations for Smoluchoski’s coagulation equation. Ann. Appl. Probab. 11 1137–1165.
  • [5] Fournier, N. and Giet, J.-S. (2004). Convergence of the Marcus–Lushnikov process. Methodol. Comput. Appl. Probab. 6 219–231.
  • [6] Ikeda, N. and Watanabe, S. (1981). Stochastic Differential Equations and Diffusion Processes. North-Holland Mathematical Library 24. North-Holland, Amsterdam.
  • [7] Grosskinsky, S., Klingenberg, C. and Oelschläger, K. (2003). A rigorous derivation of Smoluchowski’s equation in the moderate limit. Stoch. Anal. Appl. 22 113–141.
  • [8] Jeon, I. (1998). Existence of gelling solutions for coagulation-fragmentation equations. Comm. Math. Phys. 194 541–567.
  • [9] Hammond, A. and Rezakhanlou, F. (2006). Kinetic limit for a system of coagulating planar Brownian particles. J. Stat. Phys. 124 997–1040.
  • [10] Hammond, A. and Rezakhanlou, F. (2007). The kinetic limit of a system of coagulating Brownian particles. Arch. Ration. Mech. Anal. 185 1–67.
  • [11] Kipnis, C. and Landim, C. (1999). Scaling Limits of Interacting Particle Systems. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 320. Springer, Berlin.
  • [12] Lang, R. and Xanh, N. X. (1980). Smoluchowski’s theory of coagulation in colloids holds rigorously in the Boltzmann–Grad-limit. Z. Wahrsch. Verw. Gebiete 54 227–280.
  • [13] Lushnikov, A. (1978). Certain new aspects of the coagulation theory. Izv. Math. 14 738–743.
  • [14] Marcus, A. H. (1968). Stochastic coalescence. Technometrics 10 133–143.
  • [15] Norris, J. R. (1999). Smoluchowski’s coagulation equation: Uniqueness, nonuniqueness and a hydrodynamic limit for the stochastic coalescent. Ann. Appl. Probab. 9 78–109.
  • [16] Norris, J. R. (2006). Notes on Brownian coagulation. Markov Process. Related Fields 12 407–412.
  • [17] Revuz, D. and Yor, M. (1999). Continuous Martingales and Brownian Motion, 3rd ed. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 293. Springer, Berlin.
  • [18] Van Smoluchowski, M. (1916). Drei vorträge über diffusion, brownsche bewegung und koagulation von kolloidteilchen. Phys. Z. 17 557–585.