The Annals of Probability

A pure jump Markov process associated with Smoluchowski's coagulation equation

Madalina Deaconu, Nicolas Fournier, and Etienne Tanré

Full-text: Open access


The aim of the present paper is to construct a stochastic process, whose law is the solution of the Smoluchowski's coagulation equation. We introduce first a modified equation, dealing with the evolution of the distribution $Q_t(dx)$ of the mass in the system. The advantage we take on this is that we can perform an unified study for both continuous and discrete models.

The integro-partial-differential equation satisfied by $\{Q_t\}_{t\geq 0}$ can be interpreted as the evolution equation of the time marginals of a Markov pure jump process. At this end we introduce a nonlinear Poisson driven stochastic differential equation related to the Smoluchowski equation in the following way: if $X_t$ satisfies this stochastic equation, then the law of $X_t$ satisfies the modified Smoluchowski equation. The nonlinear process is richer than the Smoluchowski equation, since it provides historical information on the particles.

Existence, uniqueness and pathwise behavior for the solution of this SDE are studied. Finally, we prove that the nonlinear process X can be obtained as the limit of a Marcus-Lushnikov procedure.

Article information

Ann. Probab., Volume 30, Number 4 (2002), 1763-1796.

First available in Project Euclid: 10 December 2002

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60H30: Applications of stochastic analysis (to PDE, etc.) 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 60J75: Jump processes

Smoluchowski's coagulation equations nonlinear stochastic differential equations Poisson measures


Deaconu, Madalina; Fournier, Nicolas; Tanré, Etienne. A pure jump Markov process associated with Smoluchowski's coagulation equation. Ann. Probab. 30 (2002), no. 4, 1763--1796. doi:10.1214/aop/1039548371.

Export citation


  • [1] ALDOUS, D. J. (1999). Deterministic and stochastic models for coalescence (aggregation, coagulation): a review of the mean-field theory for probabilists. Bernoulli 5 3-48.
  • [2] BALL, J. M. and CARR, J. (1990). The discrete coagulation-fragmentation equations: existence, uniqueness, and density conservation. J. Statist. Phy s. 61 203-234.
  • [3] BEESACK, P. R. (1975). Gronwall Inequalities. Carleton Math. Lecture Notes 11. Carleton Univ.
  • [4] DEACONU, M. and TANRÉ, E. (2000). Smoluchowski's coagulation equation: probabilistic interpretation of solutions for constant, additive and multiplicative kernels. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 29 549-579.
  • [5] DESVILLETTES, L., GRAHAM, C. and MÉLÉARD, S. (1999). Probabilistic interpretation and numerical approximation of a Kac equation without cutoff. Stochastic Process. Appl. 84 115-135.
  • [6] FOURNIER, N. and MÉLÉARD, S. (2000). Existence results for 2D homogeneous Boltzmann equations without cutoff and for non-Maxwell molecules by use of Malliavin calculus. Prépublication 622 du Laboratoire de probabilités et modèles aléatoires, Paris 6 et 7.
  • [7] FOURNIER, N. and MÉLÉARD, S. (2001). A Markov process associated with a Boltzmann equation without cutoff and for non-Maxwell molecules. J. Statist. Phy s. 104 359-385.
  • [8] GRAHAM, C. and MÉLÉARD, S. (1997). Stochastic particle approximations for generalized Boltzmann models and convergence estimates. Ann. Probab. 25 115-132.
  • [9] GRAHAM, C. and MÉLÉARD, S. (1999). Existence and regularity of a solution to a Kac equation without cutoff by using the stochastic calculus of variations. Comm. Math. Phy s. 205 551-569.
  • [10] HEILMANN, O. J. (1992). Analy tical solutions of Smoluchowski's coagulation equation. J. Phy s. Ser. A 25 3763-3771.
  • [11] JACOD, J. and SHIRy AEV, A. N. (1987). Limit Theorems for Stochastic Processes. Springer, Berlin.
  • [12] JEON, I. (1998). Existence of gelling solutions for coagulation-fragmentation equations. Comm. Math. Phy s. 194 541-567.
  • [13] LUSHNIKOV, A. A. (1978). Certain new aspects of the coagulation theory. Izv. Atmos. Ocean. Phy s. 14 738-743.
  • [14] MARCUS, A. H. (1968). Stochastic coalescence. Technometrics 10 133-143.
  • [15] MÉLÉARD, S. (1996). Asy mptotic behaviour of some interacting particle sy stems; McKean- Vlasov and Boltzmann models. Probabilistic Models for Nonlinear Partial Differential Equations. Lecture Notes in Math. 1627 42-95. Springer, Berlin.
  • [16] NORRIS, J. R. (1999). Smoluchowski's coagulation equation: uniqueness, nonuniqueness and hy drody namic limit for the stochastic coalescent. Ann. Appl. Probab. 9 78-109.
  • [17] NORRIS, J. R. (2000). Cluster coagulation. Comm. Math. Phy s. 209 407-435.
  • [18] SZNITMAN, A. S. (1984). Équations de ty pe de Boltzmann, spatialement homogènes. Z. Wahrsch. Verw. Gebiete 66 559-592.
  • [19] TANAKA, H. (1978). Probabilistic treatment of the Boltzmann equation of Maxwellian molecules. Z. Wahrsch. Verw. Gebiete 46 67-105.
  • [20] TANAKA, H. (1978). On the uniqueness of Markov process associated with the Boltzmann equation of Maxwellian molecules. In Proceedings of the International Sy mposium on Stochastic Differential Equations 409-425. Wiley, New York.