Abstract
We consider a continuous coagulation-fragmentation equation, which describes the concentration c(t,x) of particles of mass x∈[0,∞) at the instant t≥0 in a model where fragmentation and coalescence phenomena occur. We associate with this equation a nonlinear pure jump stochastic process, which is a stochastic microscopic description of the phenomenon. Existence is shown in a new case, where the total rate of fragmentation is infinite, and where we allow the presence of particles of mass 0. When coalescence is weaker than fragmentation, we study the appearance of particles of mass 0. We also show how to build solutions in the converse case where all particles at initial time have a mass 0. We finally study how the appearance of small particles leads to some regularization properties.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Reference
D. J. Aldous, Deterministic and stochastic models for coalescence (aggregation, coagulation): A review of the mean-field theory for probabilists, Bernoulli 5:3-48 (1999).
J. Bertoin, Homogeneous fragmentation processes, Probab. Theory Related Fields 121: 301-318 (2001).
J. Bertoin, Self-Similar Fragmentations, prépublication 610 du laboratoire de probabilités de Paris 6.
K. Bichteler and J. Jacod, Calcul de Malliavin pour les diffusions avec sauts: existence d'une densité dans le cas unidimensionnel, Séminaire de Probabilités XVII, L.N.M. 986 (Springer, 1983), pp. 132-157.
J. M. Bismut, Calcul des variations stochastique et processus de sauts, Z. Wahrsch. Verw. Gebiete 63:147-235 (1983).
M. Deaconu and E. Tanré, Smoluchowski's coagulation equation: Probabilistic interpretation of solutions for constant, additive, and multiplicative kernels, Annali della Scuola Normale Superiore di Pisa, Serie IV, Vol. XXIX, 3, 2000.
M. Deaconu, N. Fournier, andE. Tanré,A pure jump Markov process associated with Smoluchowski's coagulation equation,to appear inAnn. Probab.
N. Fournier and S. Méléard,A weak criterion of absolute continuity for jump processes, to appear in Bernoulli.
N. Fournier, Jumping SDEs: Absolute continuity using monotonicity, Stochastic Process. Appl. 98:317-330 (2002).
C. Graham and S. Méléard, Existence and regularity of a solution to a Kac equation without cutoff by using the stochastic calculus of variations, Commun. Math. Phys. 205: 551-569 (1999).
B. Haas, Loss of Mass in Deterministic and Random Fragmentations, prépublication 730 du laboratoire de probabilités de Paris 6.
J. Jacod and A. N. Shiryaev, Limit Theorems for Stochastic Processes (Springer, 1987).
B. Jourdain, Nonlinear Processes Associated with the Discrete Smoluchowski Coagulation Fragmentation Equation, Prépublication du CERMICS, 207, 2001.
P. Laurençcot, On a class of coagulation fragmentation equations, J. Differential Equations 167:245-274 (2000).
J. R. Norris, Smoluchowski's coagulation equation: uniqueness, non-unique-ness and hydrodynamic limit for the stochastic coalescent, Ann. Appl. Probab. 9:78-109 (1999).
H. Tanaka, Probabilistic treatment of the Boltzmann equation of Maxwellian molecules, Z. Wahr. Verw. Gebiete 46:67-105 (1978).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Fournier, N., Giet, JS. On Small Particles in Coagulation-Fragmentation Equations. Journal of Statistical Physics 111, 1299–1329 (2003). https://doi.org/10.1023/A:1023060417976
Issue Date:
DOI: https://doi.org/10.1023/A:1023060417976