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Marcus-Lushnikov processes, Smoluchowski’s and Flory’s models. (English) Zbl 1169.60027
The paper studies the connection between a stochastic coalescence model, the Marcus-Lushnikov process, and two deterministic coagulation equations, the Smoluchowski and Flory equations. The Marcus-Lushnikov process is a finite stochastic particle system in which each particle is entirely characterized by its mass. Each pair of particles with masses $$x$$ and $$y$$ merges into a single particle at a given rate $$K(x,y)$$. The paper studies strongly gelling kernels behaving as $$K(x,y) = x^\alpha y + x y^\alpha$$ for some $$\alpha \in (0,1]$$. For such kernels it is well-known that gelation occurs, i.e. giant particles emerge. Two possible models for the hydrodynamic limit as the number of particles goes to infinity arise: the Smoluchowski equation, in which the giant particles are inert, and the Flory equation, in which the giant particles interact with the finite ones. The paper shows these limits in the Skorokhod topology using a suitable cut-off coagulation kernel in the Marcus-Lushnikov process. Furthermore, the asymptotic behaviour of the largest particle in the Marcus-Lushnikov process without cut-off is studied, and it is shown that there is only one giant particle. This single particle represents, asymptotically, the lost mass of the solution to the Flory equation. The paper is concluded with a small simulation study, where the theoretical results are demonstrated.

##### MSC:
 60K35 Interacting random processes; statistical mechanics type models; percolation theory 45K05 Integro-partial differential equations 60G57 Random measures 60H30 Applications of stochastic analysis (to PDEs, etc.) 60F99 Limit theorems in probability theory
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##### References:
 [1] Aldous, D.J., Emergence of the giant component in special marcus – lushnikov processes, Random structures algorithms, 12, 179-196, (1998) · Zbl 1002.60564 [2] Aldous, D.J., Deterministic and stochastic models for coalescence (aggregation, coagulation): A review of the Mean-field theory for probabilists, Bernoulli, 5, 3-48, (1999) · Zbl 0930.60096 [3] Escobedo, M.; Mischler, S.; Perthame, B., Gelation in coagulation and fragmentation models, Comm. math. phys., 231, 157-188, (2002) · Zbl 1016.82027 [4] Ethier, S.N.; Kurtz, T.G., Markov processes, characterization and convergence, (1986), Wiley & Sons · Zbl 0592.60049 [5] Fournier, N.; Giet, J.S., Convergence of the marcus – lushnikov process, Methodol. comput. appl. probab., 6, 219-231, (2004) · Zbl 1229.45012 [6] Jeon, I., Existence of gelling solutions for coagulation – fragmentation equations, Comm. math. phys., 194, 541-567, (1998) · Zbl 0910.60083 [7] Lushnikov, A., Some new aspects of coagulation theory, Izv. akad. nauk SSSR, ser. fiz. atmosfer. I okeana, 14, 738-743, (1978) [8] Marcus, A., Stochastic coalescence, Technometrics, 10, 133-143, (1968) [9] Norris, J.R., Smoluchowski’s coagulation equation: uniqueness, non-uniqueness and hydrodynamic limit for the stochastic coalescent, Ann. appl. probab., 9, 78-109, (1999) · Zbl 0944.60082
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