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Stochastic Semigroups and Coagulation Equations

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Abstract

A general class of bilinear systems of discrete or continuous coagulation equations is considered. It is shown that their solutions can be approximated by the solutions of appropriate stochastic systems describing the coagulation process in terms of stochastic semigroups.

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REFERENCES

  1. J. M. Ball and J. Carr, “The discrete coagulation-fragmentation equations: existence, uniqueness and density conservation,” J. Statist. Phys., 61, 203–234 (1990).

    Article  MathSciNet  Google Scholar 

  2. F. Guias, Coagulation-Fragmentation Processes: Relations between Finite Particle Models and Differential Equations, Preprint 98-41 (SFB 359), Heidelberg (1998).

  3. M. Lachowicz, P. Laurencot, and D. Wrzosek, “On the Oort-Hulst-Safronov coagulation equation and its relation to the Smoluchowski equation,” SIAM J. Math. Anal., 34, No.6, 1399–1421 (2003).

    Article  MathSciNet  Google Scholar 

  4. M. Lachowicz and D. Wrzosek, “A nonlocal coagulation-fragmentation model,” Appl. Math. (Warsaw), 27, No.1, 45–66 (2000).

    MathSciNet  Google Scholar 

  5. M. Smoluchowski, “Versuch einer mathematischen Theorie der kolloiden Losungen,” Z. Phys. Chem., 92, 129–168 (1917).

    Google Scholar 

  6. M. Lachowicz and D. Wrzosek, “Nonlocal bilinear equations. Equilibrium solutions and diffusive limit,” Math. Models Methods Appl. Sci., 11, 1375–1390 (2001).

    Article  MathSciNet  Google Scholar 

  7. E. Jager and L. Segel, “On the distribution of dominance in a population of interacting anonymous organisms,” SIAM J. Appl. Math., 52, 1442–1468 (1992).

    MathSciNet  Google Scholar 

  8. L. Arlotti and N. Bellomo, “Population dynamics with stochastic interaction,” Transp. Theory Statist. Phys., 24, 431–443 (1995).

    MathSciNet  Google Scholar 

  9. L. Arlotti and N. Bellomo, “Solution of a new class of nonlinear kinetic models of population dynamics,” Appl. Math. Lett., 9, 65–70 (1996).

    Article  MathSciNet  Google Scholar 

  10. L. Arlotti, N. Bellomo, and M. Lachowicz, “Kinetic equations modelling population dynamics,” Transp. Theory Statist. Phys., 29, 125–139 (2000).

    MathSciNet  Google Scholar 

  11. E. Geigant, K. Ladizhansky, and A. Mogilner, “An integrodifferential model for orientational distribution of F-Actin in cells,” SIAM J. Appl. Math., 59, No.3, 787–809 (1998).

    MathSciNet  Google Scholar 

  12. M. Lachowicz, “From microscopic to macroscopic description for generalized kinetic models,” Math. Models Methods Appl. Sci., 12, No.7, 985–1005 (2002).

    Article  MATH  MathSciNet  Google Scholar 

  13. M. Lachowicz, “On bilinear kinetic equations. Between micro and macro descriptions of biological populations,” Banach Center Publ., 63, 217–230 (2004).

    MATH  MathSciNet  Google Scholar 

  14. M. Lachowicz, General Population Systems. Macroscopic Limit of a Class of Stochastic Semigroups (to appear).

  15. M. Deaconu and N. Fournier, “Probabilistic approach of some discrete and continuous coagulation equation with diffusion,” Stochast. Process. Appl., 101, 83–111 (2002).

    MathSciNet  Google Scholar 

  16. P. Donnelly and S. Simons, “On the stochastic approach to cluster size distribution during particle coagulation,” J. Phys. A: Math. Gen., 26, 2755–2767 (1993).

    Article  MathSciNet  Google Scholar 

  17. R. Lang and N. Xanh, “Smoluchowski's theory of coagulation in colloids holds rigorously in the Boltzmann-Grad limit,” Z. Wahrscheinlichkeitstheor. Verw. Geb., 54, 227–280 (1980).

    Article  Google Scholar 

  18. M. Lachowicz, “Describing competitive systems at the level of interacting individuals,” Proc. Eighth Nat. Conf. Appl. Math. Biol. Medicine (Lajs, 25–28, Sept. 2002) (2002), pp. 95–100.

  19. M. Lachowicz, “From microscopic to macroscopic descriptions of complex systems,” Comp. Rend. Mecanique (Paris), 331, 733–738 (2003).

    MATH  Google Scholar 

  20. M. Lachowicz and M. Pulvirenti, “A stochastic particle system modeling the Euler equation,” Arch. Ration. Mech. Anal., 109, No.1, 81–93 (1990).

    Article  MathSciNet  Google Scholar 

  21. V. N. Kolokoltsov, “Hydrodynamic limit of coagulation-fragmentation type models of k-nary interacting particles,” J. Statist. Phys. (to appear).

  22. V. N. Kolokoltsov, “On extension of mollified Boltzmann and Smoluchowski equations to particle systems with a k-nary interaction,” Rus. J. Math. Phys. (to appear).

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Authors and Affiliations

  1. Institute of Applied Mathematics and Mechanics, Warsaw University, Warsaw, Poland

    M. Lachowicz

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  1. M. Lachowicz
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Published in Ukrains'kyi Matematychnyi Zhurnal, Vol. 57, No. 6, pp. 770–777, June, 2005.

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Lachowicz, M. Stochastic Semigroups and Coagulation Equations. Ukr Math J 57, 913–922 (2005). https://doi.org/10.1007/s11253-005-0239-y

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  • DOI: https://doi.org/10.1007/s11253-005-0239-y

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