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November  2016, 21(9): 3029-3052. doi: 10.3934/dcdsb.2016086

Mean field limit with proliferation

Franco Flandoli 1, and  Matti Leimbach 2,

1. 

Universita di Pisa, Dipartimento Matematica, Largo Bruno Pontecorvo 5, C.A.P. 56127, Pisa, Italy
2. 

Technische Universität Berlin, Institut für Mathematik, Straße des 17. Juni 136, D-10623, Berlin, Germany

Received  October 2015 Revised  March 2016 Published  October 2016

An interacting particle system with long range interaction is considered. Particles, in addition to the interaction, proliferate with a rate depending on the empirical measure. We prove convergence of the empirical measure to the solution of a parabolic equation with non-local nonlinear transport term and proliferation term of logistic type.
Keywords: interacting particle system., limit dynamics, semilinear parabolic equation, Mean field, branching.
Mathematics Subject Classification: Primary: 60K35, 35K57; Secondary: 60F17, 35K5.
Citation: Franco Flandoli, Matti Leimbach. Mean field limit with proliferation. Discrete & Continuous Dynamical Systems - B, 2016, 21 (9) : 3029-3052. doi: 10.3934/dcdsb.2016086
References:
[1]

R. E. Baker and M. J. Simpson, Correcting mean-field approximations for birth-death-movement processes, Phys. Rev. E, 82 (2010), 041905, 12 pp. doi: 10.1103/PhysRevE.82.041905.  Google Scholar

[2]

L. Banas, Z. Brzezniak, M. Neklyudov and A. Prohl, Stochastic Ferromagnetism: Analysis and Numerics, De Gruyter Studies in Mathematics, 58. De Gruyter, Berlin, 2014.  Google Scholar

[3]

M. Bodnar and J. J. L. Velazquez, Derivation of macroscopic equations for individual cell-based models: A formal approach, Math. Meth. Appl. Sci., 28 (2005), 1757-1779. doi: 10.1002/mma.638.  Google Scholar

[4]

Z. Brzezniak, M. Ondrejat and E. Motyl, Invariant measure for the stochastic Navier-Stokes equations in unbounded 2D domains, preprint,, , ().   Google Scholar

[5]

Z. Brzezniak and E. Motyl, Existence of a martingale solution of the stochastic Navier-Stokes equations in unbounded 2D and 3D-domains, J. Diff. Eq., 254 (2013), 1627-1685. doi: 10.1016/j.jde.2012.10.009.  Google Scholar

[6]

F. Flandoli and D. Gatarek, Martingale and stationary solutions for stochastic Navier-Stokes equations, preprint, Probab. Theory Relat. Fields, 102 (1995), 367-391. doi: 10.1007/BF01192467.  Google Scholar

[7]

J. L. Lions, Quelques Methodes de Resolution des Problemes aux Limites non Lineaires, Dunod, Gauthier-Villars, Paris, 1969.  Google Scholar

[8]

S. Meleard and V. Bansaye, Some Stochastic Models for Structured Populations: Scaling Limits and Long Time Behavior, Stochastics in Biological Systems, 1.4. Springer, Cham; MBI Mathematical Biosciences Institute, Ohio State University, Columbus, OH, 2015, arXiv:1506.04165v1. doi: 10.1007/978-3-319-21711-6.  Google Scholar

[9]

M. Metivier, Quelques problèmes liés aux systèmes infinis de particules et leurs limites, Séminaire de Probabilités, (Strasbourg) 20 (1986), 426-446. doi: 10.1007/BFb0075734.  Google Scholar

[10]

G. Nappo and E. Orlandi, Limit laws for a coagulation model of interacting random particles, Annales de l'I.H.P. section B, 24 (1988), 319-344.  Google Scholar

[11]

K. Oelschläger, A law of large numbers for moderately interacting diffusion processes, Zeitschrift fur Wahrsch. Verwandte Gebiete, 69 (1985), 279-322. doi: 10.1007/BF02450284.  Google Scholar

[12]

K. Oelschläger, On the Derivation of Reaction-Diffusion Equations as Limit Dynamics of Systems of Moderately Interacting Stochastic Processes, Probab. Th. Rel. Fields, 82 (1989), 565-586. doi: 10.1007/BF00341284.  Google Scholar

[13]

R. Philipowski, Interacting diffusions approximating the porous medium equation and propagation of chaos, Stoch. Proc. Appl., 117 (2007), 526-538. doi: 10.1016/j.spa.2006.09.003.  Google Scholar

[14]

A. Stevens, The derivation of chemotaxis equations as limit dynamics of moderately interacting stochastic many-particle systems, SIAM J. Appl. Math., 61 (2000), 183-212. doi: 10.1137/S0036139998342065.  Google Scholar

[15]

S. He, J. Wang and J. A. Yan, Semimartingale Theory and Stochastic Calculus, Beijing: Science Press, 1992.  Google Scholar

[16]

P. Hinow, P. Gerlee, L. J. McCawley, V. Quaranta, M. Ciobanu, S. Wang, J. M. Graham, B. P. Ayati, J. Claridge, K. R. Swanson, M. Loveless and A. R. A. Anderson, A spatial model of tumor-host interaction: Application of chemoterapy, Math. Biosc. Engin., 6 (2009), 521-546.  Google Scholar

[17]

K. Uchiyama, Pressure in classical Statistical Mechanics and interacting Brownian particles in multi-dimensions, Ann. H. Poincaré, 1 (2000), 1159-1202. doi: 10.1007/PL00001025.  Google Scholar

[18]

S. R. S. Varadhan, Scaling limit for interacting diffusions, Comm. Math. Phys., 135 (1991), 313-353. doi: 10.1007/BF02098046.  Google Scholar

show all references

References:
[1]

R. E. Baker and M. J. Simpson, Correcting mean-field approximations for birth-death-movement processes, Phys. Rev. E, 82 (2010), 041905, 12 pp. doi: 10.1103/PhysRevE.82.041905.  Google Scholar

[2]

L. Banas, Z. Brzezniak, M. Neklyudov and A. Prohl, Stochastic Ferromagnetism: Analysis and Numerics, De Gruyter Studies in Mathematics, 58. De Gruyter, Berlin, 2014.  Google Scholar

[3]

M. Bodnar and J. J. L. Velazquez, Derivation of macroscopic equations for individual cell-based models: A formal approach, Math. Meth. Appl. Sci., 28 (2005), 1757-1779. doi: 10.1002/mma.638.  Google Scholar

[4]

Z. Brzezniak, M. Ondrejat and E. Motyl, Invariant measure for the stochastic Navier-Stokes equations in unbounded 2D domains, preprint,, , ().   Google Scholar

[5]

Z. Brzezniak and E. Motyl, Existence of a martingale solution of the stochastic Navier-Stokes equations in unbounded 2D and 3D-domains, J. Diff. Eq., 254 (2013), 1627-1685. doi: 10.1016/j.jde.2012.10.009.  Google Scholar

[6]

F. Flandoli and D. Gatarek, Martingale and stationary solutions for stochastic Navier-Stokes equations, preprint, Probab. Theory Relat. Fields, 102 (1995), 367-391. doi: 10.1007/BF01192467.  Google Scholar

[7]

J. L. Lions, Quelques Methodes de Resolution des Problemes aux Limites non Lineaires, Dunod, Gauthier-Villars, Paris, 1969.  Google Scholar

[8]

S. Meleard and V. Bansaye, Some Stochastic Models for Structured Populations: Scaling Limits and Long Time Behavior, Stochastics in Biological Systems, 1.4. Springer, Cham; MBI Mathematical Biosciences Institute, Ohio State University, Columbus, OH, 2015, arXiv:1506.04165v1. doi: 10.1007/978-3-319-21711-6.  Google Scholar

[9]

M. Metivier, Quelques problèmes liés aux systèmes infinis de particules et leurs limites, Séminaire de Probabilités, (Strasbourg) 20 (1986), 426-446. doi: 10.1007/BFb0075734.  Google Scholar

[10]

G. Nappo and E. Orlandi, Limit laws for a coagulation model of interacting random particles, Annales de l'I.H.P. section B, 24 (1988), 319-344.  Google Scholar

[11]

K. Oelschläger, A law of large numbers for moderately interacting diffusion processes, Zeitschrift fur Wahrsch. Verwandte Gebiete, 69 (1985), 279-322. doi: 10.1007/BF02450284.  Google Scholar

[12]

K. Oelschläger, On the Derivation of Reaction-Diffusion Equations as Limit Dynamics of Systems of Moderately Interacting Stochastic Processes, Probab. Th. Rel. Fields, 82 (1989), 565-586. doi: 10.1007/BF00341284.  Google Scholar

[13]

R. Philipowski, Interacting diffusions approximating the porous medium equation and propagation of chaos, Stoch. Proc. Appl., 117 (2007), 526-538. doi: 10.1016/j.spa.2006.09.003.  Google Scholar

[14]

A. Stevens, The derivation of chemotaxis equations as limit dynamics of moderately interacting stochastic many-particle systems, SIAM J. Appl. Math., 61 (2000), 183-212. doi: 10.1137/S0036139998342065.  Google Scholar

[15]

S. He, J. Wang and J. A. Yan, Semimartingale Theory and Stochastic Calculus, Beijing: Science Press, 1992.  Google Scholar

[16]

P. Hinow, P. Gerlee, L. J. McCawley, V. Quaranta, M. Ciobanu, S. Wang, J. M. Graham, B. P. Ayati, J. Claridge, K. R. Swanson, M. Loveless and A. R. A. Anderson, A spatial model of tumor-host interaction: Application of chemoterapy, Math. Biosc. Engin., 6 (2009), 521-546.  Google Scholar

[17]

K. Uchiyama, Pressure in classical Statistical Mechanics and interacting Brownian particles in multi-dimensions, Ann. H. Poincaré, 1 (2000), 1159-1202. doi: 10.1007/PL00001025.  Google Scholar

[18]

S. R. S. Varadhan, Scaling limit for interacting diffusions, Comm. Math. Phys., 135 (1991), 313-353. doi: 10.1007/BF02098046.  Google Scholar

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