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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 and 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.

[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.

[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.

[4]

Z. Brzezniak, M. Ondrejat and E. Motyl, Invariant measure for the stochastic Navier-Stokes equations in unbounded 2D domains, preprint, arXiv:1502.02637.

[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.

[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.

[7]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[15]

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

[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.

[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.

[18]

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

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.

[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.

[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.

[4]

Z. Brzezniak, M. Ondrejat and E. Motyl, Invariant measure for the stochastic Navier-Stokes equations in unbounded 2D domains, preprint, arXiv:1502.02637.

[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.

[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.

[7]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[15]

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

[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.

[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.

[18]

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

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