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Mean field limit with proliferation
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 |
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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