Partial differential equation models in the socio-economic sciences

Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences

Volumn 372, Issue 2028, 2014, Pages

  Burger, Martin ^a   Caffarelli, Luis ^b   Markowich, Peter A ^c  

^a UNIVERSITY OF MÜNSTER   (Germany)

^b UNIVERSITY OF TEXAS AT AUSTIN   (United States)

^c KING ABDULLAH UNIVERSITY OF SCIENCE AND TECHNOLOGY   (Saudi Arabia)

Author keywords

Maths; Partial differential equations; Socio economics

Indexed keywords

ECONOMICS; PARTIAL DIFFERENTIAL EQUATIONS;

INTEGRAL PART; MATHS; PARTIAL DIFFERENTIAL EQUATION MODELS; PARTIAL DIFFERENTIAL EQUATIONS (PDES); SCIENCE AND ENGINEERING; SOCIO-ECONOMICS;

MATHEMATICAL MODELS;

EID: 84907863442     PISSN: 1364503X     EISSN: None     Source Type: Journal    
DOI: 10.1098/rsta.2013.0406     Document Type: Article

References (67)

1. Chatterjee AK, Chakrabarti B, Manna SS. 2004 Pareto law in a kinetic model of market with random saving propensity. Physica A 335, 155-163. (doi:10.1016/j.physa.2003.11.014)
2. Bouchaud JP, Mezard M. 2000 Wealth condensation in a simple model of economy. Physica A 282, 536-545. (doi:10.1016/S0378-4371(00)00205-3)
3. Cordier S, Pareschi L, Toscani G. 2005 On a kinetic model for a simple market economy. J. Stat. Phys. 120, 253-277. (doi:10.1007/s10955-005-5456-0)
4. Düring B, Matthes D, Toscani G. 2008 Kinetic equations modelling wealth redistribution: a comparison of approaches. Phys. Rev. E 78, 056103. (doi:10.1103/PhysRevE.78.056103)
5. Pareschi L, Toscani G. 2014 Wealth distribution and collective knowledge: a Boltzmann approach. Phil. Trans. R. Soc. A 372, 20130396. (doi:10.1098/rsta.2013.0396)
6. Degond P, Liu J-G, Ringhofer C. 2014 Evolution of wealth in a non-conservative economy driven by local Nash equilibria. Phil. Trans. R. Soc. A 372, 20130394. (doi:10.1098/rsta.2013.0394)
7. Sznajd-Weron K, Sznajd J. 2000 Opinion evolution in closed community. Int. J. Mod. Phys. C 11, 1157-1165. (doi:10.1142/S0129183100000936)
8. Boudin L, Salvarani F. 2009 A kinetic approach to the study of opinion formation. ESAIM: Math. Model. Numer. Anal. 43, 507-522. (doi:10.1051/m2an/2009004)
9. Toscani G. 2006 Kinetic models of opinion formation. Commun. Math. Sci. 4, 481-496. (doi:10.4310/CMS.2006.v4.n3.a1)
10. Düring B, Markowich P, Pietschmann J-F, Wolfram M-T. 2009 Boltzmann and Fokker-Planck equations modelling opinion formation in the presence of strong leaders. Proc. R. Soc. A 465, 3687-3708. (doi:10.1098/rspa.2009.0239)
11. Albi G, Pareschi L, Zanella M. 2014 Boltzmann-type control of opinion consensus through leaders. Phil. Trans. R. Soc. A 372, 20140138. (doi:10.1098/rsta.2014.0138)
12. Cucker F, Smale S. 2007 Emergent behavior in flocks. IEEE Trans. Autom. Control 52, 852-862. (doi:10.1109/TAC.2007.895842)
13. Carrillo JA, Fornasier M, Rosado J, Toscani G. 2010 Asymptotic flocking dynamics for the kinetic Cucker-Smale model. SIAM J. Math. Anal. 42, 218-236. (doi:10.1137/090757290)
14. Ha SY, Lattanzio C, Rubino B, Slemrod M. 2011 Flocking and synchronization of particle models. Q. Appl. Math. 69, 91-103. (doi:10.1090/S0033-569X-2010-01200-7)
15. Degond P, Motsch S. 2008 Continuum limit of self-driven particles with orientation interaction. Math. Models Methods Appl. Sci. 18, 1193-1215. (doi:10.1142/S0218202508003005)
16. Shen J. 2007 Cucker-Smale flocking under hierarchical leadership. SIAM J. Appl. Math. 68, 694-719. (doi:10.1137/060673254)
17. Barbaro AB, Degond P. 2012 Phase transition and diffusion among socially interacting self-propelled agents. (http://arxiv.org/abs/1207.1926)
18. Degond P, Frouvelle A, Liu JG. 2013 Macroscopic limits and phase transition in a system of self-propelled particles. J. Nonlinear Sci. 23, 427-456. (doi:10.1007/s00332-012-9157-y)
19. Frouvelle A, Liu JG. 2012 Dynamics in a kinetic model of oriented particles with phase transition. SIAM J. Math. Anal. 44, 791-826. (doi:10.1137/110823912)
20. Tadmor E, Tan C. 2014 Critical thresholds in flocking hydrodynamics with non-local alignment. Phil. Trans. R. Soc. A 372, 20130401. (doi:10.1098/rsta.2013.0401)
21. Grünbaum D, Okubo A. 1994 Modelling social animal aggregations. In Frontiers in mathematical biology (ed. SA Levin). Lecture Notes in Biomathematics, vol. 100, pp. 296-325. Berlin, Germany: Springer. (doi:10.1007/978-3-642-50124-1-18)
22. Mogilner A, Edelstein-Keshet L. 1999 A non-local model for a swarm. J. Math. Biol. 38, 534-570. (doi:10.1007/s002850050158)
23. Morale D, Capasso V, Oelschläger K. 2005 An interacting particle system modelling aggregation behavior: from individuals to populations. J. Math. Biol. 50, 49-66. (doi:10.1007/s00285-004-0279-1)
24. Bedrossian J. 2011 Intermediate asymptotics for critical and supercritical aggregation equations and Patlak-Keller-Segel models. Commun. Math. Sci. 9, 1143-1161. (doi:10.4310/CMS.2011.v9.n4.a11)
25. Bertozzi AL, Slepčev D. 2010 Existence and uniqueness of solutions to an aggregation equation with degenerate diffusion. Commun. Pure Appl. Anal. 9, 1617-1637. (doi:10.3934/cpaa.2010.9.1617)
26. Burger M, Capasso V, Morale D. 2007 On an aggregation model with long and short range interactions. Nonlinear Anal. Real World Appl. 8, 939-958. (doi:10.1016/j.nonrwa.2006.04.002)
27. Leverentz AJ, Topaz CM, Bernoff AJ. 2009 Asymptotic dynamics of attractive-repulsive swarms. SIAM J. Appl. Dyn. Syst. 8, 880-908. (doi:10.1137/090749037)
28. Bedrossian J. 2011 Global minimizers for free energies of subcritical aggregation equations with degenerate diffusion. Appl. Math. Lett. 24, 1927-1932. (doi:10.1016/j.aml.2011.05.022)
29. Burger M, Di Francesco M. 2008 Large time behavior of nonlocal aggregation models with nonlinear diffusion. Netw. Heterog. Media 3, 749-785. (doi:10.3934/nhm.2008.3.749)
30. Burger M, Di Francesco M, Franek M. 2013 Stationary states of quadratic diffusion equations with long-range attraction. Commun. Math. Sci. 11, 709-738. (doi:10.4310/CMS.2013.v11.n3.a3)
31. Carrillo JA, Chipot M, Huang Y. 2014 On global minimizers of repulsive-attractive power-law interaction energies. Phil. Trans. R. Soc. A 372, 20130399. (doi:10.1098/rsta.2013.0399)
32. Aubin JP. 1998 Optima and equilibria: an introduction to nonlinear analysis. New York, NY: Springer.
33. Dockner E (ed.). 2000 Differential games in economics and management science. Cambridge, UK: Cambridge University Press.
34. Friedman A. 2013 Differential games. New York, NY: Dover Publications.
35. Lasry JM, Lions PL. 2007 Mean field games. Jpn. J. Math. 2, 229-260. (doi:10.1007/s11537-007-0657-8)
36. Huang M, Malhame RP, Caines PE. 2006 Large population stochastic dynamic games: closed-loop McKean-Vlasov systems and the Nash certainty equivalence principle. Commun. Inform. Syst. 6, 221-251. (doi:10.4310/CIS.2006.v6.n3.a5)
37. Guéant O, Lasry J-M, Lions P-L. 2011 Mean field games and applications. In Paris-Princeton lectures on mathematical finance 2010. Lecture Notes in Mathematics, vol. 2003, pp. 205-266. Berlin, Germany: Springer. (doi:10.1007/978-3-642-14660-2-3)
38. Burger M, Di Francesco M, Markowich P, Wolfram MT. 2014 Mean field games with nonlinear mobilities in pedestrian dynamics. Discr. Contin. Dyn. Syst. B 19, 1311-1333. (doi:10.3934/dcdsb.2014.19.1311)
39. Achdou Y, Capuzzo-Dolcetta I. 2010 Mean field games: numerical methods. SIAM J. Numer. Anal. 48, 1136-1162. (doi:10.1137/090758477)
40. Gomes D, Saúde J. 2014 Mean field games - a brief survey. Dyn. Games Appl. 4, 110-154. (doi:10.1007/s13235-013-0099-2)
41. Lachapelle A, Wolfram M-T. 2011 On a mean field game approach modeling congestion and aversion in pedestrian crowds. Transp. Res. Part B: Methodol. 45, 1572-1589. (doi:10.1016/j.trb.2011.07.011)
42. Cardaliaguet P. 2010 Notes on mean field games from P.L. Lions' lectures at Collège de France. See https://www.ceremade.dauphine.fr/~cardalia/MFG100629.pdf.
43. Cardaliaguet P. 2014 Mean field games systems of first order. (http://arxiv.org/abs/1401.1789)
44. Cardaliaguet P, Lasry JM, Lions PL, Porretta A. 2012 Long time average of mean field games. Netw. Heterogeneous Media 7, 279-301. (doi:10.3934/nhm.2012.7.279)
45. Cardaliaguet P, Lasry JM, Lions PL, Porretta A. 2013 Long time average of mean field games with a nonlocal coupling. SIAM J. Control Optim. 51, 3558-3591. (doi:10.1137/120904184)
46. Carmona R, Delarue F. 2013 Probabilistic analysis of mean-field games. SIAM J. Control Optim. 51, 2705-2734. (doi:10.1137/120883499)
47. Carmona R, Delarue F. 2013 Mean-field forward-backward stochastic differential equations. Elect. Commun. Probab. 18, 1-15. (doi:10.1214/ECP.v18-2446)
48. Evans LC. 2010 Adjoint and compensated compactness methods for Hamilton-Jacobi PDE. Arch. Rational Mech. Anal. 197, 1053-1088. (doi:10.1007/s00205-010-0307-9)
49. Schulte JM. 2010 Adjoint methods for Hamilton-Jacobi equations. Master thesis, WWU Münster.
50. Tran HV. 2011 Adjoint methods for static Hamilton-Jacobi equations. Calc. Var. Partial Differ. Equ. 41, 301-319. (doi:10.1007/s00526-010-0363-x)
51. Achdou Y, Buera FJ, Lasry J-M, Lions P-L, Moll B. 2014 Partial differential equation models in macroeconomics. Phil. Trans. R. Soc. A 372, 20130397. (doi:10.1098/rsta.2013.0397)
52. Gomes D, Velho RM, Wolfram M-T. 2014 Socio-economic applications of finite state mean field games. Phil. Trans. R. Soc. A 372, 20130405. (doi:10.1098/rsta.2013.0405)
53. Spohn H. 1980 Kinetic equations from Hamiltonian dynamics: Markovian limits. Rev. Mod. Phys. 52, 569-615. (doi:10.1103/RevModPhys.52.569)
54. Carmona R, Delarue F, Lachapelle A. 2012 Control of McKean-Vlasov dynamics versus mean field games. Math. Financ. Econ. 7, 131-166. (doi:10.1007/s11579-012-0089-y)
55. Fornasier M, Piccoli B, Rossi F. 2014 Mean-field sparse optimal control. Phil. Trans. R. Soc. A 372, 20130400. (doi:10.1098/rsta.2013.0400)
56. Blanchet A, Carlier G. 2014 From Nash to Cournot-Nash equilibria via the Monge-Kantorovich problem. Phil. Trans. R. Soc. A 372, 20130398. (doi:10.1098/rsta.2013.0398)
57. Chayes L, González MdM, Gualdani MP, Kim I. 2009 Global existence and uniqueness of solutions to a model of price formation. SIAM J. Math. Anal. 41, 2107-2135. (doi:10.1137/090753346)
58. González MDM, Gualdani MP. 2011 Asymptotics for a free boundary model in price formation. Nonlinear Anal. 74, 3269-3294. (doi:10.1016/j.na.2011.02.005)
59. Markowich PA, Matevosyan N, Pietschmann J-F, Wolfram M-T. 2009 On a parabolic free boundary equation modeling price formation. Math. Models Methods Appl. Sci. 19, 1929-1957. (doi:10.1142/S0218202509003978)
60. Caffarelli LA, Markowich PA, Pietschmann JF. 2011 On a price formation free boundary model by Lasry and Lions. C. R. Math. Acad. Sci. Paris 349, 621-624. (doi:10.1016/j.crma.2011.05.011)
61. Caffarelli LA, Markowich PA, Wolfram M-T. 2011 On a price formation free boundary model by Lasry and Lions: the Neumann problem. C. R. Math. Acad. Sci. Paris 349, 841-844. (doi:10.1016/j.crma.2011.07.006)
62. Burger M, Caffarelli L, Markowich PA, Wolfram M-T. 2013 On a Boltzmann-type price formation model. Proc. R. Soc. A 469, 20130126. (doi:10.1098/rspa.2013.0126)
63. Burger M, Caffarelli L, Markowich P, Wolfram MT. 2014 On the asymptotic behaviour of a Boltzmann-type price formation model. Commun. Math. Sci. 12, 1353-1361. (doi:10.4310/CMS.2014.v12.n7.a10).
64. Markowich PA, Souganidis P, Teichmann J, Wolfram MT. In preparation. On a modified Lasry-Lions price formation free boundary model with fluctuating markets.
65. Berestycki H, Monneau R, Scheinkman JA. 2014 A non-local free boundary problem arising in a theory of financial bubbles. Phil. Trans. R. Soc. A 372, 20130404. (doi:10.1098/rsta.2013.0404)
66. Engbers R, Burger M, Capasso V. 2014 Inverse problems in geographical economics: parameter identification in the spatial Solow model. Phil. Trans. R. Soc. A 372, 20130402. (doi:10.1098/rsta.2013.0402)
67. Woodworth JT, Mohler GO, Bertozzi AL, Brantingham PJ. 2014 Non-local crime density estimation incorporating housing information. Phil. Trans. R. Soc. A 372, 20130403. (doi:10.1098/rsta.2013.0403)