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Articles

A variational approach to nonlinear and interacting diffusions

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Pages 717-748 | Received 11 Mar 2019, Accepted 14 Apr 2019, Published online: 12 May 2019

References

  • Coppel, W. A. (1965). Stability and Asymptotic Behavior of Differential Equations. Boston, Mass: D. C. Heath and Co.
  • Martin, R. H. Jr.(1970). Bounds for solutions of a class of nonlinear differential equations. J. Differ. Eq. 8(3):416–430. DOI: 10.1016/0022-0396(70)90015-X.
  • Lyapunov, A. M. (1992). The General Problem of the Stability of Motion. CRC Press.
  • Khalil, H. K. (2002). Nonlinear Systems. Prentice Hall.
  • Carverhill, A. P., Elworthy, K. D. (1983). Flows of stochastic dynamical systems: The functional analytic approach. Z Wahrscheinlichkeitstheorie Verw. Gebiete 65(2):245–267. DOI: 10.1007/BF00532482.
  • Norris, J. R. (1986). Simplified Malliavin calculus. Séminaire Probab. 20:101–130.
  • Graham, C. (1992). McKean-Vlasov, Ito-Skorohod equations and nonlinear diffusions with discrete jumps. Stoch. Proc. Appl. 40(1):69–82. DOI: 10.1016/0304-4149(92)90138-G.
  • Huang, X., Röckner, M., Wang, F. Y. (2017). Nonlinear Fokker-Planck Equations for Probability Measures on Path Space and Path-Distribution Dependent SDEs. arXiv Preprint arXiv1709:00556.
  • McKean, H. P. (1966). A class of markov processes associated with nonlinear parabolic equations. Proc. Nat. Acad. Sci. USA 56(6):1907–1911. DOI: 10.1073/pnas.56.6.1907.
  • McKean, H. P. (1967). Propagation of chaos for a class of non-linear parabolic equations. In: Stochastic Differential Equations, Lecture Series in Differential Equations, Session 7, Catholic University. Arlington, VA: Air Force Office of Scientific Research, pp. 41–57.
  • Benedetto, D., Caglioti, E., Pulvirenti, M. (1997). A kinetic equation for granular media. Esaim: M2AN. 31(5):615–641. DOI: 10.1051/m2an/1997310506151.
  • Benedetto, D., Caglioti, E., Carrillo, E., Pulvirenti, M. (1998). A non-Maxwellian steady distribution for one-dimensional granular media. J. Statist. Phys. 91(5/6):979–990. DOI: 10.1023/A:1023032000560.
  • Toscani, G. (2000). One-dimensional kinetic models of granular flows. RAIRO Modél. Math. Anal. Numér. 34(6):1277–1291.
  • Villani, C. (2002). A survey of mathematical topics in the collisional kinetic theory of gases. In: Handbook of Mathematical Fluid Dynamics, Vol. 1, no 71–305, pp. 3–8.
  • Tamura, Y. (1984). On asymptotic behaviors of the solution of a nonlinear diffusion equation. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 31(1):195–221.
  • Tamura, Y. (1987). Free energy and the convergence of distributions of diffusion processes of McKean type. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 34(2):443–484.
  • Benachour, S., Roynette, B., Talay, D., Vallois, P. (1998). Nonlinear self-stabilizing processes, part I. Existence, invariant probability, propagation of chaos. Stoch. Proc. Appl. 75(2):173–201. DOI: 10.1016/S0304-4149(98)00018-0.
  • Benachour, S., Roynette, B., Vallois, P. (1998). Nonlinear self-stabilizing processes, part II: Convergence to invariant probability. Stoch. Proc. Appl. 75(2):203–224. DOI: 10.1016/S0304-4149(98)00019-2.
  • Calvez, V., Carrillo, J. (2012). Refined asymptotics for the subcritical Keller-Segel system and related functional inequalities. Proc. Amer. Math. Soc. 140(10):3515–3530. DOI: 10.1090/S0002-9939-2012-11306-1.
  • Carrillo, J. A., Toscani, G. (2005). Wasserstein metric and large-time assymptotics of nonlinear diffusion equations. In: New Trends in Mathematical Physics. Singapore: World Scientific.
  • Malrieu, F. (2001). Logarithmic sobolev inequalities for some nonlinear PDE’s. Stoch. Proc. Appl. 95(1):109–132. DOI: 10.1016/S0304-4149(01)00095-3.
  • Malrieu, F. (2003). Convergence to equilibrium for granular media equations and their Euler schemes. Ann. Appl. Probab. 13(2):540–560. DOI: 10.1214/aoap/1050689593.
  • Carrillo, J. A., McCann, R. J., Villani, C. (2006). Contractions in the 2-Wasserstein length space and thermalization of granular media. Arch. Rational Mech. Anal. 179(2):217–263. DOI: 10.1007/s00205-005-0386-1.
  • Cordero-Erausquin, D., Gangbo, W., Houdré, C. (2004). Inequalities for generalized entropy and optimal transportation. In: Recent Advances in the Theory and Applications of Mass Transport, Contemp. Math, Vol. 353. Providence: A. M. S.
  • Bolley, F., Gentil, I., Guillin, A. (2013). Uniform convergence to equilibrium for granular media. Arch. Rational Mech. Anal. 208(2):429–445. DOI: 10.1007/s00205-012-0599-z.
  • Cattiaux, P., Guillin, A., Malrieu, F. (2007). Probabilistic approach for granular media equations in the non uniformly convex case. Probab. Theory Relat. Fields 140(1–2):19–40. DOI: 10.1007/s00440-007-0056-3.
  • Otto, F. (2001). The geometry of dissipative evolution equations: The porous medium equation. Commun. Part. Diff. Eq. 26(1–2):101–174. DOI: 10.1081/PDE-100002243.
  • Bakry, D., Emery, M. (1985). Diffusions hypercontractives. In: Séminaire de Probabilités, XIX. 1983/84. Lecture Notes in Mathematics 1123, Springer; pp. 177–206.
  • Otto, F., Villani, V. (2000). Generalization of an inequality by talagrand, and links with the logarithmic Sobolev inequality. J. Funct. Anal. 173(2):361–400. DOI: 10.1006/jfan.1999.3557.
  • Tugaut, J. (2013). Convergence to the equilibria for self-stabilizing processes in double well landscape. Ann. Probab. 41(3A):1427–1460. DOI: 10.1214/12-AOP749.
  • Tugaut, J. (2014). Self-stabilizing processes in multi-wells landscape in Rd –Invariant probabilities. J. Theor. Probab. 27(1):27–57. DOI: 10.1007/s10959-012-0435-2.
  • Tugaut, J. (2012). Exit problem of McKean-Vlasov diffusions in convex landscapes. Electron. J. Probab. 17(no. 76):1–26.
  • Tugaut, J. (2016). A simple proof of a kramers’ type law for self-stabilizing diffusions. Electron. Commun. Probab. 21.
  • Tugaut, J. (2011). McKean-Vlasov diffusions: From the asynchronization to the synchronization. CR Math. Acad. Sci. 349(17–18):983–986. DOI: 10.1016/j.crma.2011.08.002.
  • Tugaut, J. (2018). Convergence in wasserstein distance for self-stabilizing diffusion evolving in a double-well landscape. CR Math. 356(6):657–660. DOI: 10.1016/j.crma.2018.04.020.
  • Duong, M. H., Tugaut, J. (2018). The Vlasov-Fokker-Planck equation in non-convex landscapes: Convergence to equilibrium. Electron. Commun. Probab. 23.
  • Herrmann, S., Tugaut, J. (2010). Non-uniqueness of stationary measures for self-stabilizing processes. Stoch. Proc. Appl. 120(7):1215–1246. DOI: 10.1016/j.spa.2010.03.009.
  • Herrmann, S., Tugaut, J. (2010). Stationary measures for self-stabilizing processes: Asymptotic analysis in the small noise limit. Electron. J. Probab. 15:2087–2116. DOI: 10.1214/EJP.v15-842.
  • Herrmann, S., Tugaut, J. (2012). Self-stabilizing processes: Uniqueness problem for stationary measures and convergence rate in the small-noise limit. ESAIM: PS. Statist. 16:277–305. DOI: 10.1051/ps/2011152.
  • Da Prato, G., Menaldi, J. L., Tubaro, L. (2007). Some results of backward ito formula. Stoch. Anal. Appl. 25(3):679–703. DOI: 10.1080/07362990701283045.
  • Arnaudon, M., Coulibaly, A., Thalmaier, A. (2010). Horizontal diffusions in C1 path space. Séminaire de Probabilités XLIII. In: Lecture Notes in Mathematics 2006, Springer; pp. 73–94.
  • Arnaudon, M., Thalmaier, A., Wang, F. Y. (2006). Harnack inequality and heat kernel estimates on manifolds with curvature unbounded below. Bull. Sci. Math. 130(3):223–233. DOI: 10.1016/j.bulsci.2005.10.001.
  • Feng-Yu, W. (2014). Analysis for Diffusion Processes on Riemannian Manifolds, Advanced Series on Statistical Science and Applied Probability, Vol. 18. World Scientific.

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