zbMATH — the first resource for mathematics

Linear and nonlinear dissipative dynamics. (English) Zbl 1376.37115
Summary: In this paper we introduce and study new dissipative dynamics for large interacting systems.

37L05 General theory of infinite-dimensional dissipative dynamical systems, nonlinear semigroups, evolution equations
37A25 Ergodicity, mixing, rates of mixing
37A50 Dynamical systems and their relations with probability theory and stochastic processes
Full Text: DOI
[1] Albeverio, S.; Kondratiev, Y. G.; Räckner, M., Symmetrizing measures for infinite dimensional diffusions: an analytic approach, (Hildebrandt, Stefan; etal., Geometric Analysis and Nonlinear Partial Differential Equations, (2003), Springer Berlin), 475-486 · Zbl 1033.60083
[2] Al-Rashed, M.; Zegarliński, B., Monotone norms and Finsler structure in noncommutative spaces, Infin. Dimens. Anal. Qu., 17, No. 4, (2014), 1450029; http://dx.doi.org/10.1142/S0219025714500295. · Zbl 1320.46047
[3] Baudoin, F.; Hairer, M.; Teichmann, J., Ornstein-Uhlenbeck processes on Lie groups, J. Func. Analysis, 255, 877-890, (2008) · Zbl 1151.58018
[4] Baudoin, F.; Teichmann, J., Hypoellipticity in infinite dimensions and an application in interest rate theory, Ann. Appl. Probab., 15, 1765-1777, (2005) · Zbl 1081.60039
[5] Bakry, D.; Emery, M., Diffusions hypercontractives, (Sém. de Probab. XIX, Lecture Notes in Math, 1123, (1985), Springer Berlin), 177-206
[6] Bakry, D.; Baudoin, F.; Bonnefont, M.; Chafai, D., On gradient bounds for the heat kernel on the Heisenberg group, J. Func. Analysis, 255, 1905-1938, (2008) · Zbl 1156.58009
[7] Blanchard, Ph.; Hellmich, M.; Ługiewicz, P.; Olkiewicz, R., Quantum dynamical semigroups for finite and infinite Bose systems, J. Math. Phys., 48, (2007), Continuity and generators of dynamical semigroups for infinite Bose systems, J. Func. Analysis 256 (2009), 1453-1475. · Zbl 1121.81085
[8] Bobkov, S. G.; Goetze, F., Exponential integrability and transportation cost related to logarithmic Sobolev inequalities, J. Func. Analysis, 163, 1-28, (1999), http://www.idealibrary.com · Zbl 0924.46027
[9] Bodineau, T.; Zegarliński, B., Hypercontractivity via spectral theory, Infin. Dimens. Anal. Qu., 3, No. 1, 15-31, (2000) · Zbl 1040.47503
[10] Carbone, R.; Martinelli, A., Logarithmic Sobolev inequalities in non-commutative algebras, Infin. Dimens. Anal. Qu., 18, 1550011, (2015), http://dx.doi.org/10.1142/S0219025715500113 · Zbl 1341.46040
[11] Carbone, R.; Sasso, E., Hypercontractivity for a quantum Ornstein-Uhlenbeck semi-group, Probab. Theory Relat. Fields, 140, (2008), no. 3-4, 505-522. · Zbl 1138.81405
[12] Cipriani, F., Dirichlet forms and Markovian semigroups on standard forms of von Neumann algebras, J. Funct. Anal., 147, (1997), no. 2, 259-300; Dirichlet Forms on Noncommutative Spaces, pp. 161-276 in Quantum Potential Theory, Michael Schuermann & Uwe Franz (Eds)., Lecture Notes in Math. 1954, Springer 2008. · Zbl 0883.47031
[13] Choi, Veni; Ko, Chul-Ki; Park, Yong-Moon, Quantum Markovian semigroups on quantum spin system: Glauber dynamics, J. Korean Math. Soc., 45, 1075-1087, (2008), http://dx.doi.org/10.4134/JKMS.2008.45.4.1075 · Zbl 1158.46047
[14] Da Prato, G.; Zabczyk, J., Stochastic Equations in Infinite Dimensions, (1992), Cambridge Univ. Press · Zbl 0761.60052
[15] Dragoni, F.; Kontis, V.; Zegarliński, B., Ergodicity of Markov semigroups with Hörmander type generators in infinite dimensions, J. Potential Analysis, 37, 199-227, (2011), arXiv:1012.0257v1. · Zbl 1257.47047
[16] Driver, B. K.; Melcher, T., Hypoelliptic heat kernel inequalities on Lie groups, Stoch. Process. Appl., 118, 368-388, (2008), Hypoelliptic heat kernel inequalities on the Heisenberg group, J. Func. Analysis 221 (2005), 340-365. · Zbl 1138.22008
[17] Fougéres, P.; Roberto, C.; Zegarliński, B., Sub-Gaussian measures and associated semilinear problems, Rev. Mat. Iberoam., 28, (2012), no. 2, 305-350. http://rmi.rsme.es/index.php?option=com_contentview=sectionlayout=blogid=7Itemid=95lang=en · Zbl 1248.35227
[18] P. Fougéres, I. Gentil and B. Zegarliński: Application of logarithmic Sobolev inequality to Cauchy problems of reaction-diffusion equations, [hal-00987370]
[19] P. Fougéres, I. Gentil and B. Zegarliński: Ergodicity and smoothing for reaction-diffusion equations via application of logarithmic Sobolev inequality, work in progress.
[20] Fougéres, P.; Zegarliński, B., Semi-linear problems in infinite dimensions, J. Func. Analysis, 228, 39-88, (2005) · Zbl 1100.35113
[21] Goderis, D.; Maes, C., Constructing quantum dissipations and their reversible states from classical interacting spin systems, Ann. Inst. Henri Poincaré, 55, 805-828, (1991) · Zbl 0742.60106
[22] Guionnet, A.; Zegarliński, B., Lectures on logarithmic Sobolev inequalities, (Séminaire de Probabilités, XXXVI, 1-134, Lecture Notes in Math, 1801, (2003), Springer Berlin) · Zbl 1125.60111
[23] Hebisch, W.; Zegarliński, B., Coercive inequalities on metric measure spaces, J. Func. Analysis, 258, 814-851, (2010), http://dx.doi.org/10.1016/j.jfa.2009.05.016 · Zbl 1189.26032
[24] J. Inglis: Coercive inequalities for generators of Hormander type, PhD Thesis, Imperial College 2010.
[25] Inglis, J.; Kontis, V.; Zegarliński, B., From U-bounds to isoperimetry with applications to H-type groups, J. Func. Analysis, 260, 76-116, (2011) · Zbl 1210.26017
[26] Inglis, J.; Papageorgiou, I., Logarithmic Sobolev inequalities for infinite dimensional hoermander type generators on the Heisenberg group, J. Pot. Analysis, 31, 79-102, (2009) · Zbl 1180.22012
[27] Temme, K.; Pastawski, F.; Kastoryano, M. J., Hypercontractivity of quasi-free quantum semigroups, J. Phys. A: Math. Theor., 47, 405303, (2014), arXiv:1403.5224 · Zbl 1298.81122
[28] Kastoryano, M. J.; Temme, K., Quantum logarithmic Sobolev inequalities and rapid mixing, J. Math. Phys., 54, (2013), 052202, arXiv:1207.3261 · Zbl 1379.81021
[29] Kontis, V.; Ottobre, M.; Zegarliński, B., Markov semigroups with hypocoercive-type generator in infinite dimensions I: ergodicity and smoothing, J. Func. Analysis, 270, 3173-3223, (2016), http://arxiv.org/abs/1306.6452arXiv:1306.6452; Markov semigroups with hypocoercive-type generator in Infinite Dimensions II: Applications, Infin. Dimens. Anal. Qu. 2016, to appear; http://arxiv.org/abs/1306.6453arXiv:1306.6453 · Zbl 1341.47055
[30] Ługiewicz, P.; Zegarliński, B., Coercive inequalities for Hörmander type generators in infinite dimensions, J. Func. Anal., 247, 438-476, (2007) · Zbl 1128.58009
[31] Olkiewicz, R.; Ługiewicz, P.; Zegarliński, B., Nonlinear Markov semigroups on C^*-algebras, Infin. Dimens. Anal. Qu., 16, No. 1, 1350004, (2013), http://dx.doi.org/10.1142/S0219025713500045 · Zbl 1297.47064
[32] Ługiewicz, P.; Olkiewicz, R.; Zegarliński, B., Ergodic properties of diffusion-type quantum dynamical semigroups, J. Phys. A: Math. Theor., 43, 425207, (2010) · Zbl 1200.81098
[33] Lihu, Xu; Zegarliński, B., Existence and exponential mixing of infinite white α-stable systems with unbounded interactions, Electronic J. Probab., 15, 1994-2018, (2010), Ergodicity of finite and infinite dimensional α-stable systems, Stoch. Anal. Appl. 27 (2009), 797-824. · Zbl 1221.37160
[34] Li, H.-Q., Estimation optimale du gradient du semi-groupe de la chaleur sur le groupe de Heisenberg, J. Func. Analysis, 236, 369-394, (2006) · Zbl 1106.22009
[35] Liggett, T. M., Interacting Particle Systems; Stochastic Interacting Systems: Contact, Voter and Exclusion Processes, (1999), Springer · Zbl 0949.60006
[36] Taku Matsui: Interacting Particle Systems on Non-Commutative Spaces, pp.115-124 in M. Fannes, C. Maes and A. Verbeure (Eds.), On Three Levels: Micro-, Meso and Macro-phenomena in Physics, Plenum (1994), Proceedings of the ASI/ARW Workshop in Leuven, 19-23 July 1993; Purification and uniqueness of quantum Gibbs states, Commun. Math. Phys. 162 (1994), 321-332.
[37] V. P. Maslov: Nonlinear averages in economics, Mathematical Notes, 2005 - Springer; Nonlinear averaging axioms in financial mathematics and stock price dynamics, Theory of Probability & Its Applications, 2004 - SIAM; Quantum economics, Russian J. Math. Phys. 2006.
[38] Majewski, A. W.; Olkiewicz, R.; Zegarliński, B., Dissipative dynamics for quantum spin systems on a lattice, J. Phys. A: Math. Gen., 31, 2045, (1998), http://dx.doi.org/10.1088/0305-4470/31/8/015 · Zbl 0917.46059
[39] Olkiewicz, R.; Żaba, M., Dynamics of microcavity polaritions in the Markovian limit, Phys. Lett., A 372, 3176-3183, (2008), Dynamics of a degenerate parametric oscillator in a squeezed reservoir, Phys. Lett. A 372 (2008), 4985-4989. · Zbl 1220.81212
[40] Olkiewicz, R.; Zegarliński, B., Hypercontractivity in non-commutative L_p spaces, J. Func. Analysis, 161, 246-285, (1999) · Zbl 0923.46066
[41] Papageorgiou, I., The logarithmic Sobolev inequality for Gibbs measures on infinite product of Heisenberg groups, Markov Processes Relat. Fields, 20, 705-749, (2014), A note on the modified Log-Sobolev inequality, J. Potential Anal. 35 (2011), 3, 275-286; Concentration inequalities for Gibbs measures, Infin. Dimens. Anal. Qu. 14 (2011), 1, 79-104; The Logarithmic Sobolev Inequality in Infinite dimensions for Unbounded Spin systems on the Lattice with non Quadratic Interactions, Markov Processes Relat. Fields 16 (2010), 447-484. · Zbl 1315.60027
[42] Park, Y. M., Construction of Dirichlet forms on standard forms of von Neumann algebras, Infin. Dimens. Anal. Qu., 3, (2000), no. 1, 1-14; Ergodic property of Markovian semigroups on standard forms of von Neumann algebras, J. Math. Physics 46 (2005), 113507. · Zbl 1037.46502
[43] Röckner, M., L_p-analysis of finite and infinite dimensional diffusion operators, stochastic PDE’s and Kolmogorov’s equations in infinite dimensions, (Da Prato, Giuseppe, Lect. Notes Math., vol. 1715, (1999), Springer Berlin), 65-116, An analytic approach to Kolmogorov’s equations in infinite dimensions and probabilistic consequences, XIVth International Congress on Mathematical Physics, World Scientific, 2005, invited talk at the International Congress on Mathematical Physics (ICMP) 2003, pp. 520-526.
[44] Stroock, D. W.; Zegarliński, B., The equivalence of the logarithmic Sobolev inequality and the dobrushin-shlosman mixing condition, Commun. Math. Phys., 144, 303-323, (1992), The Logarithmic Sobolev Inequality for Discrete Spin Systems on a Lattice, Commun. Math. Phys. 149 (1992), 175-193. · Zbl 0745.60104
[45] Xu, Lihu; Olkiewicz, R.; Zegarliński, B., Nonlinear problems in infinite interacting particle systems, Infin. Dimens. Anal. Qu., 11, 179, (2008), http://dx.doi.org/10.1142/S0219025708003063 · Zbl 1160.47060
[46] Xu, Lihu; Zegarliński, B., Ergodicity of the finite and infinite dimensional α-stable systems, J. Stoch. Analysis Appl., 27, 797-824, (2009), Existence and Exponential mixing of infinite white α-stable Systems with unbounded interactions, Electronic J. Probability 15 (2010), Paper no. 65, pages 1994-2018. · Zbl 1181.60150
[47] F. Zak: Exponential ergodicity of infinite system of interacting diffusions, preprint. · Zbl 1350.60106
[48] Zegarliński, B., On log-Sobolev inequalities for infinite lattice systems, Lett. Math. Phys., 20, 173-182, (1990), Log-Sobolev inequalities for infinite one dimensional lattice systems, Commun. Math. Phys. 133 (1990), 147-162; Dobrushin uniqueness theorem and logarithmic Sobolev inequalities, J. Funct. Anal. 105 (1992), 77-111; The strong decay to equilibrium for the stochastic dynamics of an unbounded spin system on a lattice, Commun. Math. Phys. 175 (1996), 401-432. · Zbl 0717.47015
[49] Zegarliński, B., Analysis on extended Heisenberg group, Annales de la Faculté des Sciences de Toulouse, XX, Nr 2, 379-405, (2011) · Zbl 1253.47028
[50] B. Zegarliński: Crystallographic Groups of Hörmander Fields. I, http://arxiv.org/abs/1306.6453hal-01160736
[51] Zegarliński, B., (Analysis of classical and quantum interacting particle systems. Quantum interacting particle systems, (2002), River Edge, NJ, Publisher: World Sci. Publ.), 241-336
[52] Zegarliński, B., Linear and nonlinear concentration phenomena, Markov Processes Relat. Fields, 16, 753-782, (2010) · Zbl 1241.47035
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.