Стохастичнi динамiки неперервних систем

D. Finkelshtein

Анотація


Стаття являє собою вступ до теорiї випадкових (маркiвських) еволюцiй нескiнченних систем елементiв, розташованих у евклiдовому просторi. Розглянуто необхiднi теоретичнi побудови для дослiдження таких еволюцiй та наведено вiдповiдний огляд лiтератури, що включає, зокрема, приклади застосування цiєї теорiї до численних моделей математичної фiзики, бiологiї, екологiї, медицини, соцiологiї, економiки тощо. Робота є дещо переробленою версiєю першої частини докторської дисертацiї автора (д.ф.-м.н.,  2014).

Повний текст:

PDF

Посилання


Ju. M. Berezanskij, Ju. G. Kondrat'ev. Spektral'nye metody v beskonechnomernom analize. Kiev: Naukova dumka, 1988.

N. N. Bogoljubov. Problemy dinamicheskoj teorii v statisticheskoj fizike. Moskva-Leningrad: OGIZ, Gostehizdat, 1946.

A. M. Vershik, I. M. Gel'fand, M. I. Graev. Predstavlenija gruppy diffeomorfizmov. UMN, 30(6(186)):3–50, 1975.

Dzh. V. Gibbs. Osnovnye principy statisticheskoj mehaniki. Reguljarnaja i haoticheskaja dinamika, 2002.

R. L. Dobrushin, Ja. G. Sinaj, Ju. M. Cuhov. Dinamicheskie sistemy- 2. Itogi nauki i tehn. Ser. Sovrem. probl. mat. Fundam. napravlenija, chapter Dinamicheskie sistemy statisticheskoj mehaniki, pages 235–284. Moskva: VINITI, 1985.

K. Cherchin'jani. Teorija i prilozhenija uravnenija Bol'cmana. Moskva: Mir, 1978.

Neravnovesnye javlenija. Uravnenie Bol'cmana // Pod red. D. L. Libovica i E. U. Montrolla. Moskva: Mir, 1986.

S. Albeverio, Y. Kondratiev, and M. R¨ockner. Analysis and geometry on configuration spaces. J. Funct. Anal., 154(2):444–500, 1998.

S. Albeverio, Y. Kondratiev, and M. R¨ockner. Analysis and geometry on configuration spaces: the Gibbsian case. J. Funct. Anal., 157(1):242–291, 1998.

D. Aldous. Interacting particle systems as stochastic social dynamics. Bernoulli, 19(4):1122–1149, 2013.

W. Arendt, A. Grabosch, G. Greiner, U. Groh, H. P. Lotz, U. Moustakas, R. Nagel, F. Neubrander, and U. Schlotterbeck. One-parameter semigroups of positive operators, volume 1184 of Lecture Notes in Mathemati- cs. Springer-Verlag, Berlin, 1986. x+460 pp.

V. P. Belavkin and V. N. Kolokolttsov. On a general kinetic equation for manyparticle systems with interaction, fragmentation and coagulation. R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 459(2031):727–748, 2003.

V. P. Belavkin, V. P. Maslov, and S. E` . Tariverdiev. The asymptotic dynamics of a system with a large number of particles described by Kolmogorov-Feller equations. Teoret. Mat. Fiz., 49(3):298–306, 1981.

Y. M. Berezanski˘ı. The generalized moment problem associated with correlation measures. Funktsional. Anal. i Prilozhen., 37(4):86–91, 2003.

Y. Berezansky, Y. Kondratiev, T. Kuna, and E. Lytvynov. On a spectral representation for correlation measures in configuration space analysis. Methods Funct. Anal. Topology, 5(4):87–100, 1999.

Y. M. Berezansky and D. A. Mierzejewski. The investigation of a generalized moment problem associated with correlation measures. Methods Funct. Anal. Topology, 13(2):124–151, 2007.

C. Berns, Y. Kondratiev, Y. Kozitsky, and O. Kutoviy. Kawasaki dynamics in continuum: micro- and mesoscopic descriptions. J. of Dyn and Diff. Eqn., 25(4):1027–1056, 2013.

C. Berns, Y. Kondratiev, and O. Kutoviy. Construction of a state evoluti- on for kawasaki dynamics in continuum. Analysis and Mathematical Physics, 3(2):97–117, 2013.

C. Boldrighini, Y. Kondratiev, R. Minlos, A. Pellegrinotti, and E. Zhizhina. Random jumps in evolving random environment. Markov Process. Related Fields, 14(4):543–570, 2008.

B. Bolker and S. W. Pacala. Using moment equations to understand stochastically driven spatial pattern formation in ecological systems. Theor. Popul. Biol., 52(3):179–197, 1997.

D. G. Bossomaier, Terry R. J. and Green, editor. Complex systems. Cambridge University Press, Cambridge, 2000. vi+413 pp.

N. R. Campbell. The study of discontinuous problem. Proc. Cambridge Philos. Soc., 15:117–136, 1909.

N. R. Campbell. Discontinuities in light emission. Proc. Cambridge Phi- los. Soc., 15:310–328, 1910.

C. Cercignani, V. I. Gerasimenko, and D. Y. Petrina. Manyparticle dynamics and kinetic equations, volume 420 of Mathematics and its Applications. Kluwer Academic Publishers Group, Dordrecht, 1997. viii+244 pp.

N. Champagnat, R. Ferri`ere, and S. M´el´eard. From individual stochastic processes to macroscopic models in adaptive evolution. Stoch. Models, 24 (suppl. 1):2–44, 2008.

A. C.-L. Chian. Complex systems approach to economic dynamics, volume 592 of Lecture Notes in Economics and Mathematical Systems. Springer, Berlin, 2007. x+101 pp.

A. De Masi, I. Merola, E. Presutti, and Y. Vignaud. Potts models in the continuum. Uniqueness and exponential decay in the restricted ensembles. J. Stat. Phys., 133(2):281–345, 2008.

A. De Masi, I. Merola, E. Presutti, and Y. Vignaud. Coexistence of ordered and disordered phases in Potts models in the continuum. J. Stat. Phys., 134(2):243–306, 2009.

A. De Masi and E. Presutti. Mathematical methods for hydrodynamic limits, volume 1501 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 1991. x+196 pp.

X. Descombes, R. Minlos, and E. Zhizhina. Object extraction using a stochastic birth-and-death dynamics in continuum. J. Math. Imaging Vision, 33(3):347–359, 2009.

U. Dieckmann and R. Law. Relaxation projections and the method of moments. In The Geometry of Ecological Interactions, pages 412–455. Cambridge University Press, Cambridge, UK, 2000.

R. Durrett, T. M. Liggett, F. Spitzer, and A.-S. Sznitman. Interacting particle systems at Saint-Flour. Probability at Saint-Flour. Springer, Heidelberg, 2012. viii+331 pp.

R. Durrett and J. Mayberry. Evolution in predator-prey systems. Stochastic Process. Appl., 120(7):1364–1392, 2010.

D. Filonenko, D. Finkelshtein, and Y. Kondratiev. On two-component contact model in continuum with one independent component. Methods Funct. Anal. Topology, 14(3):209–228, 2008.

D. Finkelshtein. Functional evolutions for homogeneous stationary death- immigration spatial dynamics. Methods Funct. Anal. Topology, 17(4): 300–318, 2011.

D. Finkelshtein. On convolutions on configuration spaces. I. Spaces of finite configurations. Ukrainian Math. J., 64(11):1752–1775, 2013.

D. Finkelshtein. On convolutions on configuration spaces. II. Spaces of locally finite configurations. Ukrainian Math. J., 64(12):1919–1944, 2013.

D. Finkelshtein and Y. Kondratiev. Regulation mechanisms in spatial stochastic development models. J. Stat. Phys., 136(1):103–115, 2009.

D. Finkelshtein, Y. Kondratiev, and Y. Kozitsky. Glauber dynamics in continuum: a constructive approach to evolution of states. Discrete and Cont. Dynam. Syst. - Ser A., 33(4):1431–1450, 4 2013.

D. Finkelshtein, Y. Kondratiev, Y. Kozitsky, and O. Kutoviy. Stochastic evolution of a continuum particle system with dispersal and competiti- on: Micro- and mesoscopic description. The European Physical Journal Special Topics, 216(1):107–116, 2013.

D. Finkelshtein, Y. Kondratiev, Y. Kozitsky, and O. Kutoviy. The statisti- cal dynamics of a spatial logistic model and the related kinetic equation. Math. Models Methods Appl. Sci., 25(2):343–370, 2015.

D. Finkelshtein, Y. Kondratiev, and O. Kutoviy. Individual based model with competition in spatial ecology. SIAM J. Math. Anal., 41(1):297–317, 2009.

D. Finkelshtein, Y. Kondratiev, and O. Kutoviy. Vlasov scaling for stochastic dynamics of continuous systems. J. Stat. Phys., 141(1):158– 178, 2010.

D. Finkelshtein, Y. Kondratiev, and O. Kutoviy. Vlasov scaling for the Glauber dynamics in continuum. Infin. Dimens. Anal. Quantum Probab. Relat. Top., 14(4):537–569, 2011.

D. Finkelshtein, Y. Kondratiev, and O. Kutoviy. Correlation functions evolution for the Glauber dynamics in continuum. Semigroup Forum, 85: 289–306, 2012.

D. Finkelshtein, Y. Kondratiev, and O. Kutoviy. Semigroup approach to birth-and-death stochastic dynamics in continuum. J. of Funct. Anal., 262(3):1274–1308, 2012.

D. Finkelshtein, Y. Kondratiev, and O. Kutoviy. Establishment and fecundity in spatial ecological models: statistical approach and kinetic equations. Infin. Dimens. Anal. Quantum Probab. Relat. Top., 16(2): 1350014 (24 pages), 2013.

D. Finkelshtein, Y. Kondratiev, and O. Kutoviy. An operator approach to Vlasov scaling for some models of spatial ecology. Methods Funct. Anal. Topology, 19(2):108–126, 2013.

D. Finkelshtein, Y. Kondratiev, and O. Kutoviy. Statistical dynamics of continuous systems: perturbative and approximative approaches. Arab. J. Math., 2014.

D. Finkelshtein, Y. Kondratiev, O. Kutoviy, and E. Lytvynov. Binary jumps in continuum. I. Equilibrium processes and their scaling limits. J. Math. Phys., 52:063304:1–25, 2011.

D. Finkelshtein, Y. Kondratiev, O. Kutoviy, and E. Lytvynov. Binary jumps in continuum. II. Non-equilibrium process and a Vlasov-type scali- ng limit. J. Math. Phys., 52:113301:1–27, 2011.

D. Finkelshtein, Y. Kondratiev, O. Kutoviy, S. Molchanov, and E. Zhizhina. Density behavior of spatial birth-and-death stochastic evolution of mutating genotypes under selection rates. Russ. J. Math. Phys., 21 (4):450–459, 2014.

D. Finkelshtein, Y. Kondratiev, O. Kutoviy, and M. J. Oliveira. Dynami- cal Widom–Rowlinson model and its mesoscopic limit. J. Stat. Phys., 158(1):57–86, 2015.

D. Finkelshtein, Y. Kondratiev, O. Kutoviy, and E. Zhizhina. An approxi- mative approach for construction of the Glauber dynamics in continuum. Math. Nachr., 285(2–3):223–235, 2012.

D. Finkelshtein, Y. Kondratiev, O. Kutoviy, and E. Zhizhina. On an aggregation in birth-and-death stochastic dynamics. Nonlinearity, 27: 1105–1133, 2014.

D. Finkelshtein, Y. Kondratiev, and E. Lytvynov. Equilibrium Glauber dynamics of continuous particle systems as a scaling limit of Kawasaki dynamics. Random Oper. Stoch. Equ., 15(2):105–126, 2007.

D. Finkelshtein, Y. Kondratiev, and M. J. Oliveira. Markov evolutions and hierarchical equations in the continuum. I. One-component systems. J. Evol. Equ., 9(2):197–233, 2009.

D. Finkelshtein, Y. Kondratiev, and M. J. Oliveira. Glauber dynamics in the continuum via generating functionals evolution. Complex Analysis and Operator Theory, 6(4):923–945, 2012.

D. Finkelshtein, Y. Kondratiev, and M. J. Oliveira. Kawasaki dynamics in the continuum via generating functionals evolution. Methods Funct. Anal. Topology, 18(1):55–67, 2012.

D. Finkelshtein, Y. Kondratiev, and M. J. Oliveira. Markov evolutions and hierarchical equations in the continuum. II: Multicomponent systems. Reports Math. Phys., 71(1):123–148, 2013.

D. Finkelshtein and M. J. Oliveira. A survey on bogoliubov generati- ng functionals for interacting particle systems in the continuum. In C. Bernardin and P. Gonc¸alves, editors, From Particle Systems to Partial Differential Equations, volume 75 of Springer Proceedings in Mathematics & Statistics, pages 161–177. Springer-Verlag Berlin Heidelberg, 2014.

G. B. Folland. Real analysis. Pure and Applied Mathematics (New York). John Wiley & Sons Inc., New York, second edition, 1999. xvi+386 pp.

N. Fournier and S. Meleard. A microscopic probabilistic description of a locally regulated population and macroscopic approximations. The Annals of Applied Probability, 14(4):1880–1919, 2004.

M. Fukushima, Y. O¯ shima, and M. Takeda. Dirichlet forms and symmetric Markov processes, volume 19 of de Gruyter Studies in Mathematics. Walter de Gruyter & Co., Berlin, 1994. x+392 pp.

N. L. Garcia and T. G. Kurtz. Spatial birth and death processes as solutions of stochastic equations. ALEA Lat. Am. J. Probab. Math. Stat., 1:281–303 (electronic), 2006.

N. L. Garcia and T. G. Kurtz. Spatial point processes and the projection method. Progress in Probability, 60:271–298, 2008.

H.-O. Georgii and O. H¨aggstro¨m. Phase transition in continuum Potts models. Comm. Math. Phys., 181(2):507–528, 1996.

H.-O. Georgii, S. Miracle-Sole, J. Ruiz, and V. A. Zagrebnov. Mean-field theory of the Potts gas. J. Phys. A, 39(29):9045–9053, 2006.

J. W. Gibbs. Elementary principles in statistical mechanics: developed wi- th especial reference to the rational foundation of thermodynamics. Dover publications Inc., New York, 1960. xviii+207 pp.

E. Gl¨otzl. Time reversible and Gibbsian point processes. I. Markovian spatial birth and death processes on a general phase space. Math. Nachr., 102:217–222, 1981.

E. Glo¨tzl. Time reversible and Gibbsian point processes. II. Markovian particle jump processes on a general phase space. Math. Nachr., 106: 63–71, 1982.

M. Grothaus, Y. Kondratiev, E. Lytvynov, and M. R¨ockner. Scaling limit of stochastic dynamics in classical continuous systems. Ann. Probab., 31 (3):1494–1532, 2003.

R. A. Holley and D. W. Stroock. Nearest neighbor birth and death processes on the real line. Acta Math., 140(1-2):103–154, 1978.

C. Kipnis and C. Landim. Scaling limits of interacting parti- cle systems, volume 320 of Grundlehren der Mathematischen Wi- ssenschaften. Springer-Verlag, Berlin, 1999. xvi+442 pp.

V. N. Kolokoltsov. Kinetic equations for the pure jump models of k- nary interacting particle systems. Markov Process. Related Fields, 12(1): 95–138, 2006.

Y. Kondratiev and T. Kuna. Harmonic analysis on configuration space. I. General theory. Infin. Dimens. Anal. Quantum Probab. Relat. Top., 5 (2):201–233, 2002.

Y. Kondratiev and T. Kuna. Correlation functionals for Gibbs measures and Ruelle bounds. Methods Funct. Anal. Topology, 9(1):9–58, 2003.

Y. Kondratiev, T. Kuna, and M. J. Oliveira. On the relations between Poissonian white noise analysis and harmonic analysis on configuration spaces. J. Funct. Anal., 213(1):1–30, 2004.

Y. Kondratiev, T. Kuna, and M. J. Oliveira. Holomorphic Bogoliubov functionals for interacting particle systems in continuum. J. Funct. Anal., 238(2):375–404, 2006.

Y. Kondratiev, O. Kutoviy, and E. Lytvynov. Diffusion approximation for equilibrium Kawasaki dynamics in continuum. Stochastic Process. Appl., 118(7):1278–1299, 2008.

Y. Kondratiev, O. Kutoviy, and R. Minlos. On non-equilibrium stochastic dynamics for interacting particle systems in continuum. J. Funct. Anal., 255(1):200–227, 2008.

Y. Kondratiev, O. Kutoviy, and S. Pirogov. Correlation functions and invariant measures in continuous contact model. Infin. Dimens. Anal. Quantum Probab. Relat. Top., 11(2):231–258, 2008.

Y. Kondratiev, O. Kutoviy, and E. Zhizhina. Nonequilibrium Glauber- type dynamics in continuum. J. Math. Phys., 47(11):113501, 17, 2006.

Y. Kondratiev and E. Lytvynov. Glauber dynamics of continuous particle systems. Ann. Inst. H. Poincar´e Probab. Statist., 41(4):685–702, 2005.

Y. Kondratiev, E. Lytvynov, and M. R¨ockner. Infinite interacting diffusi- on particles. I. Equilibrium process and its scaling limit. Forum Math., 18(1):9–43, 2006.

Y. Kondratiev, E. Lytvynov, and M. R¨ockner. Equilibrium Kawasaki dynamics of continuous particle systems. Infin. Dimens. Anal. Quantum Probab. Relat. Top., 10(2):185–209, 2007.

Y. Kondratiev, E. Lytvynov, and M. R¨ockner. Non-equilibrium stochastic dynamics in continuum: The free case. Cond. Matter Phys., 11(4(56)): 701–721, 2008.

Y. Kondratiev, R. Minlos, and E. Zhizhina. One-particle subspace of the Glauber dynamics generator for continuous particle systems. Rev. Math. Phys., 16(9):1073–1114, 2004.

Y. Kondratiev, R. Minlos, and E. Zhizhina. Self-organizing birth-and- death stochastic systems in continuum. Rev. Math. Phys., 20(4):451–492, 2008.

Y. Kondratiev, E. Pechersky, and S. Pirogov. Markov process of muscle motors. Nonlinearity, 21(8):1929–1936, 2008.

Y. Kondratiev, A. Rebenko, and M. R¨ockner. On diffusion dynamics for continuous systems with singular superstable interaction. J. Math. Phys., 45(5):1826–1848, 2004.

Y. Kondratiev and A. Skorokhod. On contact processes in continuum. Infin. Dimens. Anal. Quantum Probab. Relat. Top., 9(2):187–198, 2006.

Y. Kondratiev and E. Zhizhina. Spectral analysis of a stochastic Ising model in continuum. J. Stat. Phys., 129(1):121–149, 2007.

T. Kuna. Studies in configuration space analysis and applications. Bonner Mathematische Schriften [Bonn Mathematical Publications], 324. Universita¨t Bonn Mathematisches Institut, Bonn, 1999. ii+187 pp. Dissertation, Rheinische Friedrich-Wilhelms-Universita¨t Bonn, Bonn, 1999.

T. Kuna, Y. Kondratiev, and J. L. da Silva. Marked Gibbs measures via cluster expansion. Methods Funct. Anal. Topology, 4(4):50–81, 1998.

J. Lebowitz and E. Lieb. Phase transition in a continuum classical system with finite inteiactions. Physics Letters A, 39:98–100, 1972.

A. Lenard. Correlation functions and the uniqueness of the state in classical statistical mechanics. Comm. Math. Phys., 30:35–44, 1973.

A. Lenard. States of classical statistical mechanical systems of infinitely many particles. I. Arch. Rational Mech. Anal., 59(3):219–239, 1975.

A. Lenard. States of classical statistical mechanical systems of infinitely many particles. II. Characterization of correlation measures. Arch. Rational Mech. Anal., 59(3):241–256, 1975.

S. A. Levin. Complex adaptive systems: exploring the known, the unknown and the unknowable. Bulletin of the AMS, 40(1):3–19, 2002.

T. M. Liggett. Interacting particle systems. Springer-Verlag, New York, 1985ю xv+488 pp.

T. M. Liggett. Stochastic interacting systems: contact, voter and exclusi- on processes, volume 324 of Grundlehren der Mathematischen Wi- ssenschaften. Springer-Verlag, Berlin, 1999. xii+332 pp.

H. P. Lotz. Uniform convergence of operators on L∞ and similar spaces. Math. Z., 190(2):207–220, 1985.

Z. M. Ma and M. R¨ockner. Introduction to the theory of (nonsymmetric) Dirichlet forms. Universitext. Springer-Verlag, Berlin, 1992. vi+209 pp.

K. Matthes, J. Kerstan, and J. Mecke. Infinitely divisible point processes. John Wiley & Sons, Chichester-New York-Brisbane, 1978. xii+532 pp.

J. Mecke. Eine charakteristische Eigenschaft der doppelt stochastischen Poissonschen Prozesse. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 11:74–81, 1968.

U. Murrell, David J. and Dieckmann and R. Law. On moment closures for population dynamics in continuous space. Journal of Theoretical Biology, 229(3):421 – 432, 2004.

G. Nicolis and C. Nicolis. Foundations of complex systems. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2007. xiv+328 pp. Nonlinear dynamics, statistical physics, information and prediction.

O. Ovaskainen, D. Finkelshtein, O. Kutoviy, S. Cornell, B. Bolker, and Y. Kondratiev. A mathematical framework for the analysis of spatial- temporal point processes. Theoretical Ecology, 7(1):101–113, 2014.

G. Papanicolaou, editor. Hydrodynamic behavior and interacting particle systems, volume 9 of The IMA Volumes in Mathematics and its Appli- cations. Springer-Verlag, New York, 1987. x+210 pp.

K. R. Parthasarathy. Probability measures on metric spaces. Probability and Mathematical Statistics, No. 3. Academic Press Inc., New York, 1967. xi+276 pp.

M. D. Penrose. Existence and spatial limit theorems for lattice and conti- nuum particle systems. Prob. Surveys, 5:1–36, 2008.

D. Y. Petrina, V. I. Gerasimenko, and P. V. Malyshev. Mathematical foundations of classical statistical mechanics, volume 8 of Advanced Studies in Contemporary Mathematics. Taylor & Francis, London, second edi- tion, 2002. x+338 pp.

C. Preston. Spatial birth-and-death processes. Bull. Inst. Internat. Stati- st., 46(2):371–391, 405–408, 1975.

X. Qi. A functional central limit theorem for spatial birth and death processes. Adv. in Appl. Probab., 40(3):759–797, 2008.

A. Quarteroni, L. Formaggia, and A. Veneziani, editors. Complex systems in biomedicine. Springer-Verlag Italia, Milan, 2006. xiv+292 pp.

M. R¨ockner. Stochastic analysis on configuration spaces: basic ideas and recent results. In New directions in Dirichlet forms, volume 8 of AMS/IP Stud. Adv. Math., pages 157–231. Amer. Math. Soc., Providence, RI, 1998.

D. Ruelle. Cluster property of the correlation functions of classical gases. Rev. Modern Phys., 36:580–584, 1964.

D. Ruelle. Statistical mechanics: Rigorous results. W. A. Benjamin, Inc., New York-Amsterdam, 1969. xi+219 pp.

D. Ruelle. Superstable interactions in classical statistical mechanics. Comm. Math. Phys., 18:127–159, 1970.

D. Ruelle. Existence of a phase transition in a continuous classical system. Phys. Rev. Letters, 27:1040–1041, 1971.

H. Shimomura. Poisson measures on the configuration space and unitary representations of the group of diffeomorphisms. J. Math. Kyoto Univ., 34(3):599–614, 1994.

A. V. Skorohod. On the differentiability of measures which correspond to stochastic processes. I. Processes with independent increments. Teor. Veroyatnost. i Primenen, 2:417–443, 1957.

H. Spohn. Kinetic equations from Hamiltonian dynamics: Markovian limits. Rev. Modern Phys., 52(3):569–615, 1980.

Y. Takahashi. Absolute continuity of Poisson random fields. Publ. Res. Inst. Math. Sci., 26(4):629–647, 1990.

B. Widom and J. Rowlinson. New model for the study of liquid-vapor phase transitions. J. Chem. Phys., 52(4):1670–1684, 1970.


Пристатейна бібліографія ГОСТ




Посилання

  • Поки немає зовнішніх посилань.