Book contents
- Frontmatter
- Contents
- Preface
- Basic definitions, notation and abbreviations
- 1 Introduction
- Part I Tools from Markov process theory
- Part II Nonlinear Markov processes and semigroups
- Part III Applications to interacting particles
- Appendices
- A Distances on measures
- B Topology on càdlàg paths
- C Convergence of processes in Skorohod spaces
- D Vector-valued ODEs
- E Pseudo-differential operator notation
- F Variational derivatives
- G Geometry of collisions
- H A combinatorial lemma
- I Approximation of infinite-dimensional functions
- J Bogolyubov chains, generating functionals and Fock-space calculus
- K Infinite-dimensional Riccati equations
- References
- References
References
from Appendices
Published online by Cambridge University Press: 05 July 2014
Book contents
- Frontmatter
- Contents
- Preface
- Basic definitions, notation and abbreviations
- 1 Introduction
- Part I Tools from Markov process theory
- Part II Nonlinear Markov processes and semigroups
- Part III Applications to interacting particles
- Appendices
- A Distances on measures
- B Topology on càdlàg paths
- C Convergence of processes in Skorohod spaces
- D Vector-valued ODEs
- E Pseudo-differential operator notation
- F Variational derivatives
- G Geometry of collisions
- H A combinatorial lemma
- I Approximation of infinite-dimensional functions
- J Bogolyubov chains, generating functionals and Fock-space calculus
- K Infinite-dimensional Riccati equations
- References
- References
Summary
A summary is not available for this content so a preview has been provided. Please use the Get access link above for information on how to access this content.
- Type
- Chapter
- Information
- Nonlinear Markov Processes and Kinetic Equations , pp. 360 - 372Publisher: Cambridge University PressPrint publication year: 2010
References
[1] , (eds.). Quantum interacting particle systems. In: Proc. Volterra-CIRM Int. School, Trento, 2000, QP-PQ: Quantum Probability and White Noise Analysis, vol. 14, World Scientific, 2002.
[2] , and . Sur le comportement asymptotique du noyau associé à une diffusion dégénéré. C.R. Math. Rep. Acad. Sci. Canada 22:4 (2000), 151–159.Google Scholar
[3] , . Stochastic integrals and the Lévy-Ito decomposition theorem on separable Banach spaces. Stoch. Anal. Appl. 23:2 (2005), 217–253.Google Scholar
[4] . Deterministic and stochastic models for coalescence (aggregation and coagulation): a review of the mean-field theory for probabilists. Bernoulli 5:1 (1999), 3–48.Google Scholar
[5] . Coagulation-fragmentation processes. Arch. Ration. Mech. Anal. 151 (2000), 339–366.Google Scholar
[6] . Continuous-Time Markov Chains. Probability and its Applications. Springer Series in Statistics. Springer, 1991.
[7] . Probability and Information.Cambridge University Press, 1996.
[8] . Lévy Processes and Stochastic Calculus. Cambridge Studies in Advanced Mathematics, vol. 93. Cambridge University Press, 2004.
[10] . On the Boltzmann equation. Parts I and II. Arch. Ration. Mech. Anal. 45 (1972), 1–35.Google Scholar
[11] . L∞ Estimates for the spatially-homogeneous Boltzmann equation. J. Stat. Phys. 31:2 (1983), 347–361.Google Scholar
[12] . Lektsii o kineticheskikh uravneniyakh (in Russian) (Lectures on kinetic equations). Nauka, Moscow, 1992.
[13] , . On a connection between the solution of the Boltzmann equation and the solution of the Landau-Fokker-Planck equation (in Russian). Mat. Sb. 181:4 (1990), 435-446; English translation in Math. USSR Sb. 69:2 (1991), 465–478.Google Scholar
[14] . Sensitivity for Smoluchovski equation. Preprint 2009. http://www. statslab.cam.ac.uk/ismael/files/Sensitivity.pdf.
[15] , . Fundamentals of Stochastic Filtering. Stochastic Modelling and Applied Probability, vol. 60. Springer, 2009.
[16] . Statistical Dynamics. Matter out of Equilibrium.Imperial College Press, 1997.
[17] , . The discrete coagulation-fragmentation equations: existence, uniqueness and density conservation. J. Stat. Phys. 61 (1990), 203–234.Google Scholar
[18] . Uniqueness in law for pure jump type Markov processes. Prob. Theory Relat. Fields 79 (1988), 271–287.Google Scholar
[19] , . Systems of equations driven by stable processes. Prob. Theory Relat. Fields 134 (2006), 175–214.Google Scholar
[20] , , . Limit theorems for coupled continuous time random walks. Ann. Prob. 32:1B (2004), 730–756.Google Scholar
[21] . Quantum branching processes and nonlinear dynamics of multi-quantum systems (in Russian). Dokl. Acad. Nauk SSSR 301:6 (1988), 1348–1352.Google Scholar
[23] , . Stochastic evolutions as boundary value problems. In: Infinite Dimensional Analysis and Quantum Probability, RIMS Kokyuroku 1227 (2001), 83–95.
[24] , . Stochastic evolution as interaction representation of a boundary value problem for Dirac type equation. Inf. Dim. Anal., Quantum Prob. Relat. Fields 5:1 (2002), 61–92.Google Scholar
[25] , . On general kinetic equation for many particle systems with interaction, fragmentation and coagulation. Proc. Roy. Soc. London A 459 (2003), 727–748.Google Scholar
[26] , . Uniformization method in the theory of nonlinear hamiltonian systems of Vlasov and Hartree type (in Russian). Teoret. i Matem. Fizika 33:1 (1977), 17–31. English translation in Theor. Math. Phys. 43:3 (1977), 852–862.Google Scholar
[27] . Dynamic Programming. Princeton University Press and Oxford University Press, 1957.
[28] . Developpement asymptotique du noyau de la chaleur sur la diagonale. Ann. Inst. Fourier 39:1 (1989), 73–99.Google Scholar
[29] . Asymptotic formulas for symmetric stable semigroups. Exp. Math. 12 (1994), 381–384.Google Scholar
[30] , , , , , . Fractionally stable distributions. In: , (eds.), Stochastic Models ofPlasma Turbulence (in Russian), Moscow State University, 2003, pp. 291-360. English translation in V. Korolev, N. Skvortsova (eds.), Stochastic Models ofStructural Plasma Turbulence, VSP, 2006, pp. 175-244.
[31] , , . Limit theorems for continuous-time random walks in the double array limit scheme. J. Math. Sci. (NY) 138:1 (2006), 5348–5365.Google Scholar
[32] , . Stochastic differential equations with polar-decomposed Levy measures and applications to stochastic optimization. Fron. Math. China 2:4 (2007), 539–558.Google Scholar
[33] . Lévy Processes. Cambridge Tracts in Mathematics, vol. 121, Cambridge University Press, 1996.
[34] . Random Fragmentation and Coagulation Processes. Cambridge Studies in Advanced Mathematics, vol. 102, Cambridge University Press, 2006.
[35] . Stochastic Integration with Jumps. Encyclopedia of Mathematics and Applications, Cambridge University Press, 2002.
[36] , , . Malliavin Calculus for Processes with Jumps. Stochastic Monographs, vol. 2, Gordon and Breach, 1987.
[37] , . Global existence versus blow up for some models of interacting particles. Colloq. Math. 106:2 (2006), 293–303.Google Scholar
[38] . Convergence of Probability Measures. Wiley, 1968.
[39] , . Potential Theory – An Analytic Approach to Balayage. Universitext, Springer, 1986.
[40] , . Some theorems on stable processes. Trans. Amer. Math. Soc. 95 (1960), 263–273.Google Scholar
[41] . The theory of the nonlinear spatially uniform Boltzmann equation for Maxwell molecules. Sov. Sci. Rev. C, Math. Phys. Rev. 7 (1988), 111–233.Google Scholar
[42] . Problems of the Dynamic Theory in Statistical Physics. Moscow, 1946 (in Russian).
[43] , , . Semi-groupes de Feller sur une variété a bord compacte et problèmes aux limites intégro-différentiels du second ordre donnant lieu au principe du maximum. Ann. Inst. Fourier, Grenoble 18:2 (1968), 369–521.Google Scholar
[44] , . Geometric Inequalities. Springer, 1988.
[45] . Problèmes mathématique dans la théorie cinétique des gaz. Almquist and Wiksells, 1957.
[46] , . Nonlinear Stochastic Integrators, Equations and Flows. Stochatic Monographs, vol. 6, Gordon and Breach, 1990.
[47] , , . The Mathematical Theory of Dilute Gases. Springer, 1994.
[48] . A priori estimates for quantum dynamic semigroups (in Russian). Teoret. Mat. Fiz 134:2 (2003), 185-190; English translation in Theor. Math. Phys. 134:2 (2003), 160–165.Google Scholar
[49] , . Sufficient conditions for conservativity of minimal quantum dynamic semigroups. J. Funct. Anal. 118 (1993), 131–153.Google Scholar
[50] , . Sufficient conditions for conservativity of minimal quantum dynamic semigroups. J. Funct. Anal. 153 (1998), 382–104.Google Scholar
[51] , . Existence of solutions to coagulation-fragmentation systems with diffusion. Transport Theory Statist. Phys. 25 (1996), 503–513.Google Scholar
[52] . Sur la forme integro-différentiélle du générateur infinitésimal d'un semi-groupe de Feller sur une variété. In: Sém. Théorie du Potentiel, 1965–1966. Exposé 3.
[53] , , . Long-time behaviour and self-similarity in a coagulation equation with input of monomers. Markov Proc. Relat. Fields 12 (2006), 367–398.Google Scholar
[54] , . Approximate McKean-Vlasov representations for a class of SPDEs. To appear in Stochastics.
[55] . Riccati equations for stable well-posed linear systems: the generic case. SIAMJ. Control Optim. 42: 5 (2003), 1681-1702 (electronic).Google Scholar
[56] . Quantum Theory of Open Systems. Academic Press, 1976.
[57] . Heat Kernels and Spectral Theory. Cambridge University Press, 1992.
[58] . Critical dynamics and fluctuations for a mean-field model of cooperative behavior. J. Stat. Phys. 31: 1 (1983), 29–85.Google Scholar
[59] . Measure-valued Markov processes. In: (ed.), Proc. Ecole d'Eté de probabilités de Saint-Flour XXI, 1991. Springer Lecture Notes in Mathematics, vol. 1541, 1993, pp. 1-260.
[60] et al.Generalized Mehler semigroups and catalytic branching processes with immigration. Potential Anal. 21:1 (2004), 75–97.Google Scholar
[61] , . Mathematical Methods for Hydrodynamic Limits. Springer, 1991.
[62] , , . A pure jump Markov process associated with Smoluchovski's coagulation equation. Ann. Prob. 30:4 (2002), 1763–1796.Google Scholar
[63] , , . Rate of convergence of a stochastic particle system for the Smoluchovski coagulation equation. Methodol. Comput. Appl. Prob. 5:2 (2003), 131–158.Google Scholar
[64] . Feynman-Kac Formulae. Genealogical and Interacting particle Systems with Applications. Probability and its Application. Springer, 2004.
[65] , . On the spatially homogeneous Landau equation for hard potentials. Part I. Comm. Partial Diff. Eq. 25 (2000), 179–259.Google Scholar
[66] , . Unimodality, Convexity, and Applications. Academic Press, 1988.
[67] , . Constructions of diffusions on path spaces and loop spaces of compact riemannian manifolds. C.R. Acad. Sci. Paris, Ser. I 320 (1995), 1249–1254.Google Scholar
[68] , . Existence, uniqueness and mass conservation for the coagulation-fragmentation equation. Math. Meth. Appl. Sci. 19 (1996), 571–591.Google Scholar
[69] . Superdiffusions and Positive Solutions of Nonlinear Partial Differential Equations. University Lecture Series, vol. 34, American Mathematical Society, 2004.
[70] , . Stochastic particle approximation to Smoluchovski's coagulation equation. Ann. Appl. Prob. 11:4 (2001), 1137–1165.Google Scholar
[71] . Global boundedness of moments of solutions of the Boltzmann equation for forces of inifinite range. Arch. Ration. Mech. Anal. 82 (1983), 1–12.Google Scholar
[72] , . Self-preservation and gelation during turbulance induced coagulation. J. Aerosol Sci. 37:2 (2006), 123–142.Google Scholar
[73] . An Introduction to Superprocesses. University Lecture Series, vol. 20, American Mathematical Society, 2000.
[74] , . Markov Processes – Characterization and Convergence. Wiley Series in Probability and Mathematical Statistics, Wiley, 1986.
[75] , . Feller semigroups obtained by variable order subordination. Rev. Mat. Comput. 20:2 (2007), 293–307.Google Scholar
[76] . An Introduction to Probability. Theory and Applications, second edition, vol. 2. John Wiley and Sons, 1971.
[77] , . Local properties of self-similar solutions to Smoluchowski's coagulation equation with sum kernels. Proc. Roy. Soc. Edinburgh. A 136 :3 (2006), 485–508.Google Scholar
[78] . Functional Integration and Partial Differential Equations. Princeton University Press, 1985.
[80] . The scaling limit behavior of periodic stable-like processes. Bernoulli 21:3 (2006), 551–570.Google Scholar
[81] , , . Dirichlet Forms and Symmetric Markov Processes. de Gruyter, 1994.
[82] . On the McKean-Vlasov limit for interacting diffusions. Math. Nachri. 137 (1988), 197–248.Google Scholar
[83] , . Uniform convergence in some limit theorem for multiple particle systems. Stochastic Proc. Appl. 72 (1997), 47–72.Google Scholar
[84] . Game Theory Evolving. Princeton University Press, 2000.
[85] . Sur l'equation de Boltzmann homogène et sa relation avec l'equation de Landau-Fokker-Planck. C.R. Acad. Sci. Paris 324, 265-270.
[86] , . A classification of disintegrations of measures. In: Measures and Measurable Dynamics. Contemporary Mathematics, vol. 94, American Mathematical Society, 1989, 147-158.
[87] , . Chaos hypothesis for a system interacting through shared resources. Prob. Theory Relat. Fields 100 (1994), 157–173.Google Scholar
[88] , . Stochastic particle approximations for generalized Boltzmann models and convergence estimates. Ann. Prob. 25:1 (1997), 115–132.Google Scholar
[89] . Existence and regularity of a weak function-solution for some Landau equations with a stochastic approach. Stoch. Proc. Appl. 101 (2002), 303–325.Google Scholar
[90] . Landau equation for some soft potentials through a probabilistic approach. Ann. Appl. Prob. 13:2 (2003), 515–539.Google Scholar
[91] , , . Estimates for the density of a nonlinear Landau process. J. Funct. Anal. 238 (2006), 649–677.Google Scholar
[92] . Lp -properties for the nonlinear spatially homogeneous Boltzmann equation. Arch. Ration. Mech. Anal. 92 (1986), 23–57.Google Scholar
[93] . Global Lp-properties for the spatially homogeneous Boltzmann equation. Arch. Ration. Mech. Anal. 103 (1988), 1–38.Google Scholar
[94] . Lectures on Continuous-Time Markov Control Processes. Aportaciones Matematicas, vol. 3, Sociedad Matematica Mexicana, Mexico, 1994.
[95] , , . Discrete-Time Markov Control Processes. Basic Optimality Criteria. Applications of Mathematics, vol. 30. Springer, 1996.
[96] , . Evolutionary Games and Population Dynamics. Cambridge University Press, 1998.
[97] . The martingale problem for a class of pseudo differential operators. Math. Ann. 300 (1994), 121–147.Google Scholar
[98] , . On the Dirichlet problem for pseudodifferential operators generating Feller semigroups. J. Funct. Anal. 137:1 (1996), 19–48.Google Scholar
[99] . Conditionally positive definite functions in quantum probability (in Russian). In: Itogi Nauki i Tekniki. Modern Problems of Mathematics, vol. 36, 1990, pp. 103-148.
[100] , , . Large population stochastic dynamic games: closed-loop McKean-Vlasov systems and the Nash certainty equivalence principle. Commun. Inf. Syst. 6:3 (2006), 221–251.Google Scholar
[101] , , . Well-posed quasi-linear second-order hyperbolic systems with applications to nonlinear elastodynamics and general relativity. Arch. Ration. Mech. Anal. 63:3 (1976), 273–294.Google Scholar
[102] , Diffusion equations. Translations of Mathematical Monographs, vol. 114. American Mathematical Society, 1992.
[103] . Pseudo-Differential Operators and Markov Processes, vols. I, II, III. Imperial College London Press, 2001, 2002, 2005.
[104] , . Lévy-type processes and pseudodifferential operators. In: et al. (eds), Lévy Processes, Theory and Applications, Birkhäuser, 2001, pp. 139-168.
[105] et al.Non-local (semi-)Dirichlet forms generated by pseudo differential operators. In: et al. (eds.), Dirichlet Forms and Stochastic Processes, Proc. Int. Conf. Beijing 1993, de Gruyter, 1995, pp. 223-233.
[106] , . Probability Essentials. Springer, 2004.
[107] , . Limit Theorems for Stochastic Processes. Springer, 1987. Second edition, 2003.
[109] . Existence of gelling solutions for coagulation-fragmentation equations. Commun. Math. Phys. 194 (1998), 541–567.Google Scholar
[110] . Construction of the Brownian motion and the Orstein-Uhlenbeck Process in a Riemannian manifold. Z. Wahrsch. verw. Gebiete 44 (1978), 71–87.Google Scholar
[111] , . Weak convergence of sequence of semimartingales with applications to multitype branching processes. Adv. Appl. Prob. 18 (1986), 20–65.Google Scholar
[112] . Nonlinear Dirichlet forms. In: New Directions in Dirichlet Forms, American Mathematical Society/IP Studies in Advanced Mathematics, vol. 8, American Mathematical Society, 1998, pp. 1-47.
[113] . Probability and Related Topics in Physical Science. Interscience, 1959.
[114] . Foundations of Modern Probability, second edition. Springer, 2002.
[115] , . Brownian Motion and Stochastic Calculus. Springer, 1998.
[116] . Quasi-linear equations of evolution, with applications to partial differential equations. In: Spectral Theory and Differential Equations, Proc. Symp. Dundee, 1974, Lecture Notes in Mathematics, vol. 448, Springer, 1975, pp. 25-70.
[117] . Le processes d'Ornstein-Uhlenbeck sur l'espace des chemins et le probleme des martingales. J. Funct. Anal. 144 (1997), 20–45.Google Scholar
[118] . Sur la crosissance locale des prosessus stochastiques homogènes à acroissements indépendants. Isvestia Akad. Nauk SSSR, Ser. Math. (1939), 487–508.
[119] , . On Markov processes generated by pseudodifferential operator of variable order. Osaka J. Math. 34 (1997), 319–335.Google Scholar
[120] , . Scaling Limits of Interacting Particle Systems. Grundlehren der Mathematischen Wissenschaften, vol. 320, Springer, 1999.
[121] . Parabolic pseudo-differentiable equations, supersingular integrals and Markov processes (in Russian). Izvestia Akad. Nauk, Ser. Matem. 52:5 (1988), 909–934. English translation in Math. USSR Izv. 33:2 (1989), 233–259.Google Scholar
[122] , , . A stochastic method for solving Smolu-chowski's coagulation equation. Math. Comput. Simulation 49 (1999), 57–79.Google Scholar
[123] . On linear, additive, and homogeneous operators in idempo-tent analysis. In: and : (eds.), Idempotent Analysis, Advances in Soviet Mathematics, vol. 13, 1992, pp. 87-101.
[124] . Semiclassical Analysis for Diffusions and Stochastic Processes. Springer Lecture Notes in Mathematics, vol. 1724, Springer, 2000.
[125] . Symmetric stable laws and stable-like jump-diffusions. Proc. London Math. Soc. 3 80 (2000), 725–768.Google Scholar
[126] . Small diffusion and fast dying out asymptotics for super-processes as non-Hamiltonian quasi-classics for evolution equations. Electronic J. Prob., http://www.math.washington.edu/ ejpecp/ 6 (2001), paper 21.
[127] . Measure-valued limits of interacting particle systems with k-nary interactions I. Prob. Theory Relat. Fields 126 (2003), 364–394.Google Scholar
[128] . On extension of mollified Boltzmann and Smoluchovski equations to particle systems with a k-nary interaction. Russian J. Math. Phys. 10 3 (2003), 268–295.Google Scholar
[129] . Measure-valued limits of interacting particle systems with k-nary interactions II. Stoch. Stoch. Rep. 76 1 (2004), 45–58.Google Scholar
[130] . On Markov processes with decomposable pseudo-differential generators. Stoch. Stoch. Rep. 76 1 (2004), 1–44.Google Scholar
[131] . Hydrodynamic limit of coagulation-fragmentation type models of k-nary interacting particles. J. Stati. Phys. 115: 5/6 (2004), 1621–1653.Google Scholar
[132] . Kinetic equations for the pure jump models of k-nary interacting particle systems. Markov Proc. Relat. Fields 12 (2006), 95–138.Google Scholar
[133] . On the regularity of solutions to the spatially homogeneous Boltzmann equation with polynomially growing collision kernel. AdvancedStud. Contemp. Math. 12 (2006), 9–38.Google Scholar
[134] . Nonlinear Markov semigroups and interacting Lévy type processes. J. Stat. Phys. 126:3 (2007), 585–642.Google Scholar
[135] . Generalized continuous-time random walks (CTRW), subordination by hitting times and fractional dynamics. arXiv:0706.1928v1[math.PR] 2007. Probab. Theory Appl. 53:4 (2009), 594–609.Google Scholar
[136] . The central limit theorem for the Smoluchovski coagulation model. arXiv:0708.0329v1[math.PR] 2007. Prob. Theory Relat. Fields 146:1 (2010), 87. Published online, http://dx.doi.org/10.1007/s00440-008-0186-2.CrossRefGoogle Scholar
[137] . The Lévy-Khintchine type operators with variable Lips-chitz continuous coefficients generate linear or nonlinear Markov processes and semigroupos. To appear in Prob. Theory. Relat. Fields.
[138] , , . Fractional stable distributions. J. Math. Sci. (N.Y.) 105:6 (2001), 2570–2577.Google Scholar
[139] , . Introduction to the Analysis of Many Agent Systems of Competition and Cooperation (Game Theory for All). St Petersburg University Press, 2008 (in Russian).
[140] , . Understanding Game Theory. World Scientific, 2010.
[141] , . Idempotent Analysis and its Application to Optimal Control. Moscow, Nauka, 1994 (in Russian).
[142] , . Idempotent Analysis and its Applications. Kluwer, 1997.
[143] , , . Transience and non-explosion of certain stochastic newtonian systems. Electronic J. Prob. 7 (2002), paper no. 19.Google Scholar
[144] . On the martingale problem for generators of stable processes with perturbations. Osaka J. Math. 21 (1984), 113–132.Google Scholar
[145] , , . Mathematical Foundation of Risk Theory. Moscow, Fismatlit, 2007 (in Russian).
[146] et al.Some methods of the analysis of time characteristics of catastrophes in nonhomogeneous flows of extremal events. In: (ed.), Sistemi i Sredstva Informatiki. Matematicheskie Modeli v Informacionnich Technologiach, Moscow, RAN, 2006, pp. 5-23 (in Russian).
[147] , . A study of the nonlinear breakage equations: analytical and asymptotic solutions. J. Phys. A 33 (2000), 1221–1232.Google Scholar
[148] . Asymptotic distribution of continuous-time random walks: a probabilistic approach. J. Stat. Phys. 81:3/4 (1995), 777–792.Google Scholar
[149] , . A Monte Carlo method for identification and sensitivity analysis of coagulation processes. J. Comput. Phys. 200 (2004), 50–59.Google Scholar
[150] . Stochastic Flows and Stochastic Differential Equations. Cambridge Studies in Advanced Mathematics, vol. 24, Cambridge University Press, 1990.
[151] , . Particle representations for a class of nonlinear SPDEs. Stochastic Proc. Appl. 83:1 (1999), 103–126.Google Scholar
[152] , . Numerical solutions for a class of SPDEs with application to filtering. In: Stochastics in Finite and Infinite Dimensions, Trends in Mathematics, Birkhäuser, 2001, pp. 233-258.
[153] . Introductory Lectures on Fluctuations of Lévy Processes with Applications. Universitext, Springer, 2006.
[154] . Stochastic semigroups and coagulation equations. Ukrainian Math. J. 57:6 (2005), 913–922.Google Scholar
[155] , , . On the Oort-Hulst-Savronov coagulation equation and its relation to the Smoluchowski equation. SIAM J. Math. Anal. 34 (2003), 1399–1421.Google Scholar
[156] , . The continuous coagulation-fragmentation equations with diffusion. Arch. Ration. Mech. Anal. 162 (2002), 45–99.Google Scholar
[157] , . The discrete coagulation equations with collisional breakage. J. Stat. Phys. 104: 1/2 (2001), 193–220.Google Scholar
[158] . Uniform upper bounds for hypoelliptic kernels with drift. J. Math. Kyoto University 34:2 (1994), 263–271.Google Scholar
[159] , (eds.). Non-Equilibrium Phenomena I: The Boltzmann Equation. Studies in Statistical Mechanics, vol. X, North-Holland, 1983.
[160] . Main equations of the kinetic theory from the point of view of random processes (in Russian). J. Exp. Theoret. Phys. 5 (1935), 211–231.Google Scholar
[161] , . Perturbations of generalized Mehler semigroups and applications to stochastic heat equation with Levy noise and singular drift. Potential Anal. 20:4 (2004), 317–344.Google Scholar
[162] . Interacting Particle Systems. Reprint of the 1985 original. Classics in Mathematics, Springer, 2005.
[163] . On the Generators of quantum dynamic semigroups. Commun. Math. Phys. 48 (1976), 119–130.Google Scholar
[164] , . Solutions with increasing energy for the spatially homogeneous Boltzmann equation. Nonlinear Anal. Real World Appl. 3 (2002), 243–258.Google Scholar
[165] . Some new aspects of coagulation theory. Izv. Akad. Nauk SSSR, Ser. Fiz. Atmosfer. i Okeana 14:10 (1978), 738–743.Google Scholar
[166] , . Singular self-preserving regimes of coagulation processes. Phys. Rev. E 65 (2002).Google Scholar
[167] , . Introduction to the Theory of Non-Symmetric Dirichlet Forms. Springer, 1992.
[168] . Analytic Treatment of One-Dimensional Markov Processes. Springer, 1968.
[170] . Nonlinear Operators and Differential Equations in Banach Spaces. Wiley, 1976.
[171] , . Mathematical Theory of Entropy. Addison-Wesley, 1981.
[172] . Perturbation Theory and Asymptotical Methods. Moscow State University Press, 1965 (in Russian). French Translation, Dunod, Paris, 1972.
[173] . Complex Markov Chains and Functional Feynman Integrals. Moscow, Nauka, 1976 (in Russian).
[174] . Nonlinear averaging axioms in financial mathematics and stock price dynamics. Theory Prob. Appl. 48:04 (2004), 723–733.Google Scholar
[175] . Quantum Economics. Moscow, Nauka, 2006 (in Russian).
[176] , . Geometric Asymptotics for Nonlinear PDE. I. Translations of Mathematical Monographs, vol. 202, American Mathematical Society, 2001.
[177] , . Asymptotics of the Kolmogorov-Feller equation for systems with a large number of particles. Itogi Nauki i Techniki. Teoriya veroyatnosti, vol. 19, VINITI, Moscow, 1982, pp. 85-125 (in Russian).
[178] . Existence and uniqueness theorems for the Boltzmann equation. In: (ed.), Encyclopaedia of Mathematical Sciences, vol. 2, Springer, 1989, pp. 254-278.
[179] . Nonlinear Evolution Equations: Kinetic Approach. World Scientific, 1993.
[180] . A new fundamental solution for differential Riccati equations arising in control. Automatica (J. IFAC) 44:4 (2008), 920–936.Google Scholar
[181] . A class of Markov processes associated with nonlinear parabolic equations. Proc. Nat. Acad. Sci. 56 (1966), 1907–1911.Google Scholar
[182] . An exponential formula for solving Boltzmann's equation for a Maxwellian gas. J. Combin. Theory 2:3 (1967), 358–382.Google Scholar
[183] , . Limit Distributions for Sums of Independent Random Vectors. Wiley Series in Probability and Statistics, John Wiley and Son, 2001.
[184] , . Limit theorems for continuous-time random walks with infinite mean waiting times. J. Appl. Prob. 41 (2004), 623–638.Google Scholar
[185] . Convergence of the fluctuations for interacting diffusions with jumps associated with Boltzmann equations. Stocha. Stoch. Rep. 63: 3-4 (1998), 195–225.Google Scholar
[186] , . The random walk's guide to anomalous diffusion: a fractional dynamic approach. Phys. Rep. 339 (2000), 1–77.Google Scholar
[187] . Quantum Probability for Probabilists. Springer Lecture Notes in Mathematics, vol. 1538, Springer, 1993.
[188] , . On the spatially homogeneous Boltzmann equation. Ann. Inst. H. Poincaré Anal. Non Linéaire 16:4 (1999), 467–501.Google Scholar
[189] , , . Phase transitions and spatio-temporal fluctuations in stochastic lattice Lotka-Volterra models. J. Stat. Phys. 128: 1-2 (2007), 447–483.Google Scholar
[191] , . Regularity theory for the spatially homogeneous Boltzmann equation with cut-off. Arch. Ration. Mech. Anal. 173:2 (2004), 169–212.Google Scholar
[192] . Stable-like processes: construction of the transition density and the behavior of sample paths near t = 0. Osaka J. Math. 31 (1994), 189–214.Google Scholar
[193] . Markov Chains. Cambridge University Press, 1998.
[195] . Notes on Brownian coagulation. Markov Proc. Relat. Fields 12:2 (2006), 407–412.Google Scholar
[196] . The Malliavin Calculus and Related Topics. Probability and its Applications, second edition. Springer, 2006.
[197] , , . Nonlinear problems in infinite interacting particle systems. Inf. Dim. Anal. Quantum Prob. Relat. Topics 11:2 (2008), 179–211.Google Scholar
[198] , . Stochastic Partial Differential Equations with Lévy Noise. Encyclopedia of Mathematics, Cambridge University Press, 2007.
[199] , . Cauchy problem for Bogolyubov's kinetic equations. TrudiMat. Inst. USSR Acad. Sci. 136 (1975), 370–378.Google Scholar
[200] , . On multidimensional stable processes with locally unbounded drift. Random Oper. Stoch. Eq. 3:2 (1995), 113–124.Google Scholar
[201] , . Spatial Deterministic Epidemics. Mathematical Surveys and Monographs, vol. 102, American Mathematical Society, 2003.
[202] , . Mass Transportation Problems, vols. I, II. Springer, 1998.
[203] . La methode des martingales appliquée l'etude de la convergence en loi de processus (in French). Bull. Soc. Math. France Mem. 62, 1979.Google Scholar
[204] . Sur l'existence de solutions certains problemes de semimartin-gales (in French). C. R. Acad. Sci. Paris A-B 290:18 (1980), A843-A846.Google Scholar
[205] , . Methods of Modern Mathematical Physics, vol. 1, Functional Analysis. Academic Press, 1972.
[206] , . Methods of Modern Mathematical Physics, vol. 2, Harmonic Analysis. Academic Press, 1975.
[207] , . Methods of Modern Mathematical Physics, vol. 4, Analysis of Operators. Academic Press, 1978.
[208] , , . Coexistence versus extinction in the stochastic cyclic Lotka-Volterra model. Phys. Rev. E(3) 74:5 (2006).Google Scholar
[209] , . Continuous Martingales and Brownian Motion. Springer, 1999.
[210] . Probability Theory, Stochastic Processes and Mathematical Statistics (in Russian). Moscow, Nauka, 1985. English translation: Mathematics and its Applications, vol. 344, Kluwer, 1995.
[211] , . Fragmentation-coagulation models of phytoplankton. Bull. Polish Acad. Sci. Math. 54:2 (2006), 175–191.Google Scholar
[212] . Evolution of the Pre-Planetary Cloud and the Formation of the Earth and Planets. Moscow, Nauka, 1969 (in Russian). English translation: Israel Program for Scientific Translations, Jerusalem, 1972.
[213] , . Distributions in the Physical and Engineering Sciences vol. 1, Birkhäuser, Boston, 1997.
[214] , . Fractional kinetic equations: solutions and applications. Chaos 7:4 (1997), 753–764.Google Scholar
[215] . Hypersingular Integrals and Applications. Rostov-na-Donu University Press, 1984 (in Russian).
[216] , , . Fractional Integrals and Derivatives and Their Applications. Naukla i Teknika, Minsk, 1987 (in Russian). English translation Harwood Academic.
[217] , . Stable Non-Gaussian Random Processes, Stochastic Models with Infinite Variance. Chapman and Hall, 1994.
[218] . On Feller processes with sample paths in Besov spaces. Math. Ann. 309 (1997), 663–675.Google Scholar
[219] . Convex Bodies: The Brunn-Minkowski Theory. Cambridge University Press, 1993.
[220] . Probability. Springer, 1984.
[221] , . On an existence theorem for the solutions of Bogoljubov's chain of equations (in Russian). Teoret. Mat. Fiz. 19 (1974), 344–363.Google Scholar
[222] , . Nonlinear Markov semigroups, nonlinear Dirichlet forms and applications to minimal surfaces. J. Reine Angew. Math. 562 (2003), 201–235.Google Scholar
[223] . Stochastic Equations for Complex Systems. Translated from the Russian. Mathematics and its Applications (Soviet Series), vol. 13, Reidel, 1988.
[224] . Shock Waves and Reaction-Diffusion Equations. Springer, 1983.
[225] . Large Scaling Dynamics of Interacting Particles. Springer, 1991.
[226] . Diffusion processes associated with Lévy generators. Z. Wahrsch. verw. Gebiete 32 (1975), 209–244.Google Scholar
[227] . Markov Processes from K. Ito's Perspective. Annals of Mathematics Studies. Princeton University Press, 2003.
[228] , . On degenerate elliptic-parabolic operators of second order and their associated diffusions. Commun. Pure Appl. Math. XXV (1972), 651–713.Google Scholar
[229] . . Multidimensional Diffusion Processes. Springer, 1979.
[230] . Nonlinear reflecting diffusion process and the propagation of chaos and fluctuation associated. J. Funct. Anal. 56 (1984), 311–336.Google Scholar
[231] . Equations de type de Boltzmann, spatialement homogènes. Z. Wahrsch. verw. Gebeite 66 (1984), 559–592.Google Scholar
[232] . Topics in propagation of chaos. In: Proc. Ecole d'Eté de probabilités de Saint-Flour XIX-1989. Springer Lecture Notes in Mathematics, vol. 1464, Springer, 1991, pp. 167-255.
[233] . On the existence of Feller semigroups with boundary conditions. Mem. Ameri. Math. Soc. 99 (1992), 1–65.Google Scholar
[234] . On the existence of Feller semigroups with Dirichlet conditions. Tsukuba J. Math. 17 (1993), 377–427.Google Scholar
[235] . Boundary value problems for elliptic pseudo-differential operators II. Proc. Roy. Soc. Edinburgh 127A (1997), 395–105.Google Scholar
[236] , and . Feller semigroups and degenerate elliptic operators with Wentzell boundary conditions. Stud. Math. 145: 1 (2001), 17–53.Google Scholar
[237] , (eds.). Probabilistic Models for Nonlinear Partial Differential Equations. In: Proc. Conf. at Montecatini Terme, 1995, Springer Lecture Notes in Mathematics, vol. 1627, Springer, 1996.
[238] . Purely discontinuous Markov processes with nonlinear generators and their propagation of chaos (in Russian). Teor. Verojatnost. i Primenen 15 (1970), 599–621.Google Scholar
[239] . On Markov process corresponding to Boltzmann's equation of Maxwellian gas. In: Proc. Second Japan-USSR Symp on Probability Theory, Kyoto, 1972, Springer Lecture Notes in Mathematics, vol. 330, Springer, 1973, pp. 478-489.
[240] , . Central limit theorems for a simple diffusion model of interacting particles. Hiroshima Math. J. 11 (1981), 415–423.Google Scholar
[241] , . Chance and Stability: Stable Distributions and their Applications. VSP, 1999.
[242] . Montroll-Weisse problem, fractional equations and stable distributions. Int. J. Theor. Phys. 39:8 (2000), 2087–2105.Google Scholar
[243] . Scaling limit of interacting diffusions with arbitrary initial distributions. Prob. Theory Relat. Fields 99 (1994), 97–110.Google Scholar
[244] . Continuity and representation of Gaussian Mehler semigroups. Potential Anal. 13:3 (2000), 199–211.Google Scholar
[245] . On a new class of weak solutions to the spatially homogeneous Boltzmann and Landau equations. Arch. Ration. Mech. Anal. 143 (1998), 273–307.Google Scholar
[246] . Topics in Optimal Transportation. Graduate Studies in Mathematics vol. 58, American Mathematical Society, 2003.
[247] . Stochastic-Process Limits. Springer, 2002.
[248] , . Modern Analysis, third edition. Cambridge University Press, 1920.
[249] . Mass-conservation solutions to the discrete coagulation-fragmentation model with diffusion. Nonlinear Anal. 49 (2002), 297–314.Google Scholar
[250] . Functional Analysis. Springer, 1980.
[252] . Quantum evolution as a nonlinear Markov process. Found. Phys. Lett. 15:3 (2002), 229–243.Google Scholar
[253] . Fractional kinetic equation for Hamiltonian chaos. Physica D 76 (1994), 110–122.Google Scholar
[254] . Linear and nonlinear phenomena in large interacting systems. Rep. Math. Phys. 59:3 (2007), 409–419.Google Scholar
[255] . One-Dimensional Stable Distributions. Moscow, Nauka, 1983 (in Russian). English translation: Translations of Mathematical Monographs, vol. 65, American Mathematical Society, 1986.