The preparation of these notes was supported in part by N.S.F. Grant No. MPS 72-04591, and by an Alfred P. Sloan Foundation research fellowship.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Bibliographie
A. B. Bortz, M. H. Kalos, J. L. Lebowitz, and J. Marro (1975). Time evolution of a quenched binary alloy II. Computer simulation of a three dimensional model system, Physical Review B, 12, 2000–2011.
A. B. Bortz, M. H. Kalos, J. L. Lebowitz, and M. A. Zendejas (1974). Time evolution of a quenched binary alloy. Computer simulation of a two dimensional model system, Physical Review B, 10, 535–541.
J. Chover (1975). Convergence of a local lattice process, Stochastic Processes and their Applications, 3, 115–135.
D. Dawson (1974). Information flow in discrete Markov systems, Journal of Applied Probability, 11, 594–600.
D. Dawson (1975). Synchronous and asynchronous reversible Markov systems, Canadian Mathematical Bulletin, 17, 633–649.
R. L. Dobrushin (1971). Markov processes with a large number of locally interacting components, Problems of Information Transmission, 7, 149–164, and 235–241.
R. Glauber (1963). The statistics of the stochastic Ising model, Journal of Mathematical Physics, 4, 294–307.
H. O. Georgii (1975). Canonical Gibbs states, their relation to Gibbs states, and applications to two-valued Markov chains, Z. Wahrscheinlichkeitstheorie verw. Geb., 32, 277–300.
H. O. Georgii (1976). On canonical Gibbs states and symmetric and tail events, Z. Wahrscheinlichkeitstheorie verw. Geb., 33, 331–341.
L. Gray and D. Griffeath (1976). On the uniqueness of certain interacting particle systems, Z. Wahrscheinlichkeitstheorie verw. Geb., 35, 75–86.
D. Griffeath (1975). Ergodic theorems for graph interactions, Advances in Applied Probability, 7, 179–194.
T. E. Harris (1972). Nearest-neighbor Markov interaction processes on multidimensional lattices, Advances in Mathematics, 9, 66–89.
T. E. Harris (1974). Contact interactions on a lattice, Annals of Probability, 2, 969–988.
T. E. Harris (1976). On a class of set-valued Markov Processes, Annals of Probability, 4, 175–194.
L. L. Helms (1974). Ergodic properties of several interacting Poisson particles, Advances in Mathematics, 12, 32–57.
Y. Higuchi ( ). A study on the canonical Gibbs states.
Y. Higuchi and T. Shiga (1975). Some results on Markov processes of infinite lattice spin systems, Journal of Mathematics of Kyoto University, 15, 211–229.
R. Holley (1970). A class of interactions in an infinite particle system, Advances in Mathematics, 5, 291–309.
R. Holley (1971). Free energy in a Markovian model of a lattice spin system, Communications in Mathematical Physics, 23, 87–99.
R. Holley (1971). Pressure and Helmholtz free energy in a dynamic model of a lattice gas, Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability, III, 565–578.
R. Holley (1972). An ergodic theorem for interacting systems with attractive interactions, Z. Wahrscheinlichkeitstheorie verw. Geb., 24, 325–334.
R. Holley (1972). Markovian interaction processes with finite range interactions, The Annals of Mathematical Statistics, 43, 1961–1967.
R. Holley (1974). Recent results on the stochastic Ising model, Rocky Mountain Journal of Mathematics, 4, 479–496.
R. Holley and T. M. Liggett (1975). Ergodic theorems for weakly interacting infinite systems and the voter model, The Annals of Probability, 3, 643–663.
R. Holley and D. Stroock (1976). A martingale approach to infinite systems of interacting processes, The Annals of Probability, 4, 195–228.
R. Holley and D. Stroock (1976). Applications of the stochastic Ising model to the Gibbs states, Communications in Mathematical Physics, 48, 249–265.
R. Holley and D. Stroock ( ). Dual processes and their application to infinite interacting systems, Advances in Mathematics.
R. Holley and D. Stroock (1976). L2 theory for the stochastic Ising model, Z. Wahrscheinlichkeitstheorie verw. Geb., 35, 87–101.
J. Kemeny, J. Snell, and A. Knapp (1966). Denumerable Markov Chains, Van Nostrand, Princeton, N. J.
R. Kinderman and J. Snell (1976). Markov random fields.
T. Kurtz (1969). Extensions of Trotter’s operator semigroup approximation theorems, Journal of Functional Analysis, 3, 354–375.
T. M. Liggett (1972). Existence theorems for infinite particle systems, Transactions of the A.M.S., 165,471–481.
T. M. Liggett (1973). An infinite particle system with zero range interactions, The Annals of Probability, 1, 240–253.
T. M. Liggett (1973). A characterization of the invariant measures for an infinite particle system with interactions, Transactions of the A.M.S., 179, 433–453.
T. M. Liggett (1974). A characterization of the invariant measures for an infinite particle system with interactions II, Transactions of the A.M.S., 198,201–213.
T. M. Liggett (1974). Convergence to total occupancy in an infinite particle system with interactions, The Annals of Probability, 2, 989–998.
T. M. Liggett (1975). Ergodic theorems for the asymmetric simple exclusion process, Transactions of the A.M.S., 213, 237–261.
T. M. Liggett (1976). Coupling the simple exclusion process, The Annals of Probability, 4, 339–356.
K. Logan (1974). Time reversible evolutions in statistical mechanics, Cornell University, Ph.D. dissertation.
N. Matloff ( ). Ergodicity conditions for a dissonant voting model, The Annals of Probability.
C. J. Preston (1974). Gibbs states on countable sets, Cambridge University Press.
D. Ruelle (1969). Statistical mechanics, W. A. Benjamin.
D. Ruelle (1968). Statistical mechanics of a one-dimensional lattice gas, Communications in Mathematical Physic, 9, 267–278.
D. Schwartz (1976). Ergodic theorems for an infinite particle system with births and deaths, Annals of Probability, 4
F. Spitzer (1970). Interaction of Markov processes, Advances in Mathematics, 5, 246–290.
F. Spitzer (1971). Random fields and interacting particle systems, M. A. A. Summer Seminar Notes, Williamstown, Mass.
F. Spitzer (1974). Introduction aux processus de Markov a parametres dans Zv, Lecture Notes in Mathematics 390, Springer-Verlag.
F. Spitzer (1974). Recurrent random walk of an infinite particle system, Transactions of the A.M.S., 198, 191–199.
F. Spitzer (1975). Random time evolution of infinite particle systems, Advances in Mathematics, 16, 139–143.
O. N. Stavskaya and I. I. Pyatetskii-Shapiro (1971). On homogeneous nets of spontaneously active elements, Systems Theory Res., 20, 75–88.
W. G. Sullivan (1973). Potentials for almost Markovian random fields, Communications in Mathematical Physics, 33, 61–74.
W. G. Sullivan (1974). A unified existence and ergodic theorem for Markov evolution of random fields, Z. Wahrscheinlichkeitstheorie verw. Geb., 31,47–56.
W. G. Sullivan (1976). Processes with infinitely may jumping particles, Proceedings of the A.M.S., 54, 326–330.
W. G. Sullivan (1975). Mean square relaxation times for evolution of random fields, Communications in Mathematical Physics, 40, 249–258.
W. G. Sullivan (1975). Markov processes for random fields, Communications of the Dublin Institute for Advanced Studies, series A, number 23.
W. G. Sullivan ( ). Specific information gain for interacting Markov processes.
L. N. Vasershtein (1969). Processes over denumerable products of spaces, describing large systems, Problems of Information Transmission, 3, 47–52.
D. Schwartz ( ). Applications of duality to a class of Markov processes
D. Griffeath ( ). An ergodic theorem for a class of spin systems.
R. Lang ( ). Unendlich-dimensionale Wienerprozesse mit wechselwirkung I: Existenz.
L. N. Vasershtein and A. M. Leontovich (1970). Invariant measures of certain Markov operators describing a homogeneous random medium, Problems of Information Transmission, 6, 61–69.
C. Cocozza and C. Kipnis ( ). Existence de processus Markoviens pour des systems infinis de particules.
L. Gray and D. Griffeath ( ). On the nonuniqueness of proximity processes.
R. Holley and D. Stroock ( ). Nearest neighbor birth and death processes on the real line.
Editor information
Rights and permissions
Copyright information
© 1977 Springer-Verlag
About this paper
Cite this paper
Liggett, T.M. (1977). The stochastic evolution of infinite systems of interacting particles. In: Hennequin, P.L. (eds) Ecole d’Eté de Probabilités de Saint-Flour VI-1976. Lecture Notes in Mathematics, vol 598. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0097493
Download citation
DOI: https://doi.org/10.1007/BFb0097493
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-08340-5
Online ISBN: 978-3-540-37307-0
eBook Packages: Springer Book Archive