## Asymptotic analysis of transport processes.(English)Zbl 0361.60056

### MSC:

 60J75 Jump processes (MSC2010) 60J70 Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.) 60K35 Interacting random processes; statistical mechanics type models; percolation theory 91E99 Mathematical psychology 34F05 Ordinary differential equations and systems with randomness 70K99 Nonlinear dynamics in mechanics
Full Text:

### References:

 [1] A. Kolmogoroff, Über die analytischen Methoden in der Wahrscheinlichkeitsrechnung, Math. Ann. 104 (1931), no. 1, 415 – 458 (German). · Zbl 0001.14902 [2] W. Feller, Zur Theorie der stochastischen Prozesse, Math. Ann. 113 (1936), 113-160. · Zbl 0014.22201 [3] I. I. Gikhman and A. V. Skorokhod, Introduction to the theory of random processes, Translated from the Russian by Scripta Technica, Inc, W. B. Saunders Co., Philadelphia, Pa.-London-Toronto, Ont., 1969. · Zbl 0573.60003 [4] Kenneth M. Case and Paul F. Zweifel, Linear transport theory, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1967. · Zbl 0162.58903 [5] A. Khinchine, Asymptotische Gesetze der Wahrscheinlichkeitsrechnung, Springer, Berlin, 1933. · JFM 59.1153.01 [6] H. A. Kramers, Brownian motion in a field of force and the diffusion model of chemical reactions, Physica 7 (1940), 284 – 304. · Zbl 0061.46405 [7] M. Frank Norman, Markov processes and learning models, Academic Press, New York-London, 1972. Mathematics in Science and Engineering, Vol. 84. · Zbl 0262.92003 [8] M. Iosifescu and R. Theodorescu, Random processes and learning, Springer-Verlag, New York, 1969. Die Grundlehren der mathematischen Wissenschaften, Band 150. · Zbl 0194.51101 [9] Reuben Hersh, Random evolutions: a survey of results and problems, Rocky Mountain J. Math. 4 (1974), 443 – 477. Based on lectures given by Richard Griego, Reuben Hersh, Tom Kurtz and George Papanicolaou; Papers arising from a Conference on Stochastic Differential Equations (Univ. Alberta, Edmonton, Alta., 1972). · Zbl 0366.60005 [10] Mark A. Pinsky, Multiplicative operator functionals and their asymptotic properties, Advances in probability and related topics, Vol. 3, Dekker, New York, 1974, pp. 1 – 100. · Zbl 0342.60012 [11] Thomas G. Kurtz, A limit theorem for perturbed operator semigroups with applications to random evolutions, J. Functional Analysis 12 (1973), 55 – 67. · Zbl 0246.47053 [12] Стохастические дифференциал$$^{\приме}$$ные уравнения, Издат. ”Наукова Думка”, Киев, 1968 (Руссиан). · Zbl 0169.48702 [13] R. Z. Has$$^{\prime}$$minskiĭ, Stochastic processes defined by differential equations with a small parameter, Teor. Verojatnost. i Primenen 11 (1966), 240 – 259 (Russian, with English summary). [14] Daniel W. Stroock, Some stochastic processes which arise from a model of the motion of a bacterium, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 28 (1973/74), 303 – 315. · Zbl 0282.60047 [15] L. Baggett and D. Stroock, An ergodic theorem for Poisson processes on a compact group with applications to random evolutions, J. Functional Analysis 16 (1974), 404 – 414. · Zbl 0365.60024 [16] S. Chandresekhar, Stochastic problems in physics and astronomy, Rev. Modern Phys. 15 (1943), 1 – 89. · Zbl 0061.46403 [17] A. M. Il$$^{\prime}$$in and R. Z. Has$$^{\prime}$$minskiĭ, On the equations of Brownian motion, Teor. Verojatnost. i Primenen. 9 (1964), 466 – 491 (Russian, with English summary). [18] Edward Nelson, Dynamical theories of Brownian motion, Princeton University Press, Princeton, N.J., 1967. · Zbl 0165.58502 [19] K. M. Case, ”The soluble boundary value problems of transport theory,” in The Boltzmann equation, F. A. Grunbaum (editor), Courant Inst. Lecture Notes, 1972. · Zbl 0241.35064 [20] Robert D. Richtmyer and K. W. Morton, Difference methods for initial-value problems, Second edition. Interscience Tracts in Pure and Applied Mathematics, No. 4, Interscience Publishers John Wiley & Sons, Inc., New York-London-Sydney, 1967. · Zbl 0155.47502 [21] G. C. Papanicolaou and W. Kohler, Asymptotic theory of mixing stochastic ordinary differential equations, Comm. Pure Appl. Math. 27 (1974), 641 – 668. · Zbl 0288.60056 [22] R. Z. Has$$^{\prime}$$minskiĭ, On the principle of averaging the Itô’s stochastic differential equations, Kybernetika (Prague) 4 (1968), 260 – 279 (Russian, with Czech summary). [23] W. Kohler and G. C. Papanicolaou, Limit theorems for stochastic equations with rapidly varying components (to appear). · Zbl 0319.60037 [24] Ryogo Kubo, Stochastic Liouville equations, J. Mathematical Phys. 4 (1963), 174 – 183. · Zbl 0135.45102 [25] G. S. Agarwal, ”Master equation methods in quantum optics,” in Progress in Optics. Vol. 11, E. Wolf (editor), North-Holland, Amsterdam, 1973. [26] Richard Griego and Reuben Hersh, Theory of random evolutions with applications to partial differential equations, Trans. Amer. Math. Soc. 156 (1971), 405 – 418. · Zbl 0223.35082 [27] O. A. Oleĭnik and E. V. Radkevič, Second order equations with nonnegative characteristic form, Plenum Press, New York-London, 1973. Translated from the Russian by Paul C. Fife. [28] D. Stroock and S. R. S. Varadhan, On degenerate elliptic-parabolic operators of second order and their associated diffusions, Comm. Pure Appl. Math. 25 (1972), 651 – 713. · Zbl 0344.35041 [29] H. P. McKean Jr., Stochastic integrals, Probability and Mathematical Statistics, No. 5, Academic Press, New York-London, 1969. · Zbl 0191.46603 [30] H. P. McKean, Fluctuations in the kinetic theory of gases, Comm. Pure Appl. Math. 28 (1975), no. 4, 435 – 455. [31] N. N. Bogoljubov and Ju. A. Mitropol’skiĭ, Asymptotic methods in the theory of nonlinear oscillations, 2nd rev. ed., Fizmatgiz, Moscow, 1958; English transl., Gordon and Breach, New York, 1961. MR 20 #6812; 25 #5242. [32] Harold Grad, Solution of the Boltzmann equation in an unbounded domain, Comm. Pure Appl. Math. 18 (1965), 345 – 354. · Zbl 0138.34703 [33] G. E. Uhlenbeck and L. S. Ornstein, On the theory of Brownian motion, Phys. Rev. 36(1930), 823-841. · JFM 56.1277.03 [34] Ming Chen Wang and G. E. Uhlenbeck, On the theory of the Brownian motion. II, Rev. Modern Phys. 17 (1945), 323 – 342. · Zbl 0063.08172 [35] R. Z. Has$$^{\prime}$$minskiĭ, The averaging principle for parabolic and elliptic differential equations and Markov processes with small diffusion, Teor. Verojatnost. i Primenen. 8 (1963), 3 – 25 (Russian, with English summary). [36] V. M. Volosov, Averaging in systems of ordinary differential equations, Uspehi Mat. Nauk 17 (1962), no. 6 (108), 3 – 126 (Russian). · Zbl 0119.07502 [37] Richard S. Ellis and Mark A. Pinsky, Asymptotic nonuniqueness of the Navier-Stokes equations in kinetic theory, Bull. Amer. Math. Soc. 80 (1974), no. 6, 1160 – 1164. , https://doi.org/10.1090/S0002-9904-1974-13656-6 Richard S. Ellis and Mark A. Pinsky, The first and second fluid approximations to the linearized Boltzmann equation, J. Math. Pures Appl. (9) 54 (1975), 125 – 156. Richard S. Ellis and Mark A. Pinsky, The projection of the Navier-Stokes equations upon the Euler equations, J. Math. Pures Appl. (9) 54 (1975), 157 – 181. · Zbl 0297.76063 [38] Harold Grad, Singular and nonuniform limits of solutions of the Boltzmann equation., Transport Theory (Proc. Sympos. Appl. Math., New York, 1967) Amer. Math. Soc., Providence, R.I., 1969, pp. 269 – 308. [39] Edward W. Larsen and Joseph B. Keller, Asymptotic solution of neutron transport problems for small mean free paths, J. Mathematical Phys. 15 (1974), 75 – 81. [40] Edward W. Larsen, Solutions of the steady, one-speed neutron transport equation for small mean free paths, J. Mathematical Phys. 15 (1974), 299 – 305. [41] B. White, Dissertation, New York University, Sept. 1974. [42] Специал$$^{\приме}$$ные функции и теория представлений групп, Издат. ”Наука”, Мосцощ, 1965 (Руссиан). · Zbl 0144.38003 [43] G. C. Papanicolaou and S. R. S. Varadhan, A limit theorem with strong mixing in Banach space and two applications to stochastic differential equations, Comm. Pure Appl. Math. 26 (1973), 497 – 524. · Zbl 0253.60065 [44] George Papanicolau and Joseph B. Keller, Stochastic differential equations with applications to random harmonic oscillators and wave propagation in random media, SIAM J. Appl. Math. 21 (1971), 287 – 305. · Zbl 0224.73111
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.