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Asymptotic analysis of transport processes
Author(s):
George C.
Papanicolaou
Journal:
Bull. Amer. Math. Soc.
81
(1975),
330-392.
MSC (1970):
Primary 60J75, 82A70, 60F05;
Secondary 60J70, 92A25, 34F05, 70K99
MathSciNet review:
0362523
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References:
- 1.
- A. N. Kolmogorov, Über die analytischen Methoden in der Wahrscheinlichkeitsrechnung, Math. Ann. 104 (1931), 415-458. MR 1512678
- 2.
- W. Feller, Zur Theorie der stochastischen Prozesse, Math. Ann. 113 (1936), 113-160.
- 3.
- I. I. Gihman and A. V. Skorohod, Introduction to the theory of random processes, "Nauka", Moscow, 1965, English transl., Saunders, Philadelphia, Pa., 1969. MR 33, #6689; 40 #923. MR 247660
- 4.
- K. M. Case and P. F. Zweifel, Linear transport theory, Addison-Wesley, Reading, Mass., 1967. MR 37 #1140. MR 225547
- 5.
- A. Khinchine, Asymptotische Gesetze der Wahrscheinlichkeitsrechnung, Springer, Berlin, 1933.
- 6.
- H.A. Kramers, Brownian motion in a field of force and the diffusion model of chemical reactions, Physica 7 (1940), 284-304. MR 2, 140. MR 2962
- 7.
- M. F. Norman, Markov processes and learning models, Academic Press, New York, 1972. MR 423546
- 8.
- M. Iosifescu and R. Theodorescu, Random processes and learning, Die Grundlehren der math. Wissenschaften, Band 150, Springer-Verlag, New York, 1969. MR 45 #2781. MR 293704
- 9.
- R. Hersh, Random evolutions: a survey of results and problems, Rocky Mountain Math. J. (to appear). MR 394877
- 10.
- M. Pinsky, Multiplicative operator functionals and their asymptotic properties, Advances in Probability, vol. 3, Dekker, New York (to appear). MR 368182
- 11.
- T. G. Kurtz, A limit theorem for perturbed operator semigroups with application to random evolutions, J. Functional Analysis 12 (1973), 55-67. MR 365224
- 12.
- I. I. Gihman and A. V. Skorohod, Stochastic differential equations, "Naukova Dumka", Kiev, 1968; English transl., Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 72, Springer-Verlag, Berlin and New York, 1972. MR 41 #7777. MR 263172
- 13.
- R. Z. Has'minskiĭ (Khasminskii), On stochastic processes defined by differential equations with a small parameter, Teor. Verojatnost. i Primenen. 11 (1966), 240-259= Theor. Probability Appl. 11 (1966), 211-228. MR 34 #3636. MR 203788
- 14.
- D. W. Stroock, Some stochastic processes which arise from a model of the motion of a bacterium (to appear). MR 386038
- 15.
- L. Baggett and D. W. Stroock, An ergodic theorem for Poisson processes on a compact group with applications to random evolutions, J. Functional Analysis (to appear). MR 362432
- 16.
- S. Chandrasekhar, Stochastic problems in Physics and Astronomy, Rev. Modern Phys. 15 (1943), 1-89. MR 4, 248. MR 8130
- 17.
- A. M. Il'in and R. Z. Has'minskiĭ (Khasminskiĭ), On the equations of Brownian motion, Teor. Verojatnost. i Primenen. 9 (1964), 421-444. (Russian) MR 29 #5283. MR 168018
- 18.
- E. Nelson, Dynamical theories of Brownian motion, Princeton Univ. Press, Princeton, N.J., 1967. MR 35 #5001. MR 214150
- 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.
- 20.
- R. D. Richtmyer and K. W. Morton, Difference methods for initial-value problems, 2nd ed., Interscience Tracts in Pure and Appl. Math., no. 4, Interscience, New York, 1967. MR 36 #3515. MR 220455
- 21.
- G. C. Papanicolaou and W. Kohler, Asymptotic theory of mixing stochastic stochastic ordinary differential equations, Comm. Pure Appl. Math, (to appear). MR 368142
- 22.
- R. Z. Has'minskiĭ (Khasminskii), On the principle of averaging for Itô's stochastic differential equations, Kybernetika (Prague) 4 (1968), 260-279. (Russian) MR 41 #4681. MR 260052
- 23.
- W. Kohler and G. C. Papanicolaou, Limit theorems for stochastic equations with rapidly varying components (to appear).
- 24.
- R. Kubo, Stochastic Liouville equations, J. Mathematical Phys. 4 (1963), 174-183. MR 26 #7370. MR 149885
- 25.
- G. S. Agarwal, "Master equation methods in quantum optics," in Progress in Optics. Vol. 11, E. Wolf (editor), North-Holland, Amsterdam, 1973.
- 26.
- R. J. Griego and R. Hersh, Theory of random evolutions with applications to partial differential equations, Trans. Amer. Math. Soc. 156 (1971), 405-418. MR 43 #1261. MR 275507
- 27.
- O. A. Oleĭnik and E. V. Radkeviĭ, Equations of second order with nonnegative characteristic form, Itogi Nauki Mat. Anal. 1969, VINITI, Moscow, 1971; English transl., Plenum Press, New York, 1973. MR 457908
- 28.
- D. W. Stroock and S. R. S. Varadhan, On degenerate elliptic-parabolic operators of second order and their associated diffusions, Comm. Pure Appl. Math. 25 (1972), 651-714. MR 387812
- 29.
- H. P. McKean, Jr., Stochastic integrals, Probability and Math. Statist., no. 5, Academic Press, New York, 1969. MR 40 #947. MR 247684
- 30.
- H. P. McKean, Jr., Fluctuations in the kinetic theory of gases (to appear). MR 395662
- 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.
- H. Grad, Solution of the Boltzmann equation in an unbounded domain, Comm. Pure Appl. Math. 18 (1965), 345-354. MR 32 #8913. MR 191508
- 33.
- G. E. Uhlenbeck and L. S. Ornstein, On the theory of Brownian motion, Phys. Rev. 36(1930), 823-841.
- 34.
- Ming Chen Wang and G. E. Uhlenbeck, On the theory of Brownian motion. II, Rev. Modern Phys. 17 (1945), 323-342. MR 7, 130. MR 13266
- 35.
- R. Z. Has'minskiĭ (Khasminskii), The averaging principle for parabolic and elliptic differential equations and Markov processes with small diffusion, Teor. Verojatnost. i Primenen. 8 (1963), 3-25=Theor. Probability Appl. 8 (1963), 1-21. MR 28 #4253. MR 161044
- 36.
- V. M. Volosov, Averaging in systems of ordinary differential equations, Uspehi Mat. Nauk 17 (1962), no. 6 (108), 3-126=Russian Math. Surveys 17 (1962), no. 6, 1-126. MR 26 #3976. MR 146454
- 37.
- R. S. Ellis and M. A. Pinsky, Asymptotic nonuniqueness of the Navier-Stokes equation in kinetic theory, Bull. Amer. Math. Soc. 80 (1974), 1160-1164. MR 609539
- 38.
- H. Grad, Singular and nonuniform limits of solutions of the Boltzmann equation, SIAM-AMS Proc., vol. 1, Amer. Math. Soc., Providence, R.I., 1969, pp. 269-308. MR 40 #8410. MR 255205
- 39.
- E. W. Larsen and J. B. Keller, Asymptotic solution of neutron transport problems for small mean free paths, J. Mathematical Phys. 15 (1974), 75-81. MR 339741
- 40.
- E. W. Larsen, Solutions of the steady, one-speed neutron transport equation for small mean free paths, J. Mathematical Phys. 15 (1974), 299-305. MR 345570
- 41.
- B. White, Dissertation, New York University, Sept. 1974.
- 42.
- N. Ja. Vilenkin, Special functions and the theory of group representations, "Nauka", Moscow, 1965; English transl., Transl. Math. Monographs, vol. 22, Amer. Math. Soc., Providence, R.I., 1968. MR 35 #420; 37 #5429. MR 209523
- 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. MR 383530
- 44.
- G. C. Papanicolaou and J. 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. MR 46 #8304. MR 309194
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Additional Information:
DOI:
10.1090/S0002-9904-1975-13744-X
PII:
S 0002-9904(1975)13744-X
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