On small random perturbations of some smooth dynamical systems

Yu. I. Kifer

Izvestiya. Mathematics , Tome 8 (1974) no. 5, pp. 1083-1107.

Résumé

This paper studies the problem of convergence of invariant measure which is a result of perturbation by a diffusion process with a small parameter (tending to zero) of certain smooth dynamical systems.

Bibliographie

[1] Dub Dzh. L., Veroyatnostnye protsessy, IL, M., 1956

[2] Nevelson M. B., “O povedenii invariantnoi mery diffuzionnogo protsessa s maloi diffuziei na okruzhnosti”, Teor. veroyatn. i ee primen., 9:2 (1964), 139–145 | MR

[3] Khasminskii R. Z., “O printsipe usredneniya dlya parabolicheskikh i ellipticheskikh differentsialnykh uravnenii i markovskikh protsessov s maloi diffuziei”, Teor. veroyatn. i ee primen., 8:1 (1963), 3–25 | MR

[4] Khasminskii R. Z., “O rabote avtokolebatelnoi sistemy pri vozdeistvii na nee malogo shuma”, Prikl. matem. i mekh., 27:4 (1963), 683–688 | MR

[5] Khasminskii R. Z., “O rabote konservativnoi sistemy pri vozdeistvii malogo treniya i malogo sluchainogo shuma”, Prikl. matem. i mekh., 28:5 (1964), 931–935 | MR

[6] Venttsel A. D., Freidlin M. I., “O malykh sluchainykh vozmuscheniyakh dinamicheskikh sistem”, Uspekhi matem. nauk, 25:1 (1970), 3–55 | Zbl

[7] Kifer Yu. I., “O malykh sluchainykh vozmuscheniyakh diffeomorfizmov”, Uspekhi matem. nauk, 29:3 (1974), 173–174 | MR | Zbl

[8] Auslendr L., Grin L., Khan F., Potoki na odnorodnykh prostranstvakh, Mir, M., 1966 | MR

[9] Smeil S., “Differentsiruemye dinamicheskie sistemy”, Uspekhi matem. nauk, 25:1 (1970), 113–185 | MR

[10] Ruelle D., A measure associated with Axiom $\mathrm{A}$ attractors, preprint | MR

[11] Sinai Ya. G., “Gibbsovskie mery v ergodicheskoi teorii”, Uspekhi matem. nauk, 27:4 (1972), 21–64 | MR | Zbl

[12] Sinai Ya. G., “Markovskie razbieniya i $Y$-diffeomorfizmy”, Funkts. analiz i ego prilozh., 2:1 (1968), 64–68 | MR

[13] Ruelle D., Takens F., “On the Nature of Turbulence”, Comm. Math. Phis., 20:3 (1971), 167–192 | DOI | MR | Zbl

[14] Sinai Ya. G., “Klassicheskie dinamicheskie sistemy so schetno-kratnym lebegovskim spektrom, II”, Izv. AN SSSR. Ser. matem., 30 (1966), 15–68 | MR | Zbl

[15] Brin M. I., Pesin Ya. B., “Potoki reperov na mnogoobraziyakh otritsatelnoi krivizny”, Uspekhi matem. nauk, 28:4 (1973), 209–210 | MR | Zbl

[16] Hirch and Pugh C., “Stable manifolds and hyperbolic sets”, Proc. Sympos. Pure Math., 14, 1970, 133–163 | MR

[17] Bowen R., “Markov partitions for Axiom $\mathrm{A}$ diffeomorphisms”, Amer. J. Math., 92 (1970), 725–747 | DOI | MR | Zbl

[18] Molchanov S. A., “Diffuzionnye protsessy i rimanova geometriya”, Uspekhi matem. nauk, 29:6 (1974)

[19] Bowen R., Symbolic dinamics for hyperbolic flows, preprint

[20] Aronson D. G., “The fundamental solution of a linear parabolic equations containing a small parameter”, Illinois J. Math., 3:4 (1959), 580–619 | MR | Zbl

[21] Freidlin M. I., “Funktsional deistviya dlya odnogo klassa sluchainykh protsessov”, Teor. veroyatn. i ee primen., 17:3 (1972), 536–541 | MR | Zbl

[22] Anosov D. V., Geodezicheskie potoki na zamknutykh rimanovykh mnogoobraziyakh otritsatelnoi krivizny, Tr. Matem. in-ta im. V. A. Steklova AN SSSR, 90, 1967 | MR

[23] Furstenberg H., “The structure of distal flows”, Amer. J. Math., 85:3 (1963), 477–515 | DOI | MR | Zbl

[24] Furstenberg H., “Strict ergodicity and transformations of the torus”, Amer. J. Math., 83:3 (1961), 447–515 | MR

[25] Newhouse S. T., “Hyperbolic limit sets”, Trans. Amer. Math. Soc., 167 (1972), 125–150 | DOI | MR | Zbl

[26] Furstenberg H., “The unique ergodicity of the horocycle flow”, Lecture Notes in Math., 318, 1973, 95–115 | MR | Zbl

[27] Bowen R., Ruelle D., The ergodic theory of Axiom $\mathrm{A}$ flows, preprint, 1974 | MR

[28] Rather M., “Markov partitions for Anosov flows on $n$-dimentional manifolds”, Israel J. Math., 15 (1973), 92–114 | DOI | MR

[29] Rokhlin V. A., “Lektsii po entropiinoi teorii preobrazovanii s invariantnoi meroi”, Uspekhi matem. nauk., 22:5 (1967), 5–56 | Zbl