| Title:
|
Ergodic behaviour of stochastic parabolic equations (English) |
| Author:
|
Seidler, Jan |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
47 |
| Issue:
|
2 |
| Year:
|
1997 |
| Pages:
|
277-316 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
The ergodic behaviour of homogeneous strong Feller irreducible Markov processes in Banach spaces is studied; in particular, existence and uniqueness of finite and $\sigma $-finite invariant measures are considered. The results obtained are applied to solutions of stochastic parabolic equations. (English) |
| Keyword:
|
Markov processes |
| Keyword:
|
invariant measures |
| Keyword:
|
recurrence |
| Keyword:
|
stochastic parabolic equations |
| MSC:
|
35K99 |
| MSC:
|
35R60 |
| MSC:
|
60H10 |
| MSC:
|
60H15 |
| MSC:
|
60J35 |
| idZBL:
|
Zbl 0935.60041 |
| idMR:
|
MR1452421 |
| . |
| Date available:
|
2009-09-24T10:05:04Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/127357 |
| . |
| Reference:
|
[1] J. Azéma, M. Kaplan-Duflo, D. Revuz: Récurrence fine des processus de Markov.Ann. Inst. H. Poincaré Probab. Statist. 2 (1966), 185–220. MR 0199889 |
| Reference:
|
[2] J. Azema, M. Kaplan-Duflo, D. Revuz: Mesure invariante sur les classes récurrentes des processus de Markov.Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 8 (1967), 157–181. MR 0222955, 10.1007/BF00531519 |
| Reference:
|
[3] J. Azéma, M. Duflo, D. Revuz: Mesure invariante des processus de Markov recurrents.Séminaire de Probabilités III, Lecture Notes in Math. 88, Springer-Verlag, Berlin, 1969, pp. 24–33. MR 0260014 |
| Reference:
|
[4] R. N. Bhattacharya: Criteria for recurrence and existence of invariant measures for multidimensional diffusions.Ann. Probab. 6 (1978), 541–553. Zbl 0386.60056, MR 0494525, 10.1214/aop/1176995476 |
| Reference:
|
[5] A. Chojnowska-Michalik, B. Gołdys: Existence, uniqueness and invariant measures for stochastic semilinear equations on Hilbert spaces.Probab. Theory Related Fields 102 (1995), 331–356. MR 1339737, 10.1007/BF01192465 |
| Reference:
|
[6] G. Da Prato, A. Debussche: Stochastic Cahn-Hilliard equation.Nonlinear Anal. 26 (1996), 241–263. MR 1359472, 10.1016/0362-546X(94)00277-O |
| Reference:
|
[7] G. Da Prato, J. Zabczyk: Stochastic equations in infinite dimensions.Cambridge University Press, Cambridge, 1992. MR 1207136 |
| Reference:
|
[8] C. Dellacherie, P.–A. Meyer: Probabilities and potential, Part A.North-Holland, Amsterdam, 1978. MR 0521810 |
| Reference:
|
[9] M. Duflo, D. Revuz: Propriétés asymptotiques de probabilités de transition des processus de Markov récurrents.Ann. Inst. H. Poincaré Probab. Statist. 5 (1969), 233–244. MR 0273680 |
| Reference:
|
[10] E. B. Dynkin: Osnovaniya teorii markovskikh processov.GIFML, Moskva, 1959. |
| Reference:
|
[11] E. B. Dynkin: Markovskie processy.GIFML, Moskva, 1963. |
| Reference:
|
[12] F. Flandoli, B. Maslowski: Ergodicity of the 2-D Navier-Stokes equation under random perturbations.Comm. Math. Phys. 171 (1995), 119–141. MR 1346374, 10.1007/BF02104513 |
| Reference:
|
[13] S. R. Foguel: Limit theorems for Markov processes.Trans. Amer. Math. Soc. 121 (1966), 200–209. Zbl 0203.19601, MR 0185642, 10.1090/S0002-9947-1966-0185642-8 |
| Reference:
|
[14] T. Funaki: Random motion of strings and related stochastic evolution equations.Nagoya Math. J. 89 (1983), 129–193. Zbl 0531.60095, MR 0692348, 10.1017/S0027763000020298 |
| Reference:
|
[15] R. J. Gardner, W. F. Pfeffer: Borel measures.Handbook of set-theoretic topology, North-Holland, Amsterdam 1984, pp. 961–1043. MR 0776641 |
| Reference:
|
[16] D. Gatarek, B. Gołdys: On weak solutions of stochastic equations in Hilbert spaces.Stochastics Stochastics Rep. 46 (1994), 41–51. MR 1787166, 10.1080/17442509408833868 |
| Reference:
|
[17] R. K. Getoor: Transience and recurrence of Markov processes.Séminaire de Probabilités XIV – 1978/79, Lecture Notes in Math. 784, Springer-Verlag, Berlin, 1980, pp. 397–409. Zbl 0431.60067, MR 0580144 |
| Reference:
|
[18] N. C. Jain: Some limit theorems for a general Markov process.Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 6 (1966), 206–223. Zbl 0234.60086, MR 0216580, 10.1007/BF00531804 |
| Reference:
|
[19] B. Jamison, S. Orey: Markov chains recurrent in the sense of Harris.Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 8 (1967), 41–48. MR 0215370, 10.1007/BF00533943 |
| Reference:
|
[20] R. Z. Hasminskiĭ: Èrgodiqeskie svoĭstva vozvratnyh diffuzionnyh processov i stabilizaciya rexeniĭ zadaqi Koxi dlya paraboliqeskih uravneniĭ.Teor. Veroyatnost. i Primenen. 5 (1960), 196–214. |
| Reference:
|
[21] R. Z. Hasminskiĭ: Ustoĭqivost sistem differencialnyh uravneniĭ pri sluqaĭnyh vozmuweniyah ih parametrov.Nauka, Moskva, 1969. MR 0259283 |
| Reference:
|
[22] U. Krengel: Ergodic theorems.Walter de Gruyter, Berlin-New York, 1985. Zbl 0575.28009, MR 0797411 |
| Reference:
|
[23] G. Maruyama: On the strong Markov property.Mem. Fac. Sci. Kyushu Univ. Ser. A 13 (1959), 17–29. Zbl 0093.14901, MR 0107306 |
| Reference:
|
[24] G. Maruyama, H. Tanaka: Some properties of one-dimensional diffusion processes.Mem. Fac. Sci. Kyusyu Univ. Ser. A 11 (1957), 117–141. MR 0097128 |
| Reference:
|
[25] G. Maruyama, H. Tanaka: Ergodic property of $N$-dimensional recurrent Markov processes.Mem. Fac. Sci. Kyushu Univ. Ser. A 13 (1959), 157–172. MR 0112175 |
| Reference:
|
[26] B. Maslowski: On probability distributions of solutions of semilinear stochastic evolution equations.Stochastics Stochastics Rep. 45 (1993), 17–44. Zbl 0792.60058, MR 1277360, 10.1080/17442509308833854 |
| Reference:
|
[27] B. Maslowski: Uniqueness and stability of invariant measures for stochastic differential equations in Hilbert spaces.Stochastics Stochastics Rep. 28 (1989), 85–114. Zbl 0683.60037, MR 1018545, 10.1080/17442508908833585 |
| Reference:
|
[28] B. Maslowski, J. Seidler: Ergodic properties of recurrent solutions of stochastic evolution equations.Osaka J. Math. 31 (1994), 965–1003. MR 1315015 |
| Reference:
|
[29] S. P. Meyn, R. L. Tweedie: Generalized resolvents and Harris recurrence of Markov processes.Doeblin and modern probability, Contemporary Mathematics Vol. 149, AMS, Providence, 1993, pp. 227–250. MR 1229967 |
| Reference:
|
[30] S. P. Meyn, R. L. Tweedie: Stability of Markovian processes IIContinuous time processes and sampled chains.Adv. in Appl. Probab. 25 (1993), 487–517. MR 1234294, 10.2307/1427521 |
| Reference:
|
[31] C. Mueller: Coupling and invariant measures for the heat equation with noise.Ann. Probab. 21 (1993), 2189–2199. Zbl 0795.60056, MR 1245306, 10.1214/aop/1176989016 |
| Reference:
|
[32] S. Peszat, J. Seidler: Maximal inequalities and space-time regularity of stochastic convolutions.Math. Bohem (to appear). MR 1618707 |
| Reference:
|
[33] S. Peszat, J. Zabczyk: Strong Feller property and irreducibility for diffusions on Hilbert spaces.Ann. Probab. 23 (1995), 157–172. MR 1330765, 10.1214/aop/1176988381 |
| Reference:
|
[34] M. Scheutzow: Qualitative behaviour of stochastic delay equations with a bounded memory.Stochastics 12 (1984), 41–80. Zbl 0539.60061, MR 0738934, 10.1080/17442508408833294 |
| Reference:
|
[35] J. Seidler: Da Prato-Zabczyk’s maximal inequality revisited I..Math. Bohem. 118 (1993), 67–106. Zbl 0785.35115, MR 1213834 |
| Reference:
|
[36] A. N. Shiryayev: Probability.Springer-Verlag, New York-Berlin, 1984. Zbl 0536.60001, MR 0737192 |
| Reference:
|
[37] A. V. Skorohod: Asimptotiqeskie metody teorii stohastiqeskih differentsialnyh uravneniĭ.Naukova Dumka, Kiev, 1987. |
| Reference:
|
[38] Ł. Stettner: On the existence and uniqueness of invariant measure for continuous time Markov processes.Lefschetz Center for Dynamical Systems Preprint # 86–18, April 1986. |
| Reference:
|
[39] Ł. Stettner: Remarks on ergodic conditions for Markov processes on Polish spaces.Bull. Polish Acad. Sci. Math. 42 (1994), 103–114. Zbl 0815.60072, MR 1810695 |
| Reference:
|
[40] D. W. Stroock, S. R. S. Varadhan: Diffusion processes with continuous coefficients II.Comm. Pure Appl. Math. 22 (1969), 479–530. MR 0254923, 10.1002/cpa.3160220404 |
| Reference:
|
[41] D. W. Stroock, S. R. S. Varadhan: On the support of diffusion processes with applications to the strong maximum principle.Proceedings Sixth Berkeley Symposium Math. Statist. Probab., Vol. III., Univ. of California Press, Berkeley-Los Angeles, 1972, pp. 333–359. MR 0400425 |
| Reference:
|
[42] J. Zabczyk: Structural properties and limit behaviour of linear stochastic systems in Hilbert spaces.Mathematical control theory, Banach Center Publications Vol. 14, PWN, Warsaw, 1985, pp. 591–609. Zbl 0573.93076, MR 0851253 |
| Reference:
|
[43] J. Zabczyk: Symmetric solutions of semilinear stochastic equations.Stochastic partial differential equations and applications II (Trento, 1988), Lecture Notes in Math. 1390, Springer-Verlag, Berlin, 1989, pp. 237–256. Zbl 0701.60060, MR 1019609 |
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