×
Seidler, Jan

Ergodic behaviour of stochastic parabolic equations. (English) Zbl 0935.60041

Czech. Math. J. 47, No. 2, 277-316 (1997).
The main goal of the paper is to investigate ergodic properties of Markov processes induced by semilinear stochastic equations of the form \(dX=(AX+F(X)) dt+\Sigma (X) dW\) in a Hilbert space \(H\), where \(A\) is an unbounded operator on \(H\) generating a strongly continuous semigroup, \(W\) is an \(H\)-valued Wiener process and \(F\) and \(\Sigma \) are nonlinear, \(H\)-valued and \(\mathcal L(H)\)-valued, respectively, functions on \(H\). Most of the results of the paper are however applicable to general strong Feller irreducible Markov processes taking values in Polish spaces. Criteria for existence and uniqueness of finite and \(\sigma \)-finite invariant measures are found and the relation between recurrence of the process and existence of invariant measures is studied in detail. An important ratio (or Hopf type) ergodic theorem is proven and a number of its consequences is given; in particular, convergence of transition probability kernels to the invariant measure in the norm of total variation is established. The results are applied to stochastic parabolic semilinear equations.
Reviewer: B.Maslowski (Praha)

Cited in 21 Documents

MSC:

60H15 Stochastic partial differential equations (aspects of stochastic analysis)
60J35 Transition functions, generators and resolvents

Keywords:

Markov processes; invariant measures; recurrence
PDFBibTeX XMLCite
\textit{J. Seidler}, Czech. Math. J. 47, No. 2, 277--316 (1997; Zbl 0935.60041)
Full Text: DOI EuDML

References:

[1] J. Azéma, M. Kaplan-Duflo, D. Revuz: Récurrence fine des processus de Markov. Ann. Inst. H. Poincaré Probab. Statist. 2 (1966), 185-220. · Zbl 0182.51103
[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. · Zbl 0178.20302 · doi:10.1007/BF00531519
[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. · Zbl 0316.60049
[4] R. N. Bhattacharya: Criteria for recurrence and existence of invariant measures for multidimensional diffusions. Ann. Probab. 6 (1978), 541-553. · Zbl 0386.60056 · doi:10.1214/aop/1176995476
[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. · Zbl 0859.60057 · doi:10.1007/BF01192465
[6] G. Da Prato, A. Debussche: Stochastic Cahn-Hilliard equation. Nonlinear Anal. 26 (1996), 241-263. · Zbl 0838.60056 · doi:10.1016/0362-546X(94)00277-O
[7] G. Da Prato, J. Zabczyk: Stochastic equations in infinite dimensions. Cambridge University Press, Cambridge, 1992. · Zbl 0761.60052
[8] C. Dellacherie, P.-A. Meyer: Probabilities and potential, Part A. North-Holland, Amsterdam, 1978.
[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. · Zbl 0183.47003
[10] E. B. Dynkin: Osnovaniya teorii markovskikh processov. GIFML, Moskva, 1959.
[11] E. B. Dynkin: Markovskie processy. GIFML, Moskva, 1963. · Zbl 0132.37701
[12] F. Flandoli, B. Maslowski: Ergodicity of the 2-D Navier-Stokes equation under random perturbations. Comm. Math. Phys. 171 (1995), 119-141. · Zbl 0845.35080 · doi:10.1007/BF02104513
[13] S. R. Foguel: Limit theorems for Markov processes. Trans. Amer. Math. Soc. 121 (1966), 200-209. · Zbl 0203.19601 · doi:10.2307/1994339
[14] T. Funaki: Random motion of strings and related stochastic evolution equations. Nagoya Math. J. 89 (1983), 129-193. · Zbl 0531.60095
[15] R. J. Gardner, W. F. Pfeffer: Borel measures. Handbook of set-theoretic topology, North-Holland, Amsterdam 1984, pp. 961-1043. · Zbl 0593.28016
[16] D. Gatarek, B. Gołdys: On weak solutions of stochastic equations in Hilbert spaces. Stochastics Stochastics Rep. 46 (1994), 41-51. · Zbl 0824.60052 · doi:10.1080/17442509408833868
[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
[18] N. C. Jain: Some limit theorems for a general Markov process. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 6 (1966), 206-223. · Zbl 0234.60086 · doi:10.1007/BF00531804
[19] B. Jamison, S. Orey: Markov chains recurrent in the sense of Harris. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 8 (1967), 41-48. · Zbl 0153.19802 · doi:10.1007/BF00533943
[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.
[21] R. Z. Hasminskiĭ: Ustoĭqivost sistem differencialnyh uravneniĭ pri sluqaĭnyh vozmuweniyah ih parametrov. Nauka, Moskva, 1969.
[22] U. Krengel: Ergodic theorems. Walter de Gruyter, Berlin-New York, 1985. · Zbl 0575.28009
[23] G. Maruyama: On the strong Markov property. Mem. Fac. Sci. Kyushu Univ. Ser. A 13 (1959), 17-29. · Zbl 0093.14901 · doi:10.2206/kyushumfs.13.17
[24] G. Maruyama, H. Tanaka: Some properties of one-dimensional diffusion processes. Mem. Fac. Sci. Kyusyu Univ. Ser. A 11 (1957), 117-141. · Zbl 0089.34604
[25] G. Maruyama, H. Tanaka: Ergodic property of \(N\)-dimensional recurrent Markov processes. Mem. Fac. Sci. Kyushu Univ. Ser. A 13 (1959), 157-172. · Zbl 0115.36003 · doi:10.2206/kyushumfs.13.157
[26] B. Maslowski: On probability distributions of solutions of semilinear stochastic evolution equations. Stochastics Stochastics Rep. 45 (1993), 17-44. · Zbl 0792.60058 · doi:10.1080/17442509308833854
[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 · doi:10.1080/17442508908833585
[28] B. Maslowski, J. Seidler: Ergodic properties of recurrent solutions of stochastic evolution equations. Osaka J. Math. 31 (1994), 965-1003. · Zbl 0820.60040
[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. · Zbl 0784.60066
[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. · Zbl 0781.60052 · doi:10.2307/1427521
[31] C. Mueller: Coupling and invariant measures for the heat equation with noise. Ann. Probab. 21 (1993), 2189-2199. · Zbl 0795.60056 · doi:10.1214/aop/1176989016
[32] S. Peszat, J. Seidler: Maximal inequalities and space-time regularity of stochastic convolutions. Math. Bohem · Zbl 0903.60047
[33] S. Peszat, J. Zabczyk: Strong Feller property and irreducibility for diffusions on Hilbert spaces. Ann. Probab. 23 (1995), 157-172. · Zbl 0831.60083 · doi:10.1214/aop/1176988381
[34] M. Scheutzow: Qualitative behaviour of stochastic delay equations with a bounded memory. Stochastics 12 (1984), 41-80. · Zbl 0539.60061 · doi:10.1080/17442508408833294
[35] J. Seidler: Da Prato-Zabczyk’s maximal inequality revisited I. Math. Bohem. 118 (1993), 67-106. · Zbl 0785.35115
[36] A. N. Shiryayev: Probability. Springer-Verlag, New York-Berlin, 1984. · Zbl 0536.60001
[37] A. V. Skorohod: Asimptotiqeskie metody teorii stohastiqeskih differentsialnyh uravneniĭ. Naukova Dumka, Kiev, 1987.
[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.
[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
[40] D. W. Stroock, S. R. S. Varadhan: Diffusion processes with continuous coefficients II. Comm. Pure Appl. Math. 22 (1969), 479-530. · doi:10.1002/cpa.3160220404
[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. · Zbl 0255.60056
[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
[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
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.