Lower bound technique and its applications to function systems and stochastic partial differential equations

• Autorzy: Szarek T.

Identyfikatory

DOI

10.14708/ma.v41i2.466

Warianty tytułu

PL

Metoda miary dolnej i jej zastosowanie w teorii iterowanych systemów funkcyjnych i stochastycznych równań rózniczkowych

• Języki publikacji: EN

Abstrakty

EN

We formulate some criteria for the existence of an invariant measure for Markov chains and Markov processes. We also show their application in the theory of function systems and stochastic differential equations

PL

W pracy formułujemy kryteria dla istnienia miary niezmienniczej dla łańcuchów i procesów Markowa. Następnie pokazujemy ich użyteczność w teorii iterowanych układów funkcyjnych i stochastycznych równań rózniczkowych.

Słowa kluczowe

EN

ergodicity of Markov families; invariant measures; function systems; stochastic differential equations

PL

ergodyczność; rodziny Markowa; miara niezmiennicza; iterowane; systemy funkcji; stochastyczne równania różniczkowe

• Wydawca: Polskie Towarzystwo Matematyczne
• Czasopismo: Mathematica Applicanda
• Rocznik: 2013
• Tom: Vol. 41, No. 2
• Strony: 185--198
• Opis fizyczny: Bibliogr. 22 poz.

Twórcy

autor

Szarek T.

• University of Gdańsk Institute of Mathematics Wita Stwosza 57, 80-952 Gdańsk, Poland

Bibliografia

• [1] Barnsley M.F., Demko S.G., Elton J.H. and Geronimo J.S. Invariant measures arising from iterated function systems with place dependent probabilities Ann. Inst. Henri Poincaré 24(3) (1988), 367–394. MR 0971099; Zbl 0653.60057
• [2] Billingsley, P., Convergence of Probability Measures, Wiley, New York, 1968. xii+253 pp. MR 0233396; Zbl 0172.21201
• [3] Da Prato, G. and Zabczyk, J., Stochastic Equations in Infinite Dimensions. Encyclopedia of Mathematics and Its Applications. vol. 44 Cambridge University Press, Cambridge 1992. xviii+454 pp. ISBN: 0-521-38529-6 MR 1207136 ; Zbl 0761.60052
• [4] Doeblin, W., ´ Eléments d’une théorie générale des chaines simples constantes de Markov, Ann. Ecole Norm. 57 (1940), 61–111. MR 0004409; Zbl 0303.9169
• [5] Es-Sarhir, A. and von Renesse, M.-K., Ergodicity of stochastic curve shortening flow in the plane, SIAM J. Math. Anal. 44 (1) (2012), 224–244. MR 2888287 ; Zbl 1251.47041; doi:10.1137/100798235
• [6] Ethier, S. and Kurtz, T., Markov processes: Characterization and Convergence, Wiley, New York, 1968. MR 0838085; Zbl 1089.60005
• [7] Hairer, M. and Mattingly, J., Ergodicity of the 2D Navier-Stokes equations with degenerate stochastic forcing, Ann. of Math. 164 (2006), 993–1032. MR 2259251; Zbl 1130.37038; doi:10.4007/annals.2006.164.993
• [8] Jaroszewska, J., On asymptotic equicontinuity of Markov transition functions, Statistics and Probability Letters 83 (2013), 943-951. MR 3040327; Zbl 1268.60102; doi:10.1016/j.spl.2012.10.033
• [9] Kapica, R., Szarek T. and Ślęczka, M., On a unique ergodicity of some Markov processes, Potential Anal. 36 (2012), 589-606. MR 2904635; Zbl 1244.60074; doi: 10.1007/s11118-011-9242-0
• [10] Komorowski, T., Peszat, S. and Szarek, T., On ergodicity of some Markov processes, Ann. Probab. 38 (4) (2010), 1401-1443. MR 2663632; Zbl 1214.60035; doi: 10.1214/09-AOP513
• [11] Lasota, A. and Mackey, M., Probabilistic Properties of Deterministic Systems, Cambridge University Press, Cambridge 1985. MR 1244104; Zbl 0784.58005
• [12] Lasota, A. and Szarek, T., Lower bound technique in the theory of a stochastric differential equation, J. Differential Equations 231 (2006), 513–533. MR 2287895; Zbl 05115329
• [13] Lasota, A. and Yorke, J. A., On the existence of invariant measures for piecewise monotonic transformations, Trans. Amer. Math. Soc. 186 (1973), 481–488. MR 0335758; Zbl 0298.28015; doi: 10.1090/S0002-9947-1973-0335758-1
• [14] Lasota, A. and Yorke, J. A., Lower bound technique for Markov operators and iterated function systems, Random Comput. Dynam. 2 (1994), 41–77. MR 1265226; Zbl 0804.47033; doi:10.1016/j.jde.2006.04.018
• [15] Meyn, S.P. and Tweedie, R. L., Markov Chains and Stochastic Stability, Springer-Verlag, Berlin, Heidelberg, New York, 1993. ; MR 1287609; Zbl 1165.60001; doi:10.1017/CBO9780511626630
• [16] Szarek, T., Feller processes on nonlocally spaces, Ann. Probab. 34 (2006), 1849–1863. MR 2271485; Zbl 1108.60064; doi: 10.1214/009117906000000313
• [17] Szarek, T., Invariant measures for nonexpansive Markov operators on Polish spaces, Dissert. Math. 415, (2003), p. 62. MR 1997024; Zbl 1051.37005; doi: 10.4064/dm415-0-1
• [18] Szarek, T., Ślęczka, M. and Urbański, M. On stability of velocity vectors for some passive tracer models, Bull. London Math. Soc. 42 (5) (2010), 923-936. MR 2728696; Zbl 1204.60070; doi: 10.1112/blms/bdq055
• [19] Szarek, T. and Worm, D.T.H. Ergodic measures of Markov semigroups with the e-property, Ergodic Theory Dynamical Systems 32 (3) (2012), 1117-1135. MR 2995659; Zbl 1261.37007; doi: 10.1017/S0143385711000022
• [20] Xie, B. Uniqueness of invariant measures of infinite dimensional stochastic differential equations driven by Lévy noises, Potential Anal. 36 (1) (2012), 35–66. MR 2886453; Zbl 1260.60126; doi: 10.1007/s11118-011-9220-6
• [21] Wang, J. Regularity of semigroups generated by L´evy type operators via coupling, Stochastic Process. Appl. 120 (9) (2010), 1680–1700. MR 2673970; Zbl 1204.60071; doi:10.1016/j.spa.2010.04.007
• [22] Zaharopol, R., Invariant Probabilities of Markov-Feller Operators and their Supports, Birkh¨auser, Basel 2005. MR 2128542; Zbl 1072.37007