Pointwise and Renyi dimensions of an invariant measure of random dynamical systems with jumps. (English) Zbl 1107.37010
Summary: We estimate the lower pointwise dimension and the generalized Renyi dimension of an invariant measure of random dynamical systems with jumps. It is worthwhile to note that the dimensions are a useful tool in studying the Hausdorff dimension of measures and sets. Our model generalizes Markov processes corresponding to iterated function systems and Poisson driven stochastic differential equations. It can be used as a description of many physical and biological phenomena.
MSC:
| 37A99 | Ergodic theory |
| 37H10 | Generation, random and stochastic difference and differential equations |
| 60G15 | Gaussian processes |
| 47A35 | Ergodic theory of linear operators |
References:
| [1] | M. H. A. Davis, Markov Models and Optimization (Chapman and Hall, London, 1993). · Zbl 0780.60002 |
| [2] | K. J. Falconer, Dimensions and measures of quasi self-similar sets, Dimensions and measures of quasi self-similar sets, Proc. Amer. Math. Soc. 106:543–554 (1989). · Zbl 0683.58034 · doi:10.1090/S0002-9939-1989-0969315-8 |
| [3] | K. U. Frisch, Wave propagation in random media, stability, in Probabilistic Methods in Applied Mathematics, A. T. Bharucha-Reid (Ed.), Academic Press (1986). |
| [4] | K. Horbacz, Randomly connected dynamical systems–asymptotic stability, Ann. Polon. Math. 68(1):31–50 (1998). · Zbl 0910.47003 |
| [5] | K. Horbacz, Randomly connected differential equations with Poisson type perturbations (to appear). · Zbl 1011.60036 |
| [6] | K. Horbacz, Invariant measure related with randomly connected Poisson driven differential equations, Ann. Polon. Math. 79(1):31–43 (2002). · Zbl 1011.60036 · doi:10.4064/ap79-1-3 |
| [7] | K. Horbacz, Random dynamical systems with jumps, J. Appl. Prob. 41:890–910 (2004). · Zbl 1091.47012 · doi:10.1239/jap/1091543432 |
| [8] | K. Horbacz, J. Myjak, and T. Szarek, Stability of random dynamical systems on Banach spaces (to appear). · Zbl 1121.37037 |
| [9] | K. Horbacz, J. Myjak, and T. Szarek, On stability of some general random dynamical system, J. Stat. Phys. 119:35–60 (2005). · Zbl 1125.37038 · doi:10.1007/s10955-004-2045-6 |
| [10] | J. B. Keller, Stochastic equations and wave propagation in random media, Proc. Symp. Appl. Math. 16:1456–1470 (1964). · Zbl 0202.46703 |
| [11] | Y. Kifer, Ergodic Theory of Random Transformations (Birkháuser, Basel, 1986). · Zbl 0604.28014 |
| [12] | T. Kudo and I. Ohba, Derivation of relativistic wave equation from the Poisson process, Pramana - J. Phys. 59:413–416 (2002). · doi:10.1007/s12043-002-0135-z |
| [13] | A. Lasota and J. Myjak, On a dimension of measures, Bull. Polish Acad. Math. 50(2):221–235 (2002). · Zbl 1020.28004 |
| [14] | A. Lasota and T. Szarek, Dimension of measures invariant with respect to the Wazewska partial differential equation, J. Diff. Eqs. 196(2):448–456 (2004). · Zbl 1036.35054 · doi:10.1016/j.jde.2003.10.005 |
| [15] | A. Lasota and J. Traple, Invariant measures related with Poisson driven stochastic differential equation, Stoch. Proc. and Their Appl. 106(1):81–93 (2003). · Zbl 1075.60535 |
| [16] | P. Lévy, Théorie de l” addition des variables aléatoires (Gautier-Villar, Paris, 1937). |
| [17] | J. Myjak and T. Szarek, Capacity of invariant measures related to Poisson-driven stochastic differential equation, Nonlinearity 16:441–455 (2003). · Zbl 1026.47006 · doi:10.1088/0951-7715/16/2/305 |
| [18] | Y. B. Pesin, Dimension theory in dynamical systems:Contemporary views and Applications (The University of Chicago Press, 1997). |
| [19] | G. Prodi, Teoremi Ergodici per le Equazioni della Idrodinamica, (C.I.M.E., Roma, 1960). |
| [20] | T. Szarek, The pointwise dimension for invariant measures related with Poisson-driven stochastic differential equations, Bull. Pol. Acad. Sci. Math. 50(2):241–250 (2002). · Zbl 1006.60069 |
| [21] | T. Szarek, The dimension of self-similar measures, Bull. Pol. Acad. Sci. Math. 48(3):293–302 (2000). · Zbl 0974.47006 |
| [22] | J. Traple, Markov semigroups generated by Poisson driven differential equations, Bull. Pol. Ac.: Math. 44:161–182 (1996). · Zbl 0861.45008 |
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