Kantorovich-Rubinstein maximum principle in the stability theory of Markov semigroups

• Autorzy: Gacki H.
• Języki publikacji: EN

Abstrakty

EN

A new sufficient condition for the asymptotic stability of a locally Lipschitzian Markov semigroup acting on the space of signed measures. M[sig] is proved. This criterion is applied to the semigroup of Markov operators generated by a Poisson driven stochastic differential equation.

Słowa kluczowe

EN

asymptotic stability; Poisson driven stochastic equation; Markov semigroup

PL

stabilność asymptotyczna; równanie Poissona; półgrupa Markowa

• Wydawca: Instytut Matematyczny PAN
• Czasopismo: Bulletin of the Polish Academy of Sciences. Mathematics
• Rocznik: 2004
• Tom: Vol. 52, nr 2
• Strony: 211--222
• Opis fizyczny: Bibliogr. 13 poz.

Twórcy

autor

Gacki H.

• Institute of Mathematics, Silesian University, Bankowa 14, 40-007 Katowice, Poland,

Bibliografia

• [1] H. Gacki, An application of the Kantorovich-Rubinstein maximum principle in the stability theory of Markov operators, Bull. Polish Acad. Sci. Math. 46 (1998), 215-223.
• [2] J. Hutchinson, Fractals and self-similarity, Indiana Univ. Math. J. 30 (1981), 713-747.
• [3] N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, North-Holland, Amsterdam, 1981.
• [4] L. V. Kantorovich and G. S. Rubinstein, On a space of completely additive functions, Vestnik Leningrad Univ. 13 (1958), 52-59 (in Russian).
• [5] A. Lasota, Invariant principle for discrete time dynamical systems, Univ. Iagell. Acta Math. 31 (1994), 111-127.
• [6] -, From fractals to stochastic differential equations, in: Chaos-The Interplay Between Stochastic and Deterministic Behaviour (Karpacz, 1995), P. Garbaczewski et al. (eds.), Lecture Notes in Phys. 457, Springer, Berlin, 1995, 235-255.
• [7] A. Lasota and M. C. Mackey, Chaos, Fractals, and Noise, Springer, Berlin, 1994.
• [8] A. Lasota and J. Traple, An application of the Kantorovich-Rubinstein maximum principle in the theory of the Tjon-Wu equation, J. Differential Equations 159 (1999), 578-596.
• [9] A. Lasota and J. A. Yorke, Lower bound technique for Markov operators and iterated function systems, Random Comp. Dyn. 2 (1994), 41-77.
• [10] J. Malczak, Statistical stability of Poisson driven differential equations, Bull. Polish Acad. Sci. Math. 41 (1993), 159-176.
• [11] S. T. Rachev, Probability Metrics and the Stability of Stochastic Models, Wiley, New York, 1991.
• [12] J. Traple, Markov semigroups generated by Poisson driven differential equations, Bull. Polish Acad. Sci. Math. 44 (1996), 160-182.
• [13] -, On the asymptotic stability of Markov semigroups, ibid. 44 (1996), 183-195.