On the approximation theorem of Wong-Zakai type for the Lasota operator

• Autorzy: Dawidowicz A. L. , Twardowska K.
• Języki publikacji: EN

Abstrakty

EN

We consider in this paper a stochastic evolution equation with Professor A. Lasota's operator as the infinitesimal generator of a strongly continuous semigroup of transformations and with Hammerstein operator connected with a noise being the Wiener process. We show that such evolution equation satisfies the Wong-Zakai type approximation theorem. The idea of the definition of the Lasota operator has the origin in the mathematical model of the creation and differentiation of cells in biology and medicine.

Słowa kluczowe

EN

stochastic evolution equations; Wong-Zakai approximations; Lasota operator

• Wydawca: AGH University of Science and Technology Press
• Czasopismo: Opuscula Mathematica
• Rocznik: 2010
• Tom: Vol. 30, no. 3
• Strony: 255--270
• Opis fizyczny: Bibliogr. 17 poz.

Twórcy

autor

Dawidowicz A. L.

autor

Twardowska K.

• Jagellonian University Institute of Mathematics ul. Łojasiewicza 6, 30-348 Cracow, Poland,

Bibliografia

• [1] Z. Brzeźniak, A.L. Dawidowicz, On periodic solutions of the Lasota equation, Semigroup Forum 78 (2009), 118–137.
• [2] R.F. Curtain, A.J. Pitchard, Infinite dimensional linear systems theory, Springer, Berlin, 1978.
• [3] A.L. Dawidowicz, On the existence of an invariant measure for the dynamical system generated by partial differential equation, Ann. Polon. Math. XLI (1983), 129–137.
• [4] A.L. Dawidowicz, N. Haribash, On the periodic solutions of von Foerster type equation, Univ. Iagel. Acta Math. 37 (1999), 321–324.
• [5] A.L. Dawidowicz, A. Poskrobko, On asymptotic behaviour of the dynamical systems generated by von Foerster-Lasota equations, Control and Cybernet. 35 (2006) 4, 803–813.
• [6] A.L. Dawidowicz, K. Twardowska, On an approximation theorem of Wong-Zakai type for the Lasota operator, Mat. Stosow. 8 (2007) 49, 56–65 [in Polish].
• [7] A. Deitmar, S. Echterhoff, Principles of harmonic analysis, Springer, 2009.
• [8] A. Lasota, Invariant measures and a linear model of turbulence, Rendiconti del Seminario Matematica dell’Universita di Padova 61 (1979), 39–48.
• [9] A. Lasota, J. Yorke, On the existence of invariant measures for transformations with strictly turbulent trajectories, Bull. Pol. Acad. Sci., Ser. Sci. Math. Astronom. Phys. 25 (1977), 233–238.
• [10] A. Lasota, M. Ważewska-Czyżewska, Mathematical problems of the dynamics of a system of red blood cells, Mat. Stosow. 6 (1976), 23–40 [in Polish].
• [11] K. Łoskot, Turbulent solutions of first order partial differential equation, J. Differential Equations 58 (1985) 1, 1–14.
• [12] G. Prodi, Teoremi ergodici per le equazioni della idrodinamica, C.I.M.E., Roma, 1960.
• [13] R. Rudnicki, Invariant measures for the flow of a first order partial differential equation Ergodic Theory Dynam. Systems 5 (1985) 3, 437–443.
• [14] K. Twardowska, An approximation theorem of Wong-Zakai type for nonlinear stochastic partial differential equations, Stoch. Anal. Appl. 13 (1995) 5, 601–626.
• [15] K. Twardowska, Approximation theorems of Wong-Zakai type for stochastic differential equations in infinite dimensions, Dissertationes Math., Polska Akademia Nauk, Instytut Matematyczny, vol. 325, Warszawa, 1993, 1–53.
• [16] K. Twardowska, On support theorems for stochastic nonlinear partial differential equations [in:] Stochastic Differential and Difference Equations, I. Csiszar and Gy. Michaletzky eds., Birkhauser, Boston, 1997, 309–317.
• [17] E. Wong, M. Zakai, On the convergence of ordinary integrals to stochastic integrals, Ann. Math. Statist. 36 (1965), 1560–1564.