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

Dawidowicz A. L., Twardowska K.

Opuscula Mathematica

2010

Vol. 30, no. 3

255-270

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.

On support and invariance theorems for a stochastic system of Burgers equtions

Nowak A., Twardowska K.

Demonstratio Mathematica

2006

Vol. 39, nr 3

691-710

We prove a support theorem for a stochastic version of the Burgers system formulated for the deterministic case by Burgers in [Bu 39]. The existence and uniqueness theorem for such a stochastic system was given by Zabczyk and Twardowska in [TZ 06]. In the proof of our support theorem we use a Wong-Zakai type theorem for such a system proved by Nowak in [No 05]. We generalize the method of Mackevicius ([Ma 85]}, [Ma 86]) and Gyongy ([Gy 89]) to prove the support theorem for our stochastic system of Burgers equations. We also use some considerations from Twardowska [Tw 97a]. In our proof of the invariance theorem we use some result of Jachimiak from [Ja 98].