Non-autonomous stochastic Cauchy problems in Banach spaces

• Autorzy: Mark Veraar , Jan Zimmerschied
• Języki publikacji: EN

Abstrakty

EN

We study the non-autonomous stochastic Cauchy problem on a real Banach space E,
$dU(t) = A(t)U(t)dt + B(t)dW_{H}(t)$, t ∈ [0,T], U(0) = u₀.
Here, $W_{H}$ is a cylindrical Brownian motion on a real separable Hilbert space H, $(B(t))_{t∈[0,T]}$ are closed and densely defined operators from a constant domain 𝒟(B) ⊂ H into E, $(A(t))_{t∈[0,T]}$ denotes the generator of an evolution family on E, and u₀ ∈ E. In the first part, we study existence of weak and mild solutions by methods of van Neerven and Weis. Then we use a well-known factorisation method in the setting of evolution families to obtain time regularity of the solution. In the second part, we consider the parabolic case in the setting of Acquistapace and Terreni. By means of a factorisation method in the spirit of Da Prato, Kwapień, and Zabczyk we obtain space-time regularity results for parabolic evolution families on Banach spaces. We apply this theory to several examples. In the last part, relying on recent results of Dettweiler, van Neerven, and Weis, we prove a maximal regularity result where the A(t) are as in the setting of Kato and Tanabe.

Kategorie tematyczne

• 47D06: One-parameter semigroups and linear evolution equations
• 35R60: Partial differential equations with randomness, stochastic partial differential equations
• 60H15: Stochastic partial differential equations

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 2008
• Tom: 185
• Numer: 1
• Strony: 1-34

Daty

wydano

2008

Twórcy

autor

Mark Veraar

• Delft Institute of Applied Mathematics, Technical University of Delft, P.O. Box 5031, 2600 GA Delft, The Netherlands

autor

Jan Zimmerschied

• Institut für Analysis, Universität Karlsruhe (TH), Englerstraße 2, 76128 Karlsruhe, Germany

Identyfikatory

DOI

10.4064/sm185-1-1