Non-autonomous stochastic Cauchy problems in Banach spaces

Mark Veraar¹ ; Jan Zimmerschied²

¹Delft Institute of Applied Mathematics Technical University of Delft P.O. Box 5031 2600 GA Delft, The Netherlands
²Institut für Analysis Universität Karlsruhe (TH) Englerstraße 2 76128 Karlsruhe, Germany

Studia Mathematica, Tome 185 (2008) no. 1, pp. 1-34

Résumé

We study the non-autonomous stochastic Cauchy problem on a real Banach space $E$, $$\eqalign{ d U(t) = A(t) U(t) \, dt + B(t)\, d W_H(t), \ \ t\in [0,T], U(0) = u_0. }$$ Here, $W_H$ is a cylindrical Brownian motion on a real separable Hilbert space $H$, $(B(t))_{t\in [0,T]}\!$ are closed and densely defined operators from a constant domain $\mathcal D(B)\subset H$ into $E$, $(A(t))_{t\in [0,T]}$ denotes the generator of an evolution family on $E$, and $u_0\in E$. 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.

Détail

DOI : 10.4064/sm185-1-1

Keywords: study non autonomous stochastic cauchy problem real banach space eqalign t here cylindrical brownian motion real separable hilbert space closed densely defined operators constant domain mathcal subset denotes generator evolution family first part study existence weak mild solutions methods van neerven weis well known factorisation method setting evolution families obtain time regularity solution second part consider parabolic setting acquistapace terreni means factorisation method spirit prato kwapie zabczyk obtain space time regularity results parabolic evolution families banach spaces apply theory several examples part relying recent results dettweiler van neerven weis prove maximal regularity result where setting kato tanabe