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Non-autonomous stochastic Cauchy problems in Banach spaces

Mark Veraar; Jan Zimmerschied — 2008

Studia Mathematica

We study the non-autonomous stochastic Cauchy problem on a real Banach space E, $d U (t) = A (t) U (t) d t + B (t) d W_{H} (t)$ , t ∈ [0,T], U(0) = u₀. Here, $W_{H}$ is a cylindrical Brownian motion on a real separable Hilbert space H, ${(B (t))}_{t \in [0, T]}$ are closed and densely defined operators from a constant domain (B) ⊂ H into E, ${(A (t))}_{t \in [0, T]}$ 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...

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