### Hahn-Banach type theorems for adjoint semigroups.

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Let $A=\mathrm{d}/\mathrm{d}\theta $ denote the generator of the rotation group in the space $C\left(\Gamma \right)$, where $\Gamma $ denotes the unit circle. We show that the stochastic Cauchy problem $$\mathrm{d}U\left(t\right)=AU\left(t\right)+f\mathrm{d}{b}_{t},\phantom{\rule{1.0em}{0ex}}U\left(0\right)=0,\phantom{\rule{2.0em}{0ex}}\left(1\right)$$ where $b$ is a standard Brownian motion and $f\in C\left(\Gamma \right)$ is fixed, has a weak solution if and only if the stochastic convolution process $t\mapsto {(f*b)}_{t}$ has a continuous modification, and that in this situation the weak solution has a continuous modification. In combination with a recent result of Brzeźniak, Peszat and Zabczyk it follows that (1) fails to have a weak solution for all...

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