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 $$\mathrm{d}U\left(t\right)=AU\left(t\right)+f\mathrm{d}{b}_t,U\left(0\right)=0,\left(1\right)$$ 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...

We consider the second order initial value problem in a Hilbert space, which is a singular perturbation of a first order initial value problem. The difference of the solution and its singular limit is estimated in terms of the small parameter $\epsilon .$ The coefficients are commuting self-adjoint operators and the estimates hold also for the semilinear problem.