Let $A={\mathrm d}/{\mathrm d}\theta $ denote the generator of the rotation group in the space $C(\Gamma )$, where $\Gamma $ denotes the unit circle. We show that the stochastic Cauchy problem \[ {\mathrm d}U(t) = AU(t)+ f\mathrm{d}b_t, \quad U(0)=0, \qquad \mathrm{(1)}\] where $b$ is a standard Brownian motion and $f\in C(\Gamma )$ 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 $f\in C(\Gamma )$ outside a set of the first category.
Let $H$ be a Hilbert space and $E$ a Banach space. We set up a theory of stochastic integration of ${\cal L}(H,E)$-valued functions with respect to $H$-cylindrical Liouville fractional Brownian motion with arbitrary Hurst parameter $0<\beta <1$. For $0<\beta <\frac 12$ we show that a function $\Phi \colon (0,T)\to {\cal L}(H,E)$ is stochastically integrable with respect to an $H$-cylindrical Liouville fractional Brownian motion if and only if it is stochastically integrable with respect to an $H$-cylindrical fractional Brownian motion. We apply our results to stochastic evolution equations $$ {\rm d}U(t) = AU(t) {\rm d}t + B {\rm d}W_H^\beta (t) $$ driven by an $H$-cylindrical Liouville fractional Brownian motion, and prove existence, uniqueness and space-time regularity of mild solutions under various assumptions on the Banach space $E$, the operators $A\colon \scr D(A)\to E$ and $B\colon H\to E$, and the Hurst parameter $\beta $. As an application it is shown that second-order parabolic SPDEs on bounded domains in $\mathbb R^d$, driven by space-time noise which is white in space and Liouville fractional in time, admit a mild solution if $\frac {1}{4}d<\beta <1$.