Let $\H$ be a Hilbert space and $E$ a Banach space. We set up a theory of stochastic integration of $\calL(\H,E)$-valued functions with respect to $\H$-cylindrical Liouville fractional Brownian motions with arbitrary Hurst parameter $0<\b<1$. For $0<\b<\frac12$ we show that a function $\Phi:(0,T)\to \calL(\H,E)$ 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
$$ dU(t) = AU(t)\,dt + B\, dW_\H^\beta(t)$$
driven by an $\H$-cylindrical Liouville fractional Brownian motion.
This talk is based on jointwork with Z. Brzezniak and J. van Neerven.