  In the article, integration of temporal functions in (possibly non-UMD)
Banach spaces with respect to (possibly non-Gaussian) fractional processes from
a finite sum of Wiener chaoses is treated. The family of fractional processes
that is considered includes, for example, fractional Brownian motions of any
Hurst parameter or, more generally, fractionally filtered generalized Hermite
processes. The class of Banach spaces that is considered includes a large
variety of the most commonly used function spaces such as the Lebesgue spaces,
Sobolev spaces, or, more generally, the Besov and Lizorkin-Triebel spaces. In
the article, a characterization of the domains of the Wiener integrals on both
bounded and unbounded intervals is given for both scalar and cylindrical
fractional processes. In general, the integrand takes values in the space of
$\gamma$-radonifying operators from a certain homogeneous Sobolev-Slobodeckii
space into the considered Banach space. Moreover, an equivalent
characterization in terms of a pointwise kernel of the integrand is also given
if the considered Banach space is isomorphic with a subspace of a cartesian
product of mixed Lebesgue spaces. The results are subsequently applied to
stochastic convolution for which both necessary and sufficient conditions for
measurability and sufficient conditions for continuity are found. As an
application, space-time continuity of the solution to a parabolic equation of
order $2m$ with distributed noise of low time regularity is shown as well as
measurability of the solution to the heat equation with Neumann boundary noise
of higher regularity.
