  We consider a Volterra convolution equation in $\mathbb{R}^d$ perturbed with
an additive fractional Brownian motion of Riemann-Liouville type with Hurst
parameter $H\in (0,1)$. We show that its solution solves a stochastic partial
differential equation (SPDE) in the Hilbert space of square-integrable
functions. Such an equation motivates our study of an unconventional class of
SPDEs requiring an original extension of the drift operator and its Fr\'echet
differentials. We prove that these SPDEs generate a Markov stochastic flow
which is twice Fr\'echet differentiable with respect to the initial data. This
stochastic flow is then employed to solve, in the classical sense of infinite
dimensional calculus, the path-dependent Kolmogorov equation corresponding to
the SPDEs. In particular, we associate a time-dependent infinitesimal generator
with the fractional Brownian motion. In the final section, we show some
obstructions in the analysis of the mild formulation of the Kolmogorov equation
for SPDEs driven by the same infinite dimensional noise. This problem, which is
relevant to the theory of regularization-by-noise, remains open for future
research.
