  We consider stochastic partial differential equations appearing as Markovian
lifts of matrix valued (affine) Volterra type processes from the point of view
of the generalized Feller property (see e.g., \cite{doetei:10}). We introduce
in particular Volterra Wishart processes with fractional kernels and values in
the cone of positive semidefinite matrices. They are constructed from matrix
products of infinite dimensional Ornstein Uhlenbeck processes whose state space
are matrix valued measures. Parallel to that we also consider positive definite
Volterra pure jump processes, giving rise to multivariate Hawkes type
processes. We apply these affine covariance processes for multivariate (rough)
volatility modeling and introduce a (rough) multivariate Volterra Heston type
model.
