  We study the class of continuous polynomial Volterra processes, which we
define as solutions to stochastic Volterra equations driven by a continuous
semimartingale with affine drift and quadratic diffusion matrix in the state of
the Volterra process. To demonstrate the versatility of possible state spaces
within our framework, we construct polynomial Volterra processes on the unit
ball. This construction is based on a stochastic invariance principle for
stochastic Volterra equations with possibly singular kernels. Similarly to
classical polynomial processes, polynomial Volterra processes allow for
tractable expressions of the moments in terms of the unique solution to a
system of deterministic integral equations, which reduce to a system of ODEs in
the classical case. By applying this observation to the moments of the
finite-dimensional distributions we derive a uniqueness result for polynomial
Volterra processes. Moreover, we prove that the moments are polynomials with
respect to the initial condition, another crucial property shared by classical
polynomial processes. The corresponding coefficients can be interpreted as a
deterministic dual process and solve integral equations dual to those verified
by the moments themselves. Additionally, we obtain a representation of the
moments in terms of a pure jump process with killing, which corresponds to
another non-deterministic dual process.
