  The Volterra square-root process on $\mathbb{R}_+^m$ is an affine Volterra
process with continuous sample paths. Under a suitable integrability condition
on the resolvent of the second kind associated with the Volterra convolution
kernel, we establish the existence of limiting distributions. In contrast to
the classical square-root diffusion process, here the limiting distributions
may depend on the initial state of the process. Our result shows that the
non-uniqueness of limiting distributions is closely related to the
integrability of the Volterra convolution kernel. Using an extension of the
exponential-affine transformation formula we also give the construction of
stationary processes associated with the limiting distributions. Finally, we
prove that the time marginals as well as the limiting distributions, when
restricted to the interior of the state space $\mathbb{R}_{+}^m$, are
absolutely continuous with respect to the Lebesgue measure and their densities
belong to some weighted Besov space of type $B_{1,\infty}^{\lambda}$.

Probability;Mathematical Finance