  We study the long-time behavior of affine processes on positive self-adjoiont
Hilbert-Schmidt operators which are of pure-jump type, conservative and have
finite second moment. For subcritical processes we prove the existence of a
unique limit distribution and construct the corresponding stationary affine
process. Moreover, we obtain an explicit convergence rate of the underlying
transition kernels to the limit distribution in the Wasserstein distance of
order $p\in [1, 2]$ and provide explicit formulas for the first two moments of
the limit distribution. We apply our results to the study of
infinite-dimensional affine stochastic covariance models in the stationary
covariance regime, where the stationary affine process models the instantaneous
covariance process. In this context we investigate the behavior of the implied
forward volatility smile for large forward dates in a geometric affine forward
curve model used for the modeling of forward curve dynamics in fixed income or
commodity markets formulated in the Heath-Jarrow-Morton-Musiela framework.
