This paper introduces stochastic covariance models in Hilbert spaces with stationary affine instantaneous covariance processes. We explore the applications of these models in the context of forward curve dynamics within fixed-income and commodity markets. The affine instantaneous covariance process is defined on positive selfadjoint Hilbert–Schmidt operators, and we prove the existence of a unique limit distribution for subcritical affine processes, provide convergence rates of the transition kernels in the Wasserstein distance of order p ∈ [1, 2], and give explicit formulas for the first two moments of the limit distribution. Our results allow us to introduce affine stochastic covariance models in the stationary covariance regime and to investigate the behaviour of the implied forward volatility for large forward dates in commodity forward markets.