Harris–Wasserstein Ergodic Regulation and FairnessOptimization for Time-Inhomogeneous DistributedMarkov Systems

We study \emph{distributional regulation} and long-run \emph{fairness} for time-inhomogeneous stochastic distributed systems on a Polish state space $(\mathsf{X},\mathcal{B})$ modeled by controlled place-dependent iterated function systems (IFS) with nonstationary transition mechanisms. The closed loop is represented as a (possibly) time-varying Markov chain \[X_{k+1}=w^{(k)}_{\sigma_k}(X_k),\qquad \mathbb{P}(\sigma_k=i\mid X_k=x)=p^{(k)}_i(x;\pi),\] where a feedback policy $\pi$ selects (or parameterizes) the switching probabilities subject to distributed admissibility constraints (e.g.\ locality, communication, or resource limits). Fairness and predictability are formulated as \emph{asymptotic properties of occupation measures} $\mu_k:=\frac1k\sum_{j=1}^k\delta_{X_j}$ relative to a prescribed target law $\nu^\star$ (or a convex fairness set) in Wasserstein distance.Our main contribution is a Harris-type ergodic optimization framework that yields \emph{verifiable} quantitative guarantees under weak regularity: we impose a drift condition toward a petite set and a minorization condition that hold uniformly over the admissible policy class, allowing for non-contractive dynamics and state-dependent switching. Under these hypotheses, we establish (i) existence and uniqueness of a policy-dependent invariant distribution in the time-homogeneous case and, in the time-inhomogeneous case, convergence to a well-defined ergodic average; (ii) almost-sure convergence of $\mu_k$ in $W_1$ and explicit finite-time bounds on $\mathbb{E}[W_1(\mu_k,\nu_\pi)]$ driven by a quantitative mixing rate; and (iii) continuity and stability of the invariant law (and ergodic averages) with respect to policy perturbations in a kernel metric compatible with $W_1$ under uniform moment control. Building on these results, we formulate fairness regulation as the optimization problem $\inf_{\pi\in\Pi} W_1(\nu_\pi,\nu^\star)$ (or distance to a fairness set) and provide a constructive synthesis scheme based on stochastic approximation of the steady-state objective using single-trajectory data, with guarantees of convergence to stationary points under the derived ergodic bounds. The proposed theory provides a measure-theoretic bridge between long-run fairness constraints and feedback design for distributed stochastic systems beyond Wasserstein-contractive regimes.