Ergodic Shaping in Networked Stochastic Systems: Control Design via Optimal Transport and Entropic Criteria

<div xmlns="http://www.tei-c.org/ns/1.0"> We initiate a framework for the control-theoretic design of long-term statistical behaviour in networks of stochastic dynamical systems. Specifically, we address the problem of ergodic shaping: designing local feedback laws and coupling structures so that the invariant distribution of the networked system approximates a desired target distribution. Our setting includes agents evolving via iterated function systems (IFS) or controlled stochastic recursions with state-dependent randomness, interacting over a graph. We introduce a variational formulation of the ergodic shaping problem based on minimizing the relative entropy or Wasserstein distance between the induced invariant distribution and a prescribed target. We establish conditions under which ergodic shaping is feasible, characterize the set of realizable stationary distributions, and analyze optimal control laws within a class of feedback strategies. Our results combine tools from ergodic theory, optimal transport, stochastic control, and convex variational analysis. This work positions invariant measure design as a central control objective in distributed stochastic systems. It provides the theoretical foundations for applications in fair resource allocation, decentralized learning, and statistical regulation of networked agents subject to uncertainty. </div>