Exact random centering and hybrid fluctuation limits for interacting reinforced processes under critical Markov switching

Abstract

We study two interacting reinforced occupation processes driven by a common two-state Markov environment whose switching probabilities decrease at the same rate as the stochastic-approximation gain. At this critical scale, the environmental motion persists in the first-order limit, which is a telegraph-driven piecewise deterministic Markov process. Consequently, centering at a deterministic equilibrium does not separate the order-one environmental response from the intrinsic square-root fluctuations. To overcome this obstruction, we introduce an exact discrete random centering that removes the accumulated environmental forcing before rescaling. An exact decomposition into collective and synchronization coordinates then separates the two contraction mechanisms of the system. Under the contraction condition corresponding to the square-root regime, we prove joint functional convergence of the randomly centered coordinates and the first-order background to a stationary hybrid PDMP--diffusion system. Its stationary entrance law is characterized through stochastic convolutions over the remote logarithmic past. Conditionally on the complete environmental path, the two fluctuation modes are independent centered Gaussian processes with a common state-dependent volatility and different contraction rates. We also determine the invariant distribution of the first-order component, derive explicit stationary second moments, and obtain a closed-form correlation formula that recovers interaction information not visible in the synchronized first-order limit.