AI 中文总结
针对临界马尔可夫切换下交互强化过程，提出精确随机中心化与坐标分解，证明混合PDMP-扩散极限并刻画平稳性质，恢复交互信息。
AI 中文摘要
我们研究由共同的两状态马尔可夫环境驱动的两个交互强化占有过程，其切换概率以与随机近似增益相同的速率递减。在此临界尺度下，环境运动保留在一阶极限中，该极限是一个由电报驱动的分段确定性马尔可夫过程。因此，在确定性平衡处中心化并不能将一阶环境响应与内在的平方根波动分开。为克服这一障碍，我们引入一种精确的离散随机中心化，在重新缩放之前移除累积的环境强迫。精确分解为集体坐标和同步坐标后，系统的两种收缩机制得以分离。在与平方根机制对应的收缩条件下，我们证明了随机中心化坐标与一阶背景的联合函数收敛到一个平稳的混合PDMP-扩散系统。其平稳进入律通过远程对数过去的随机卷积来刻画。在完整环境路径的条件下，两种波动模式是独立的高斯过程，具有共同的状态依赖波动率和不同的收缩率。我们还确定了一阶分量的不变分布，推导了显式的平稳二阶矩，并获得了闭式相关公式，该公式恢复了在同步一阶极限中不可见的交互信息。
英文摘要
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.
