Stationary fluctuations for occupation times of the long-range voter models on lattices

In this paper, we are concerned with the long-range voter model on lattices. We prove a stationary fluctuation theorem for the occupation time of the model under a proper time-space scaling. In several cases, the fluctuation limits are driven by fractional Brownian motions with Hurst parameters in $(1/2, 1)$. The proof of our main result utilizes the martingale decomposition strategy introduced in {Kipnis1987}. A local central limit theorem of the long-range random walk, the duality relationship between the model and the long-range coalescing random walk and a fluctuation theorem of the empirical density field of the model play the key roles in the proof.