Sample path central limit theorem for the occupation time of the voter model on a lattice

In this paper, we extend the central limit theorem of the occupation time of the voter model on the lattice $\mathbb{Z}^d$ given in \cite{Cox1983} to the sample path case for $d\geq 3$. The proof of our main result utilizes the resolvent strategy and the Poisson flow strategy introduced in previous literatures, where the duality relationship between the voter model and the coalescing random walk plays the key role. For $d=2$ case, we give a conjecture about an analogue result of our main theorem.