Some limit theorems for voter model occupation times (Q1110205)

The author studies the occupation time functionals \(\int^{t}_{0}f(\eta_ s)ds\) of the voter model \(\{\eta_ t\), \(t\geq 0\}\) on \({\mathbb{Z}}^ d\). He first extends a pointwise ergodic theorem, which was proven for \(d\geq 3\) by \textit{E. D. Andjel} and \textit{C. P. Kipnis} [Probab. Theory Relat. Fields 75, 545-550 (1987; Zbl 0621.60114)], to the case \(d=2\). (It is known that the statement is false for \(d=1).\) Secondly, he proves a central limit theorem for the case \(f(\eta)=\eta (0)\) and initial distributions being either a fixed \(\eta\) of a certain class of states for \(d\geq 2\) or an extremal invariant measure for \(d\geq 3\). As is usual in the study of the voter model, the duality with a coalescing random walk is basically used.