Pointwise ergodic theorems for the symmetric exclusion process (Q1090014)

Let us place one particle in each of some sites of the integer lattice \(S={\mathbb{Z}}^ d\) at time \(t=0\). Each particle will wait an exponentially distributed time of parameter one and then attempt a jump according to a symmetric probability matrix p(x,y). The attempted jump from x to y occurs if and only if y is vacant. Let \(\eta_ t(x)\) be the indicator of the event that \(x\in S\) is occupied at time t \((\eta_ t\) is called the symmetric exclusion process on S). Let \(\mu_{\rho}\), \(0\leq \rho \leq 1\), denote the law according to which each \(x\in S\) is occupied with probability \(\rho\), independently of the others. Suppose that \(\eta_ t\) converges in distribution to \(\mu_{\rho}\) as \(t\to \infty\). Then for all continuous f \[ \lim T^{- 1}\int^{T}_{0}f(\eta_ s)ds=\int fd\mu_{\rho}\quad a.s., \] if only \(d\geq 3\) (Theorem 2.2). The condition \(d\geq 3\) can be dropped when the relative occupation time of a single site is considered, i.e. when \(f(\eta)=\eta (x)\) (Theorem 2.1).