I'm interested in the simple exclusion processes on $Z^d$ and the ergodic theorems that can be proved from the point of view of the particle. Ellen Saada proved the following in 1987 (Annals of Prob): Let $\eta_t$ be a simple exclusion process on $Z^d$ such that at time 0, there is a tagged particle at the origin, and at all sites different from 0, particles are placed according to the Bernoulli distribution with parameter p. If the transition function $p(x,y)$ is translation invariant, has finite first moment, and is not nearest neighbor in $d=1$, then Bernoulli measures on $Z^d$ are extremal invariant from the point of view of the particle.
Has there been further progress on this recently? There is an earlier paper of Ferrari (1986), and a review he wrote in 1996. But otherwise I have not been able to find too much.
More simply, if $X_t$ is the position of the tagged particle in a simple exclusion process, I am interested in the following question: under what general conditions can $X_t/t$ be shown to converge to a limit almost surely?
