  We show that representations of the Thompson group $F$ in the automorphisms
of a noncommutative probability space yield a large class of bilateral
stationary noncommutative Markov processes. As a partial converse, bilateral
stationary Markov processes in tensor dilation form yield representations of
$F$. As an application, and building on a result of K\"ummerer, we canonically
associate a representation of $F$ to a bilateral stationary Markov process in
classical probability.
