  We survey recent work that relates Pitman's transformation to a variety of
classical integrable systems, including the box-ball system, the ultra-discrete
and discrete KdV equations, and the ultra-discrete and discrete Toda lattice
equations. It is explained how this connection enables the dynamics of the
integrable systems to be initiated from infinite configurations, which is
important in the study of invariant measures. In the special case of spatially
independent and identically distributed configurations, progress on the latter
topic is also reported.
