  We first prove stochastic representation formulae for space-time harmonic
mappings defined on manifolds with evolving Riemannian metric. We then apply
these formulae to derive Liouville type theorems under appropriate curvature
conditions. Space-time harmonic mappings which are defined globally in time
correspond to ancient solutions to the harmonic map heat flow. As corollaries,
we establish triviality of such ancient solutions in a variety of different
situations.
