  We consider two nonlinear sigma models on de Sitter background which involve
the same derivative interactions as quantum gravity but without the gauge
issue. The first model contains only a single field, which can be reduced to a
free theory by a local field redefinition; the second contains two fields and
cannot be so reduced. Loop corrections in both models produce large temporal
and spatial logarithms which cause perturbation theory to break down at late
times and large distances. Many of these logarithms derive from the "tail" part
of the propagator and can be summed using a variant of Starobinsky's stochastic
formalism involving a curvature-dependent effective potential. The remaining
logarithms derive from the ultraviolet and can be summed using a variant of the
renormalization group based on a special class of curvature-dependent
renormalizations. Explicit results are derived at 1-loop and 2-loop orders.
