  We consider stochastic perturbations of PDEs which have special pattern
solutions, such as (nonlinear) travelling waves, solitons, and spiral waves. We
show orbital stability of these patterns on a timescale which is exponential in
the inverse square of the noise amplitude. We systematically treat equations
with noncommutative symmetry groups, and show how the noncommutativity affects
the motion of the pattern. This is done by introducing a new method to track
the (generalized) phase of the pattern. Furthermore, we demonstrate how orbital
stability arises from a mismatch of symmetry between the pattern and the
equation. Our phase tracking method does not rely on a Hilbert space structure.
This allows us to show stability in general Banach spaces, and to treat noise
with lower regularity than before.
