  After inflation occurs, a weakly coupled scalar field will in general not be
in thermal equilibrium but have a distribution of values determined by the
inflationary Hubble parameter. If such a field subsequently undergoes discrete
symmetry breaking, then the different degenerate vacua may not be equally
populated so the domain walls which form will be `biased' and the wall network
will subsequently collapse. Thus the cosmological domain wall problem may be
solved for sufficiently weakly coupled fields in a post-inflationary universe.
We quantify the criteria for determining whether this does happen, using a
Higgs-like potential with a spontaneously broken $Z_2$ symmetry.
