  In this paper we establish the meta-stability of travelling waves for a class
of reaction-diffusion equations forced by a multiplicative noise term. In
particular, we show that the phase-tracking technique developed in
[hamster2017,hamster2020] can be maintained over timescales that are
exponentially long with respect to the noise intensity. This is achieved by
combining the generic chaining principle with a mild version of the
Burkholder-Davis-Gundy inequality to establish logarithmic supremum bounds for
stochastic convolutions in the critical regularity regime.
