  We develop a complete and rigorous mathematical framework for the analysis of
stochastic neural field equations under the influence of spatially extended
additive noise. By comparing a solution to a fixed deterministic front profile
it is possible to realise the difference as strong solution to an
$L^2(\mathbb{R})$-valued SDE. A multiscale analysis of this process then allows
us to obtain rigorous stability results. Here a new representation formula for
stochastic convolutions in the semigroup approach to linear function-valued SDE
with adapted random drift is applied. Additionally, we introduce a dynamic
phase-adaption process of gradient type.
