  In this paper we present a general framework in which to rigorously study the
effect of spatio-temporal noise on traveling waves and stationary patterns. In
particular the framework can incorporate versions of the stochastic neural
field equation that may exhibit traveling fronts, pulses or stationary
patterns. To do this, we first formulate a local SDE that describes the
position of the stochastic wave up until a discontinuity time, at which point
the position of the wave may jump. We then study the local stability of this
stochastic front, obtaining a result that recovers a well-known deterministic
result in the small-noise limit. We finish with a study of the long-time
behavior of the stochastic wave.
