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

A general framework for stochastic traveling waves and patterns, with application to neural field equations