In this paper we present a general framework in which to rigorously study the effect of spatiotemporal 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.
All Science Journal Classification (ASJC) codes
- Modeling and Simulation
- Neural field
- Pattern formation
- Stochastic partial differential equation
- Traveling pulse
- Traveling wave