Large Deviations and Support Results for Nonlinear Schrödinger Equations with Additive Noise and Applications

Sample path large deviations for the laws of the solutions of sto-chastic nonlinear SchrÄodinger equations when the noise converges to zero arepresented. The noise is a complex additive gaussian noise. It is white in timeand colored space wise. The solutions may be global or blow-up in ¯nite time,the two cases are distinguished. The results are stated in trajectory spacesendowed with projective limit topologies. In this setting, the support of thelaw of the solution is also characterized. As a consequence, results on the lawof the blow-up time and asymptotics when the noise converges to zero areobtained. An application to the transmission of solitary waves in ¯ber opticsis also given.