Uniform Large Deviations for the Nonlinear Schrödinger Equation with Multiplicative Noise

Uniform large deviations for the laws of the paths of the solutionsof the stochastic nonlinear Schr¨odinger equation when the noise converges tozero are presented. The noise is a real multiplicative Gaussian noise. It iswhite in time and colored in space. The path space considered allows blow-upand is endowed with a topology analogue to a projective limit topology. Thusa large variety of large deviation principle may be deduced by contraction. Asa consequence, asymptotics of the tails of the law of the blow-up time whenthe noise converges to zero are obtained.