Uniform large deviations for the nonlinear Schrödinger equation with multiplicative noise

Uniform large deviations at the level of the paths for the stochastic nonlinear Schrödinger equation are presented. The noise is a real multiplicative Gaussian noise, white in time and colored in space. The trajectory space allows blow-up. It is endowed with a topology analogous to a projective limit topology. Asymptotics of the tails of the blow-up time are obtained as corollaries.