We study a nonlinear Schrödinger (NLS) equation with focusing nonlinearity, subject to a multiplicative stochastic perturbation driven by an infinite dimensional Brownian motion. Under the appropriate assumptions on the space covariance of the driving noise, previously A. de Bouard and A. Debussche established the $H^1$ local well-posedness for energy sub-critical nonlinearity, and global well-posedness in the mass-subcritical case.

In our work we study the mass-critical, intercritical and energy $\left(H^1\right)$-critical cases of stochastic NLS for a multiplicative noise, and obtain quantitative estimates on the blow-up time when the mass, energy and $L^2$-norm of the gradient of the initial condition are controlled by similar quantities of the ground state. This completes qualitative blow-up results proved by A. de Bouard and A. Debussche for energy sub-critical nonlinearities.

This is joint work with Svetlana Roudenko (FIU).