| Abstract: |
| We study nonlinear Schr\{o}dinger (NLS) equation with focusing nonlinearity, subject to additive or multiplicative stochastic
perturbations 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 $L^2$-critical, intercritical and energy ($\dot{H}^1$)-critical cases of stochastic
NLS, 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 blow-up results proved by A.~de Bouard and A.~Debussche for energy sub-critical nonlinearities. |
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