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| We present well-posedness results in the energy space $H^1(\mathbb{R}^d)$ for the stochastic nonlinear Schr\"odinger equation with linear multiplicative noise, including also the non-conservative case. The exponents of the nonlinear term obtained here are optimal for the global well-posedness, hence this work sharpens earlier well-posedness results in the conservative case. Moreover, we also study the noise effects on blowup in the non-conservative case. In contrast to the conservative case, we prove that in the non-conservative focusing mass-(super)critical case, adding a large noise one can, with high probability, prevent blowup on the bounded time interval $[0,T]$ with $T |
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