| Abstract: |
| During the talk, we will discuss the local solutions of the super-critical cubic Schr\ odinger equation (NLS) on the whole space with general differential operator. Although such problem is known to be ill-posed, we show that the random initial data yield almost sure local well-posedness. Using estimates in directional spaces, we improve and extend known results for the standard Schr\ odinger equation in various directions: higher dimensions, more general operators, weaker regularity assumptions on the initial conditions. In particular, we show that in 3D, the classical cubic NLS is stochastically, locally well-posed for any initial data with regularity in $H^\varepsilon$ for any $\varepsilon > 0$, compared to the known results $\varepsilon > \frac{1}{6}$. |
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