"In this talk we study the local well-posedness of the equation $i\partial_t u +\Delta_{G} u = |u|^{2}u$ where $\Delta_G = \partial_x^2+x^2\partial_y^2$ is the Grushin Laplacian and $u(t):\mathbb{R}^2 \to \mathbb{C}$ is the solution, to be constructed with initial data $u(0)=u_0 \in H^s_G(\mathbb{R}^d)$ (the adapted Grushin-Sobolev spaces). From a deterministic perspective, the best local well-posedness theory is in $\mathcal{C}^.([0,T),H^{\frac{3}{2}^{+}}_G)$ and the proof only uses the Sobolev embedding. Our main goal is to provide a probabilistic construction of local solutions for initial data $u_0 \in H_G^s$ where $s<3/2$. This is achieved using linear and bilinear random estimates. In the first part of the talk I will introduce the random initial data which we will consider. Then I will explain why randomisation helps to lower the well-posedness threshold: this is a general argument in the study of dispersive equations with random initial data. Then I will explain how bilinear random estimates relate to our probabilistic well-posedness problem, which we will prove if time permits. We may also discuss some extensions of our result instead. This talk is based on a joint work with Louise Gassot. "