We study the local well-posedness of the nonlinear Schr\"odinger equation associated to the Grushin operator with random initial data. To the best of our knowledge, no well-posedness result is known in the Sobolev spaces $H^k$ when $k \leq \frac{ 3}{ 2}$. In this article, we prove that there exists a large family of initial data such that, with respect to a suitable randomization in $H^k$, $k \in (1,\frac{ 3}{ 2}]$, almost-sure local well-posedness holds. The proof relies on bilinear and trilinear estimates.
Categories: math.AP