We study the cohomology of Jacobians and Hilbert schemes of points on reduced and locally planar curves, which are however allowed to be singular and reducible. We show that the cohomologies of all Hilbert schemes of all subcurves are encoded in the cohomologies of the fine compactified Jacobians of connected subcurves, via the perverse Leray filtration. We also prove, along the way, a result of independent interest, giving sufficient conditions for smoothness of the total space of the relative compactified Jacobian of a family of locally planar curves.

Sample path large deviationsfor the laws of the solutions of stochastic nonlinear Schrödinger equations when the noise converges to zero arepresented. The noise is a complex additive Gaussian noise. It iswhite in time and colored in space. The solutions may be global orblow-up in finite time, the two cases are distinguished. The results are stated in trajectory spaces endowed with topologiesanalogue to projective limit topologies. In this setting, thesupport of the law of the solution is also characterized. As aconsequence, results on the law of the blow-up time andasymptotics when the noise converges to zero are obtained. Anapplication to the transmission of solitary waves in fiber opticsis also given.