In the first talk I will give some background from the theory of singular SPDEs, more precisely I will sketch a construction of the Anderson Hamiltonian, a Schrödinger operator with an irregular random distributional potential, in 3d using the theory of paracontrolled distributions due to Gubinelli-Imkeller-Perkowski. I will also sketch how to use this construction to solve SPDEs whose linear part is given by the Anderson Hamiltonian, the most relevant one for us being the multiplicative stochstic nonlinear Schrödinger equation (NLS).
In the second part, I will discuss a recent work in this direction where global well-posedness for the multiplicative stochastic Hartree NLS in 3d is proved for some range of interactions. The analogous equation has been well studied in 2 dimensions, but ours is the first to prove such a result in three dimensions. The main tools to achieve this are the paracontrolled description of the operator, its dispersive properties (in the form of Strichartz estimates) and almost conserved energies. Finally I will motivate the study of this kind of equation by proving that (under some assumptions) it is the mean field limit of a suitably chosen N-body Quantum system.
Joint work with Franceso De Vecchi and Xiaohao Ji.
