The stochastic nonlinear Schrödinger equation in unbounded domains and non-compact manifolds

Uniqueness of martingale solutions for the stochastic nonlinear Schrödinger equation on 3d compact manifolds

“…We finish these comments with posing another interesting open problem: Is it possible to generalize the uniqueness to other geometries (in particular bounded or unbounded do-mains and non-compact manifolds). The existence in the latter cases is proved in the recent paper [31] by the second named author. In Remark 3.5 we explained why we couldn't apply the approach from the present paper to prove uniqueness for these geometries.…”

“…Instead of using a fixed point argument, we separate the proof of the existence and the uniqueness. The construction of a martingale solution has been treated by the authors in [9] and the second author in [31] based on the Hamiltonian structure of the NLS without using the Strichartz estimates. Since these ingredients are independent of the underlying geometry, the existence proof works in a more general framework including non-compact manifolds and domains with Neumann or Dirichlet boundary in arbitrary dimension.…”

Invariant measures for a stochastic nonlinear and damped 2D Schrödinger equation

“…However here we generalise that setting by dealing with a random initial data and more general diffusion terms. One should mention here that a very recent paper [Hor20] provides another generalisation [BHW19] in the direction of stochastic NLS equations on unbounded domains and non-compact manifolds.…”