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

Seminario periodico del Dipartimento di Matematica
26 maggio 2022
Orario di inizio 
Zoom platform (please contact and live streaming in Seminar room -1
Organizzato da: 
Dipartimento di Matematica
Comunità universitaria
Comunità studentesca UniTrento
Ingresso libero
Email per prenotazione: 
Luigi Amedeo Bianchi, Stefano Bonaccorsi, Michele Coghi
Università degli Studi Trento 38123 Povo (TN) - Staff Dipartimento di Matematica
+39 04 61/281508-1625-1701-3898-1980
Margherita Zanella (Politecnico di Milano)


We consider a two-dimensional stochastic nonlinear defocusing Schrödinger equation with zero-order linear damping, where the stochastic forcing term is given by a combination of a linear multiplicative noise in Stratonovich form and a nonlinear noise in Ito form. 
We work at the same time on compact Riemannian manifolds without boundary and on compact smooth domains with either Dirichlet or Neumann boundary conditions. 

We construct a martingale solution using a modified Faedo-Galerkin's method, then by means of suitable Strichartz estimates we show the pathwise uniqueness of solutions. 
Finally, we prove the existence of invariant measures by means of a version of the Krylov-Bogoliubov method, which involves the weak topology, as proposed by Maslowski and Seidler. 

Some remarks on the uniqueness in a particular case are provided as well.
The talk is based on a joint work with B. Ferrario and Z. Brzezniak.