**May 31, 2021 at 09:00 - June 04, 2021**
- BCAM

**Renato Lucà (BCAM)**

**DATES:** 31 May-04 June 2021 (5 sessions)

**TIME:** 09:00 - 11:00 (a total of 10 hours)

**LOCATION:** Online

**ABSTRACT:**
This course will be an introduction to the theory of invariant measures of Gibbs type for Hamiltonian PDEs and to the weaker notion of quasi-invariant measure.

**PROGRAMME:**
The theory of Gibbs measure for Hamiltonian PDEs has been developed since the seminal contributions of Lebowitz–Rose–Speer [LRS88] and Bourgain [Bou94], concerning the Schrödinger equation. Since then, the topic has been extensively studied by many authors, however several problems are still open. We will give an introduction to this theory, considering as models the nonlinear Schrödinger equation and the Benjamin-Bona-Mahony equation. We will also explain how, taking advantage of the existence of an invariant measure, we can deduce interesting information on the behavior of “typical solutions”, namely almost surely with respect to the measure. More precisely we will:

- construct global (and sometimes even local) typical solutions at a level of regularity for which we do not necessarily have global (resp. local) well-posedness in the deterministic sense;

- give information about the pointwise and asymptotic behavior of typical solutions

We will mainly discuss results from [Bou94, Bou96, CLS21, GLT21].

**REFERENCES:**
[Bou94] J. Bourgain. Periodic nonlinear Schrödinger equation and invariant measures. Comm. Math. Phys., 166(1):1–26, 1994.

[Bou96] J. Bourgain. Invariant measures for the 2d-defocusing nonlinear Schrödinger equation. Communications in Mathematical Physics, 176(2):421–445, Mar 1996.

[CLS21] E. Compaan, G. Staffilani, R. Lucà. Pointwise Convergence of the Schrödinger Flow. Int. Math. Res. Not., 2021(1), 596–647, 2021.

[GLT21] G.Genovese,R.Lucà, N.Tzvetkov.Transport of Gaussian measures with exponential cutoff for Hamiltonian PDEs, arXiv:2103.04408.

[LRS88] J. L. Lebowitz, H. A. Rose, and E. R. Speer. Statistical mechanics of the nonlinear Schrödinger equation. J. Statist. Phys., 50(3-4):657–687, 1988.

** *Registration is free, but mandatory before May 26th.** To sign-up go to

https://forms.gle/uwbBM4nYcWPEaw2e9 and fill the registration form.