| Abstract: | In this mini course, I will give an introduction to the theory of random data nonlinear PDE’s, on one of the most simple example of dispersive PDE’s: the one dimensional nonlinear Schrodinger equation on the line $\mathbb{R}$. More precisely, I will define essentially on $L^2 (\mathbb {R})$, the space of initial data, probability measures for which I can describe the (nontrivial) evolution by the linear flow of the Schrodinger equation $$(i\partial_t+\partial _x2)u=0, (t,x) \in\mathbb{R} \times \mathbb{R}$$ These mesures are essentially supported on $L^2( \mathbb{R})$. Then I will show that the nonlinear equation $$ (i\partial_t + \partial_x^2 ) u - |u|^{p-1} u =0, (t, x) \in \mathbb{R}\times \mathbb{R}$$ ，Is locally well posed on the support of the measure. Finally I will describe precisely the evolution by the nonlinear flow of the measure defined previously in terms of the linear evolution (quasi-invariance). Lastly I wil show how this description gives 1) (Almost sure) Global well posedness for p>1 and asymptotic behaviour of solutions (nonscattering type) 2) (Almost sure) scattering for p>3.”
This is based on joint works with L. Thomann and N. Tzvetkov, and more recently with L. Thomann. The prerequisite in probability for the course are essentially elementary probability theory. |
