course: Introduction to dispersive SPEs
speaker: Arnaud Debussche (IRMAR and ENS de Cachan, antenne de Bretagne)abstract: In these series of lectures, I will describe mathematical aspects of stochastic dispersive PDEs with a particular emphasis on the Nonlinear Schrodinger Equations (NLS).
I will start from the very beginning and explain how the Strichartz estimates allow to solve these equations in the simple case of the stochastic NLS equation with an additive noise. These tools will then be used to study the multiplicative noise case.
Once basic well-posedness results are established, we will explain some aspects on the qualitative influence of a noise on blow-up phenomena in the case of a critical or supercritical nonlinearity and show that a spatially smooth noise has a strong effect.
Then, we will prove large deviation estimates and show how these can be used to estimate the probability errors in soliton transmission in optical fibers.
Some qualitative aspects being difficult to study theoretically, numerical simulations are sometimes used and it is important to study the schemes. I will therefore devote some lectures to the numerical analysis of the stochastic NLS equations and explain how the associated infinite dimensional Kolmogorov equation is used to study the weak order of a scheme.
Finally, depending on the time spent on the preceding topics, I will adress some problems related to the Korteweg-de Vries equation and the associated solitons when a noise perturbs the dynamics.
timetable:
Mon 12 Jun, 14:30 - 16:30, Sala Conferenze Centro De Giorgi
Mon 19 Jun, 14:30 - 16:30, Sala Conferenze Centro De Giorgi
Fri 23 Jun, 14:30 - 16:30, Sala Conferenze Centro De Giorgi
Mon 26 Jun, 14:30 - 16:30, Sala Conferenze Centro De Giorgi
Fri 30 Jun, 14:30 - 16:30, Sala Conferenze Centro De Giorgi
Mon 3 Jul, 10:30 - 12:30, Sala Conferenze Centro De Giorgi
Fri 7 Jul, 10:30 - 12:30, Sala Conferenze Centro De Giorgi
Mon 10 Jul, 10:30 - 12:30, Sala Conferenze Centro De Giorgi
Mon 17 Jul, 10:30 - 12:30, Sala Conferenze Centro De Giorgi
Fri 21 Jul, 10:30 - 12:30, Sala Conferenze Centro De Giorgi
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