Stochastic nonlinear Schrödinger equations with linear multiplicative noise: the rescaling approach

Speaker(s): 
Michael Röckner (University of Bielefeld)
Date: 
Wednesday, December 18, 2013 - 6:00pm
Location: 
TU Berlin, Raum MA 041, Straße des 17. Juni 136, 10623 Berlin

We present well-posedness results for stochastic nonlinear Schrödinger equations with linear multiplicative Wiener noise including the non-conservative case. Our approach is different from the standard literature on stochastic nonlinear Schrödinger equations. By a rescaling transformation we reduce the stochastic equation to a random nonlinear Schrödinger equation with lower order terms and treat the resulting equation by a fixed point argument, based on generalizations of Strichartz estimates proved by J. Marzuola, J. Metcalfe and D. Tataru in 2008. This approach allows to improve earlier well-posedness results obtained in the conservative case by a direct approach to the stochastic Schrödinger equation. In contrast to the latter, we obtain well-posedness in the full range (1, 1 + 4/d) of admissible exponents in the non-linear part (where d is the dimension of the underlying Euclidean space), i.e. in exactly the same range as in the deterministic case.

Joint work with: Viorel Barbu (Romanian Academy) and Deng Zhang (University of Bielefeld)