Stochastic Nonlinear Schrödinger Equations with Linear Multiplicative Noise: Rescaling Approach

We prove 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 wellposedness 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. Octav Mayer Institute of Mathematics (Romanian Academy) and Al.I. Cuza University and, 700506, Iaşi, Romania. This author was supported by a grant of the Romanian National Authority for Scientific Research, DSPDE nr. 1 ERC/02.07.2012 and BiBos Research Centre. Fakultät für Mathematik, Universität Bielefeld, D-33501 Bielefeld, Germany. This research was supported by the DFG through CRC 701. Fakultät für Mathematik, Universität Bielefeld, D-33501 Bielefeld, Germany, and Institute of Applied Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences. Beijing 100190, China.