Michael Röckner (Bielefeld)
Stochastic nonlinear Schrödinger equations with linear multiplicative noise:
the rescaling approach
Joint work with: Viorel Barbu (Romanian Academy) and Deng Zhang (University of Bielefeld)
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.