Stochastic nonlinear Schrödinger equations

This paper is devoted to the well-posedness of stochastic nonlinear Schrödinger equations in the energy space H1(Rd)H1(Rd), and is a continuation of our recent work (Barbu et al., 2014). We allow general complex coefficients in the noise, thus covering both the conservative and non-conservative case, the latter being important to include quantum measurement effects in the underlying physical model. We consider both focusing and defocusing nonlinearities and prove global well-posedness in H1(Rd)H1(Rd), including also the pathwise continuous dependence on initial conditions, with exponents of the nonlinearity in exactly the same (sharp) range as in the deterministic case. In particular, in the special conservative case, this work improves earlier results in de Bouard and Debussche (2003), where the critical exponents could not be reached for d≥6d≥6 in the defocusing and for d≥4d≥4 in the focusing case respectively. Moreover, the local existence, uniqueness and blowup alternative are also established for the energy-critical case. The approach presented here is mainly based on the rescaling approach already used in Barbu et al. (2014) to study the L2L2 case and also on the Strichartz estimates established in Marzuola et al. (2008) for large perturbations of the Laplacian.