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| A nonlinear Schr\"odinger problem perturbed by multiplicative Gaussian noise will be investigated over a finite time horizon and a bounded one-dimensional domain. The appearing nonlinearity is the cubic Kerr-nonlinearity $f(z)=|z|^2 z$ for $z \in \mathbb{C}$, which has many applications in mathematical physics. Being interested in the existence and uniqueness of a variational solution, a further process will be introduced which allows to transfer the stochastic Schr\"odinger problem into a pathwise one. Exploiting the absence of noise and using Galerkin approximations and compact embedding results, one considers first a priori estimates, existence and uniqueness of a variational solution of the pathwise Schr\"odinger problem. Thereafter, it is possible to extend these results to the variational solution of the nonlinear Schr\"odinger problem with multiplicative noise. |
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