The Schr\"odinger equation with fluctuating nonlinearity in the energy space

Abstract

We study nonlinear Schr\"odinger equations with nonlinear Stratonovich noise \begin{equation*} \mathrm{d} u\,=\, i\bigl[ \Delta u \,+\, \lambda|u|^{p-1}u\bigr] \, \mathrm{d} t \,+\,i|u|^{(q-1)/2}u\circ \mathrm{d} {W}, \end{equation*} in their energy space $H^1(\mathbb R^d;\mathbb C)$. By combining the stochastic Strichartz estimates derived in [Potential Anal. 41 (2014), pp.\ 269--315] with the approach from [Ann.\ Inst.\ H.\ Poincar\'e Phys.\ Th\'eor.\ 46 (1987), pp.\ 113--129] we obtain local well-posedness for all energy-subcritical nonlinearities $p,q\in [1, 1+4/(d-2)_+)$ together with a corresponding blow-up alternative. For a linear multiplicative noise $q=1$, a real-valued noise $W$ and a defocusing nonlinearity $\lambda\le 0$, we check this blow-up condition using a bound on the energy, resulting in the global well-posedness of the equation. If both nonlinearities are mass-subcritical, i.e., $p,q\in [1, 1+4/d)$, we provide an improved blow-up criterion involving the $L^2(\mathbb R^d;\mathbb C)$-norm. Using the conservation of mass for real-valued $W$, we obtain global well-posedness also in this case. Compared to previous results on stochastic nonlinear Schr\"odinger equations, we thereby improve the range of exponents $p$ and $q$ and the spatial regularity assumption on the noise.