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arXiv · 2604.14852

The energy-critical stochastic nonlinear Schr\"odinger equation: well-posedness and blow-up

Annie Millet·Svetlana Roudenko
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Abstract

We investigate the focusing and defocusing energy-critical stochastic nonlinear Schr\"odinger equation, subject to random perturbations in the form of either additive or multiplicative (Stratonovich) noise. We establish local well-posedness for random or deterministic initial data $u_0$ in $\dot{H}^1(\mathbb{R}^n)$ or $H^1(\mathbb{R}^n)$, depending on the noise type. In the focusing case we provide quantitative estimates regarding the existence time and probability. Moreover, we derive blow-up criteria for solutions with positive energy in both cases of noise, provided that the noise intensity is sufficiently small, showing that blow-up occurs before a certain given positive time with positive probability, thus, extending deterministic results of Kenig-Merle [24] for the energy-critical NLS equation to the stochastic setting.

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Annie Millet, Svetlana Roudenko. 2026-04-16. The energy-critical stochastic nonlinear Schr\"odinger equation: well-posedness and blow-up. https://arxiv.org/abs/2604.14852

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