论文标题
由翻译不变噪声驱动的参数强迫非线性schrödinger方程的适应性
Well-posedness of a parametrically forced nonlinear Schrödinger equation driven by translation-invariant noise
论文作者
论文摘要
我们证明了在[0，\ infty）$中的任何$σ\ $ h^σ（\ mathbb {r}）$中的拟态度，该$σ\的参数强制强制迫使非线性schrödinger方程（PFNLS）在一个维度驱动的一个维度驱动的乘数级别噪声的一个维度驱动的一个维度。噪声是白色的，在太空中相关。我们首先通过定点参数构建局部温和解决方案。然后，只要$ l^2（\ mathbb {r}）$ - 解决方案的标准仍然有限，我们通过证明方程的持久性和规律性的持久性来制定爆炸标准。之后，我们使用温和的itô公式对$ l^2（\ Mathbb {r}）$ - norm进行路径估计。我们的结果还适用于由乘法不变的Stratonovich噪声驱动的标准立方NLS方程。
We prove well-posedness in $H^σ(\mathbb{R})$ for any $σ\in [0,\infty)$ of a parametrically forced nonlinear Schrödinger equation (PFNLS) in one dimension driven by multiplicative Stratonovich noise which has spatially homogeneous statistics. The noise is white in time and correlated in space. We first construct local mild solutions via a fixed-point argument. We then formulate a blow-up criterion by showing that the equation has persistence of integrability and regularity as long as the $L^2(\mathbb{R})$-norm of the solution remains finite. Afterwards we derive a pathwise estimate on the $L^2(\mathbb{R})$-norm using a mild Itô formula. Our results also apply to the standard cubic NLS equation driven by multiplicative translation-invariant Stratonovich noise.
