随机弱阻尼非线性薛定谔方程从吸引域的逃逸
Exit from a basin of attraction for stochastic weakly damped nonlinear Schrödinger equations
AI总结：
研究小振幅噪声扰动的弱阻尼非线性薛定谔方程，在$\xLtwo$或$\xHone$中从零平衡点邻域（确定性方程的不变集）逃逸的问题，给出了首次逃逸时间和逃逸点的小噪声渐近形式化刻画。
摘要：
我们考虑由小振幅噪声扰动的弱阻尼非线性薛定谔方程。小噪声要么是复加性类型，要么是实乘性类型。它在时间上是白的，在空间上是彩色的。零是确定性方程的渐近稳定平衡点。我们研究在$\xLtwo$或$\xHone$中从零邻域（由确定性方程的流不变）的逃逸。由于噪声，远离零的大涨落发生。因此，在足够大的时间尺度上，从这些吸引域的逃逸发生。给出了首次逃逸时间和逃逸点的小噪声渐近的形式化刻画。
英文摘要：
We consider weakly damped nonlinear Schrödinger equations perturbed by a noise of small amplitude. The small noise is either complex and of additive type or real and of multiplicative type. It is white in time and colored in space. Zero is an asymptotically stable equilibrium point of the deterministic equations. We study the exit from a neighborhood of zero, invariant by the flow of the deterministic equation, in $\xLtwo$ or in $\xHone$. Due to noise, large fluctuations off zero occur. Thus, on a sufficiently large time scale, exit from these domains of attraction occur. A formal characterization of the small noise asymptotic of both the first exit times and the exit points is given.
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