随机扰动反应扩散系统中平面行波的多维稳定性
Multidimensional Stability of Planar Travelling Waves for Stochastically Perturbed Reaction-Diffusion Systems
AI总结：
本文研究二维及以上空间中带乘性噪声的反应扩散系统，通过随机相位跟踪机制在指数长时间尺度上建立平面波的多维稳定性，并扩展前向积分理论以处理非 Itô 型随机积分。
摘要：
我们考虑在二维或更高维空间域上具有乘性噪声的反应扩散系统。该噪声过程在时间上是白噪声，在空间上是有色噪声，并且在平移变换下保持不变。在确定性情形下，全空间 $\mathbb R^d$ 上平面波的多维稳定性已被许多学者研究。受先前在实直线上的工作启发，我们在柱形区域上建立了平面波在时间尺度上的多维稳定性，该时间尺度相对于噪声强度呈指数级长。这是通过一种随机相位跟踪机制实现的，该机制能够在如此长的时间尺度上得以维持。我们问题的相应温和形式涉及关于可预料被积函数的随机积分，因此无法在已建立的 Itô 积分框架内加以理解。为规避这一问题，我们利用并扩展了最近发展的关于前向积分的理论。
英文摘要：
We consider reaction-diffusion systems with multiplicative noise on a spatial domain of dimension two or higher. The noise process is white in time, coloured in space, and invariant under translations. In the deterministic setting, multidimensional stability of planar waves on the whole space $\mathbb R^d$ has been studied by many. Inspired by previous works on the real line, we establish the multidimensional stability of planar waves on a cylindrical domain on time scales that are exponentially long with respect to the noise strength. This is achieved by means of a stochastic phase tracking mechanism that can be maintained over such long time scales. The corresponding mild formulation of our problem features stochastic integrals with respect to anticipating integrands, which hence cannot be understood within the well-established setting of Itô-integrals. To circumvent this problem, we exploit and extend recently developed theory concerning forward integrals.
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