确定性拉格朗日多时间动力学扩散极限中的 Lévy 面积与 Wong–Zakai 异常
Lévy areas, Wong Zakai anomalies in diffusive limits of Deterministic Lagrangian Multi-Time Dynamics
AI总结：
本文研究流体拉格朗日动力学扩散极限中噪声解释导致的稳定性差异，比较 Itô、Stratonovich 与 Wong–Zakai 极限及 Lévy 面积离散化，并借助 SALT 均匀化推导解释异常漂移的物理来源。
摘要：
随机建模要求对噪声作出解释。在本文中，我们描述一个基础流体力学问题中确定性稳定行为的丧失，其取决于噪声是以 Itô、Stratonovich 意义引入，还是以 Wong–Zakai 型极限引入。我们在含 Lévy 面积与不含 Lévy 面积的随机微分方程离散化这一更广泛背景下考察这种比较。从数值角度，我们证明使用 Lévy 面积进行高阶离散化会导致守恒面积量和角度量的丧失。这种行为在 Stratonovich 模型中并非物理上所预期的。相反，我们研究 Lie 输运随机平流（Stochastic Advection by Lie Transport）及其由均匀化理论作出的推导，该理论自然引入同类漂移修正。从均匀化角度看，Wong–Zakai 异常的定性性质具有物理动机，其产生源于快速尺度与平均尺度流体分解之间的相关性。
英文摘要：
Stochastic modelling necessitates an interpretation of noise. In this paper, we describe the loss of deterministically stable behaviour in a fundamental fluid mechanics problem, conditional to whether noise is introduced in the sense of Itô, Stratonovich or a limit of Wong-Zakai type. We examine this comparison in the wider context of discretising stochastic differential equations with and without the Lévy area. From the numerical viewpoint, we demonstrate performing higher order discretisations with the use of a Lévy area can lead to the loss of conserved area and angle quantities. Such behaviour is not physically expected in the Stratonovich model. Conversely, we study Stochastic Advection by Lie Transport and its derivation from homogenisation theory, which introduces drift corrections of the same class naturally. From the viewpoint of homogenisation, the qualitative properties of the Wong-Zakai anomaly are physically motivated as arising due to correlations from a fast and mean scale fluid decomposition.
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